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Stochastic Optimization and Control with Applications in PHEV and Inventory Management

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Title:
Stochastic Optimization and Control with Applications in PHEV and Inventory Management
Creator:
Wei, Lai
Place of Publication:
[Gainesville, Fla.]
Florida
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University of Florida
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english
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1 online resource (315 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Industrial and Systems Engineering
Committee Chair:
GUAN,YONGPEI
Committee Co-Chair:
GEUNES,JOSEPH PATRICK
Committee Members:
RICHARD,JEAN-PHILIPPE P
CARRILLO,JANICE ELLEN
Graduation Date:
8/8/2015

Subjects

Subjects / Keywords:
Carrying costs ( jstor )
Cost control ( jstor )
Cost functions ( jstor )
Electricity ( jstor )
Inventory control ( jstor )
Market prices ( jstor )
Optimal control ( jstor )
Optimal policy ( jstor )
Prices ( jstor )
Search services ( jstor )
Industrial and Systems Engineering -- Dissertations, Academic -- UF
impulse -- inventory -- phev
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Industrial and Systems Engineering thesis, Ph.D.

Notes

Abstract:
In this dissertation, we address problems related to electricity storage control policies to manage charging and discharging activities for plug-in hybrid electric vehicles. We start with the trading strategies for plug-in hybrid electric vehicles in day-ahead and real-time markets. For this topic, we first develop models for both risk-neutral and risk-averse aggregators to participate only in a real-time market. The proposed models capture the impact of the charging and discharging activities on real-time electricity prices. Then, we extend our study to the case in which aggregators participate in both the real-time and day-ahead markets. For each developed model, we analyze the properties of the optimal objective value function, prove the existence and uniqueness of the optimal policy, and explore the corresponding optimal policy structure. Moreover, through numerical studies, we explore insights on how electricity prices are influenced by charging and discharging activities. In particular, we observe that aggregated charging/discharging activities with market-impact consideration could reduce the variance of the real-time electricity prices more efficiently, as compared to individual activities. In addition, considering market impact, an aggregator tends to use less electricity storage. Finally, it is beneficial to let an aggregator control the electricity storage and participate in both the real-time and day-ahead markets, instead of participating only in the real-time market. Then, we study the electricity storage control policies for electricity reserve markets. We start with general inventory control models. We explore the optimal inventory control problem where the inventory evolves as a Brownian motion and each order has to be integral times of a fixed quantity. Under the assumption that the holding and penalty costs are convex functions, we prove that the (R,Q) policy is optimal and we further provide explicit expressions of R and Q. We also provide more general verification theorems that can be used to verify the guessed function is the optimal value function for a general class of stochastic inventory control problems. Next, we study the Brownian inventory control problem where the inventory can be both increased and decreased by integer times of a fixed based quantity Q. Under reasonable assumptions, we prove that the optimal control policy is an (S,D,Q) policy and provide explicit expressions of S, D, and the value function. Furthermore, we study the Brownian inventory control problem with piecewise linear concave control cost. The objective is to derive a control policy to minimize the discounted total costs including holding cost and control costs. We show that, depending on the basic parameters, the optimal policy is either an (s, S) policy or a new type of policy which is different from traditional ones. The new policy looks like a combination of two (s, S) policies with two waiting areas. The challenging part is to prove the existence and uniqueness of the smooth solution for a free boundary problem associated with one waiting area. Finally, we discuss how to optimally control the PHEV battery storage in electricity reserve market based on the results obtained in stochastic inventory control model. ( en )
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2015.
Local:
Adviser: GUAN,YONGPEI.
Local:
Co-adviser: GEUNES,JOSEPH PATRICK.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2016-08-31
Statement of Responsibility:
by Lai Wei.

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Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Embargo Date:
8/31/2016
Classification:
LD1780 2015 ( lcc )

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STOCHASTICOPTIMIZATIONANDCONTROLWITHAPPLICATIONSINPHEVANDINVENTORYMANAGEMENTByLAIWEIADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2015

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c2015LaiWei 2

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Tomyparents 3

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ACKNOWLEDGMENTS Iwouldliketosincerelythankmyadvisor,Dr.YongpeiGuan,forhisencouragement,patience,andsupportduringmygraduatestudiesattheUniversityofFlorida.Especially,IamgratefulthatDr.Guangivesmethefreedomtopursuemyresearchinterests.Ihavebenetedalotfromhishighstandardforgoodresearchandhishonestadvices.Inaddition,IwouldliketodeeplythankDr.JosephGeunes, Dr. Jean-PhilippeRichard,andDr.JaniceCarrilloforservingonmydissertationcommitteeandprovidingremarkableadviceandvaluablecomments.Furthermore,manythanksgotomygraduatestudentcolleaguesRuiweiJiang,QianfanWang,YiqiangSu,ChaoyueZhao,JingMa,FangHe,LeiFan,KaiPan,KezhuoZhou,JianqiuHuang,andmanyothers,who makemy graduatestudentlifeoneofmygreatestexperience.Moreover,IwouldliketothankmyroommatesBoweiCheng,ZhexuanGong,DiLi,BinHe,ChuanWang,andXiaoweiMeifortheirtruefriendship.Mostimportantly,Iwouldliketoexpressmydeepestgratitudetomyparentsfortheirunconditionalloveandsupport. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 13 2OPTIMALCONTROLOFPLUG-INHYBRIDELECTRICVEHICLES ...... 17 2.1MotivationandLiteratureReview ....................... 17 2.2Real-TimeMarket ............................... 21 2.2.1Risk-NeutralAggregator ........................ 22 2.2.2Risk-AverseAggregator ........................ 26 2.3BothDay-AheadandReal-TimeMarkets .................. 28 2.3.1NotationandAssumption ....................... 30 2.3.2RevenueFunctionforEachPeriodn ................. 32 2.3.3BellmanEquationsandMainResults ................ 33 2.3.4AlgorithmsforParticipatinginBoththeDay-AheadandReal-TimeMarkets ................................. 35 2.3.5Discussion ................................ 37 2.4FurtherInsightsExplorationandDiscussions ................ 39 2.4.1CaseStudiesforParticipatinginReal-TimeMarketOnly ...... 39 2.4.1.1OptimalPolicyComparison ................. 39 2.4.1.2MarketImpactComparisonsbetweenIndividualandAggregatorActivities ..................... 41 2.4.1.3ComparisonofRisk-NeutralandRisk-AverseAggregators ................................ 45 2.4.2BothDay-AheadandReal-TimeMarkets ............... 45 2.4.2.1ExperimentalSettingandAlgorithms ........... 46 2.4.2.2ComparisonofParticipatinginReal-TimeMarketOnlyversusinBothMarkets ................... 47 2.5ConcludingRemarks ............................. 48 3OPTIMALCONTROLOFBROWNIANINVENTORYWITHBATCHORDERING 50 3.1MotivationandLiteratureReview ....................... 50 3.2TheModelDescriptionandAssumptions .................. 51 3.3MainTheoremandExistenceofParameters ................ 53 3.4LowerBoundTheorems ............................ 54 3.5OptimalPolicyParameters .......................... 57 3.6VericationofOptimalValueFunctionandControlPolicy ......... 60 5

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3.7ConcludingRemarks ............................. 66 4BROWNIANINVENTORYCONTROLWITHBATCHCONTROLQUANTITIES 67 4.1MotivationandLiteratureReview ....................... 67 4.2TheModelDescriptionandAssumptions .................. 68 4.3MainTheorem ................................. 69 4.4LowerBoundTheorems ............................ 71 4.5OptimalPolicyParameters .......................... 74 4.5.1Preliminaryresults ........................... 76 4.5.2Reducedequations ........................... 80 4.5.3ExistenceofSandD .......................... 83 4.6VericationofOptimalValueFunctionandControlPolicy ......... 87 4.6.1Preliminaryresults ........................... 88 4.6.2LV(x)=V(x)forxSandxD ................. 89 4.6.3LV(x)>V(x)forS
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6CONCLUSIONSANDFUTURERESEARCH ................... 145 APPENDIX APROOFS ....................................... 147 A.1ProofofProposition 2.1 ............................ 147 A.2ProofofTheorem 2.1 ............................. 150 A.3ProofofProposition 2.2 ............................ 150 A.4ProofofProposition 2.3 ............................ 156 A.5ProofofTheorem 2.2 ............................. 157 A.6ProofofProposition 2.4 ............................ 158 A.7ProofsofProposition 2.5 andTheorem 2.3 ................. 162 BPROOFS ....................................... 175 B.1ProofofLemma 3.1 .............................. 175 B.2ProofofTheorem 3.2 ............................. 175 B.3ProofofTheorem 3.3 ............................. 178 B.4SomeLemmasandProofs .......................... 178 B.5ProofofLemma 3.2 .............................. 179 B.6ProofofProposition 3.1 ............................ 182 B.7ProofofLemma 3.3 .............................. 186 CPROOFS ....................................... 189 C.1ProofofLemma 4.2 .............................. 189 C.2ProofofLemma 4.3 .............................. 192 C.3ProofofLemma 4.4 .............................. 196 C.4ProofofLemma 4.5 .............................. 199 C.5ProofofLemma 4.6 .............................. 199 C.6ProofofLemma 4.7 .............................. 201 C.7ProofofLemma 4.8 .............................. 202 C.8ProofofLemma 4.9 .............................. 204 C.9ContinuedProofofProposition 4.1 ...................... 209 C.10ContinuedProofofProposition 4.2 ...................... 214 DPROOFS ....................................... 217 D.1ProofofProposition 5.1 ............................ 217 D.2ProofofProposition 5.2 ............................ 221 D.3ProofofProposition 5.3 ............................ 223 D.4ProofofProposition 5.5 ............................ 224 D.5ProofofProposition 5.5 ............................ 225 D.6ProofofProposition 5.6 ............................ 226 D.7ProofofProposition 5.7 ............................ 228 D.8ProofofProposition 5.8 ............................ 233 D.9ProofofProposition 5.9 ............................ 235 7

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D.10ProofofLemma 5.6 .............................. 236 D.11ProofofProposition 5.10 ........................... 238 D.12ProofofProposition 5.11 ........................... 241 D.13ProofofProposition 5.12 ........................... 244 D.14ProofofProposition 5.13 ........................... 249 D.15ProofofTheorem 5.1 ............................. 256 D.16ProofofLemma 5.7 .............................. 261 D.17ProofofLemma 5.8 .............................. 263 D.18ProofofLemma 5.9 .............................. 267 D.19ProofofLemma 5.10 ............................. 281 D.20ProofofLemma 5.11 ............................. 294 D.21ProofofProposition 5.15 ........................... 296 D.22ProofofProposition 5.16 ........................... 297 D.23ProofofProposition 5.18 ........................... 299 D.24ProofofProposition 5.17 ........................... 302 D.25ProofofProposition 5.19 ........................... 306 REFERENCES ....................................... 309 BIOGRAPHICALSKETCH ................................ 315 8

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LISTOFFIGURES Figure page 2-1OptimalPolicyStructure(GeneralPricevs.IndependentPrice) ......... 26 2-2ProcesstoSubmitDay-AheadandReal-TimeOffers ............... 29 2-3OptimalTrade-Down-ToLevelofElectricityStorage ................ 41 2-4ComparisonofExpectedPriceafterActionwhenK=100 ............ 43 2-5ComparisonofExpectedPriceafterActionwhenK=300 ............ 44 2-6ProtofParticipatinginReal-TimeMarketOnlyandinBothMarkets ...... 48 3-1Illustrationof(x;y) ................................. 53 3-2Illustrationof(R;Q)policy ............................. 55 4-1Illustrationofu(y;x) ................................ 70 4-2Illustrationof(S;D;Q)policy ............................ 71 4-3IllustrationofAd(x) ................................. 78 4-4IllustrationofBd(x) ................................. 80 4-5Illustrationof(z) .................................. 84 4-6Illustrationof1(z) .................................. 86 4-7BatteryControlinElectricityReserveMarket ................... 100 4-8UpperThresholdisReached ............................ 101 4-9LowerThresholdisReached ............................ 101 5-1PiecewiseLinearControlCosts .......................... 106 5-2IllustrationofV(x)whend>d ........................... 108 5-3IllustrationofV(x)whend=d ........................... 109 5-4IllustrationofV(x)whend
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B-1Existenceofy(x) .................................. 186 C-1Illustrationofd(D;x+lQ)andu(D;x)withl=4 ................ 210 C-2Illustrationofd(D;x+lQ)andu(D;x)withx=D)]TJ /F4 11.955 Tf 11.96 0 Td[(2Qandl=4 ...... 212 D-1IllustrationofQ(x),s(K),andS(K) ........................ 219 10

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophySTOCHASTICOPTIMIZATIONANDCONTROLWITHAPPLICATIONSINPHEVANDINVENTORYMANAGEMENTByLaiWeiAugust2015Chair:YongpeiGuanMajor:IndustrialandSystemsEngineering In thisdissertation,weaddressproblemsrelatedtoelectricitystoragecontrolpoliciestomanagecharginganddischargingactivitiesforplug-inhybridelectricvehicles. Westartwiththetrading strategies forplug-inhybridelectricvehiclesinday-aheadandreal-timemarkets.Forthistopic,w erstdevelopmodelsforbothrisk-neutralandrisk-averseaggregatorstoparticipateonlyinareal-timemarket.Theproposedmodelscapturetheimpactofthecharginganddischargingactivitiesonreal-timeelectricityprices.Then,weextendourstudytothecaseinwhichaggregatorsparticipateinboththereal-timeandday-aheadmarkets.Foreachdevelopedmodel,weanalyzethepropertiesoftheoptimalobjectivevaluefunction,provetheexistenceanduniquenessoftheoptimalpolicy,andexplorethecorrespondingoptimalpolicystructure.Moreover,throughnumericalstudies,weexploreinsightsonhowelectricitypricesareinuencedbycharginganddischargingactivities.Inparticular,weobservethataggregatedcharging/dischargingactivitieswithmarket-impactconsiderationcouldreducethevarianceofthereal-timeelectricitypricesmoreefciently,ascomparedtoindividualactivities.Inaddition, considering marketimpact,anaggregatortendstouselesselectricitystorage. Finally, itisbenecialtoletanaggregatorcontroltheelectricitystorageandparticipateinboththereal-timeandday-aheadmarkets, insteadofparticipatingonly in thereal-timemarket. 11

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Then ,we study theelectricitystoragecontrolpolicies for electricityreserve markets .Westartwithgeneralinventorycontrolmodels.Weexploretheoptimalinventorycontrolproblem where theinventoryevolvesasaBrownianmotionandeachorderhastobeintegraltimesofaxedquantity.Undertheassumptionthattheholdingandpenalty costsare convex functions ,weprovethatthe (R;Q)policyisoptimal andwe further provideexplicitexpressionsofRandQ.Wealsoprovidemoregeneralvericationtheoremsthatcanbeusedtoverifytheguessedfunctionis the optimalvaluefunctionforageneralclassofstochasticinventorycontrolproblems. Next ,westudytheBrownianinventorycontrolproblem where theinventorycanbebothincreasedanddecreasedbyintegertimesofa xed basedquantityQ.Underreasonableassumptions,weprovethattheoptimalcontrolpolicyis an (S;D;Q)policyandprovideexplicitexpressionsofS,D,andthevaluefunction. Furthermore ,westudytheBrownianinventorycontrolproblemwithpiecewiselinearconcavecontrolcost.The objective isto derive acontrolpolicytominimizethediscountedtotalcostsincludingholdingcostandcontrol costs .Weshowthat,dependingonthebasicparameters,theoptimalpolicyiseitheran(s;S)policyoranewtypeofpolicywhichisdifferentfromtraditional ones .Thenewpolicylookslike acombinationof two(s;S)policieswithtwowaitingareas.The challenging part istoprovetheexistenceanduniquenessofthe“smooth”solutionforafreeboundaryproblemassociatedwithonewaitingarea.Finally,wediscusshowtooptimallycontrolthePHEVbatterystorageinelectricityreservemarketbasedontheresultsobtainedin the stochasticinventorycontrolmodel. 12

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CHAPTER1INTRODUCTIONElectricity,unlikeothercommodities,istraditionallyconsiderednon-storable(cf.[ 42 ]),unlessitistransformedintootherformsofenergy,suchasoriginalpotentialenergyofthewaterthroughpumped-storagehydroelectricity.Accordingly,electricitypricesbehavedifferentlyfromthoseofothercommoditymarketsbecausepowerbalanceisrequiredatanytimeforapowergridsystem.Meanwhile,thepenetrationofrenewableenergyintothepowergridsystemhasrecentlybeenincreasingdramatically,andthistrendwillcontinuefollowingtheObamaAdministration'sgoalof20%windpowerforU.S.electricitygeneration[ 52 ]andthenewlyannouncedInternationalRenewableEnergyAgency'sgoalof30%renewableenergygenerationworldwideby2030[ 12 ].However,duetoitsintermittentnature,integrationofrenewableenergies(suchaswindandsolar)enlargesthevolatilityofelectricitypricesforderegulatedelectricitymarkets.Alargeamountofelectricitystorageisoneofafewlimitedapproachestohelpattenthepricecurveandmaintainsystemstability.Althoughpumped-storagehydroelectricitycanserveaselectricitystoragedevices,becauseoftheneedforproximitytoariver,theportionofhydroelectricpowerinmanyareasisverylimitedforwholesalemarkets,ascomparedtonuclearandthermalgenerators.However,recentpromotionofthedeploymentofplug-inhybridelectricvehicles(PHEVs)makeslarge-scaleelectricitystoragepossibleinthenearfuture.PHEVsarevehiclesthatincludebothagasolineengineandabatterystoragesystemof4kWhormore,inwhichthebatterystoragecanberechargedfromanexternalsourceandhastheabilitytosupportatleast10milesofdrivinginall-electricmode,asdescribedin[ 33 ]and[ 36 ].Ifthecapacityofthebatteryislargeenough,thevehicleoperatesasanall-electricvehicle(e.g.,NissanLeaf);otherwise,thevehiclerstoperatesinall-electricmodeandswitchestoacharge-sustainingmodeloncethegasolineengineistriggered(e.g.,ToyotaPrius)[ 11 ].Ascomparedtogas-fueled 13

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vehicles,PHEVsusedtobelesscompetitiveduetotheirhighcosts.However,PHEVshavebecomemorecompetitivethangas-fueledvehiclesforthefollowingreasons:1)recentdropinthecostofbatteries[ 19 ],2)signicantfossilfuelpriceincrementduringthelastdecade,3)exibilityandextendeddrivingrangeofferedtothecustomers(cf.[ 38 ]and[ 50 ]),and4)promotionbythegovernmentforitsfunctionofreducingtrafcemissionsandpetroleumdependence.(TheObamaAdministrationintheUnitedStates,forexample,hasproposedtoputatotalofonemillionPHEVsontheroadby2015[ 56 ].)Accordingly,thereispotentialinthenearfutureformajorvehiclemanufacturerssuchasChevrolettomass-producePHEVs,whichwillfurtherreducetheproductioncostforeachPHEV.Ascomparedtogas-fueledvehicles,PHEVshaveanadditionaladvantageofservingaselectricitystoragedevicesapartfromdailytransportationactivities[ 39 , 40 ].Sincemostvehiclesareparkedfor23hoursaday,onaverage,theirbatteriescouldbeusedtostoreandreleaseelectricity,andtheprotisestimatedtobeupto$4,000ayear[ 27 ].ThiselectricitystoragefunctionofPHEVsbenetstheelectricitysysteminthefollowingaspects: (1) Itreducesthethermalpowergenerationcostbychargingpowerduringoff-peakhoursanddischargingitduringpeakhours.Inthisway,thenet-loadduringthepeakhoursisreduced(iftheamountofelectricitydischargedisconsideredtobeanegativeload),whichleadstolowercostsfromfast-startgeneratorsusedtoservepeak-hourloads.Meanwhile,thenetloadduringoff-peakhoursisincreased,whichhelpstopreventshuttingdownslow-startthermalgeneratorsandsaveslargeshut-downcosts. (2) Becauseofitsfastresponsetimeintermsofminutes,orevenseconds,PHEVsarecapableofservingasfrequencyregulation,spinningreserve,voltagesupport,andsoon,inauxiliaryserviceformostindependentsystemoperators(ISOs)(e.g.,MISO and PJM).Thisreducestheneedforhigh-cost,fast-startgenerators.Italsoreducestheneedforkeepingthermalgeneratoroutputlowerthan that underoptimalefciencysoastomaintainacertainreserveamount. (3) Ithelpsthesystemtotakeadvantageofintermittentrenewableenergy.Duetoitsintermittentnature,renewableenergy(suchaswindandsolar)ishardtoforecastaccuratelyandincreasestheneedforreservemarkets.PHEVs,serving 14

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aselectricitystorageunitsbecauseoftheirshortreactiontime,offertheperfectsupplementofsuchintermittentenergy. (4) Asidefromreducingtrafcemissionsandservingaselectricitystorage,PHEVscanhelpreduceemissionsgeneratedfromthermalplants.BothpowersystemoperatorsandenergymarketparticipantsthusconsiderPHEVsimportantpotentialelementsintheirfuturedailyoperations:powersystemoperatorswillconsider optimalcentralcontrolonPHEVs tominimizetotalsystemoperationcostswhilemaintainingsystemreliability;energymarketparticipantswillaimtomaximizetheirtotal prots bycharginganddischargingPHEVs.Inthisdissertation,westudytheoptimalcontrolpoliciesofPHEVsinelectricitymarketsincludingelectricityreal-timemarket,electricictyday-aheadmarket,andelectricityreservemarket.InChapter 2 ,weseekto optimallycontrolthePHEVcharging/dischargingactivities formarketparticipantswiththeobjectiveofmaximizing the totaldiscountedexpectedprot.Werstconsiderparticipatinginthereal-timemarketonly bytakingadvantageofelectricitypricedifferencesatdifferenttimeperiods. Westudytheoptimalcontrolproblemforanaggregator(acontrollerowningagroupofPHEVunits)whoseaction(chargingordischarging)canaffecttheelectricityprice.Wealsotakeintoaccounttheriskattitudeoftheaggregatorandanalyzeitsinuenceontheoptimaldecision.Wethenextendourstudytothecasewheretheaggregatorparticipatesinboththereal-timeandday-aheadmarkets.Foreachmodel,weprovetheconcavityoftheaggregator'sprotfunction,explorethepropertiesofoptimalpolicies,andanalyzetheoptimalpolicystructure.Togetherwithnumerical studies ,weshowhowanaggregator'scharging/dischargingactivities,withtheconsiderationofmarketimpactindecisionmaking,inuenceelectricitypricesascomparedtothoseofindividualcontrollerswhodon'tconsidermarketimpact.Wealsoexplorehowanaggregator'srisk-averseattitudeaffectstheoptimaltradingquantityandtheoverallusageofPHEVelectricitystorage, 15

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andhowmuchextraprotcanbegeneratedbyparticipatinginboththereal-timeandday-aheadmarkets.InChapter 3 ,wefocusontheoptimalinventorycontrolproblem where thedemandevolvesasaBrownianmotionandeachorderhastobeintegraltimesofaxedquantity.Theinventorylevelcanbenegativewhenthereisabacklog.Undertheassumptionthattheholdingandpenalty costsare convex,weprovethattheoptimalcontrolpolicyis a (R;Q)policy.Moreover,weprovideexplicitexpressionsofthevaluefunctionandtheoptimalpolicyparametersRandQ.Inaddition,weprovidemoregeneralvericationtheoremsthatcanbeusedtoverify the guessedfunctionis the optimalvaluefunctionforageneralclassofstochasticinventorycontrolproblems.InChapter 4 ,westudytheBrownianinventorycontrolproblem where theinventorycanbebothincreasedanddecreasedbyintegertimesofa xedbase quantityQ.Underreasonableassumptions,weprovethattheoptimalcontrolpolicyis an (S;D;Q)policyandprovideexplicitexpressionsofS,D,andthevaluefunction.Inaddition,wediscusstheapplicationofthismodel incontrolling thePHEVbatterystorageinelectricityreservemarket.InChapter 5 ,westudytheBrownianinventorycontrolproblem where theinventorycanbeincreasedatanytimewithapositiveconstantset-upcostandapiecewiselinearconcavecontrolcosttominimizethediscountedtotalcostsincludingholdingandcontrol costs .Wederiveclosedform expressionsforthisoptimalcontrolpolicy oftheproblem.Weprovethat,dependingonthebasicparameters,theoptimalpolicyiseitheran(s;S)policyoranewtypeofpolicywhichisdifferent from traditional ones . 16

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CHAPTER2OPTIMALCONTROLOFPLUG-INHYBRIDELECTRICVEHICLES 2.1MotivationandLiteratureReviewSignicantresearchprogresshasbeenmaderecentlyonpotentiallyintegratingPHEVswithpowerelectricitysystemsandmarkets.Foreachelectricitymarket,thebenetsofPHEVstopowersystemoperatorsandmarketparticipantsaredifferent,and,accordingly,theseleadtodifferentresearchfocuses.Systemoperatorsaimto 1) performoptimalcentralcontrolonPHEVsforthebenetofthewholepowergridsystem, 2) performsensitivityanalysisofPHEVs'activities,and 3) decideandselectchargingstations.Forthecentralcontrolapproach,alarge-scaleoptimizationmodelisusuallydevelopedforthesystemoperatorwiththeobjectiveofmaximizingthetotalsocialwelfare,inwhichphysicalconstraintsofthermalgeneratorandtransmissionlinecapacitiesandpowerbalanceconstraintsareconsidered,andPHEVs'charginganddischargingactivitiesaremodeledasloadandgeneration,respectively[ 73 ].Forthisapproach,systemoperatorscontrolthecharging/dischargingactivitiesofPHEVs.Forinstance,in[ 73 ],anewunitcommitmentmodelisdevelopedforsystemoperatorstosimulatetheinteractionsamongPHEVs,windpower,anddemandresponse.FourPHEVchargingscenariosaresimulatedfortheIllinoispowersystem.Withinthiscentralcontrolframework,systemoperatorsalsoconsidertheimpactsofPHEVchargingactivitiesonthepowergridsystem,inparticular,intermsofelectricityprice(i.e.,thelocalmarginalprice(LMP)).Forinstance,in[ 74 ],acasestudybasedonPJMInterconnection(see,e.g.,[ 1 ])showsthatasmallmagnitudeofloadincreasecausedbyPHEVchargingactivitiescanincreaseLMPsignicantly.Furthermore,itisshownempiricallyin[ 75 ]thatchargingPHEVsduringoff-peakhourswillattenthedailyloadcurveandreduceLMPvolatility,andcharging c[2014]TS.Reprinted,withpermission,from[ 76 ] 17

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PHEVsduringpeakhourswillincreaseLMPvolatility.Mostrecently,in[ 67 ],theimpactsofdifferentelectricitytariffsonPHEVdrivers'chargingdecisionshavebeenempiricallystudied.Thecostand emission impactsofthesechargingpatternsarecomparedtotheidealcaseofchargingthatiscontrolledbythesystemoperator.TofacilitatePHEVcharging/dischargingactivities,governmentagenciesandcompaniesarestartingtodeploypublicchargingstations(cf.[ 53 ]).Forinstance,NRGenergyiscommittedtoinvesting$100millioninbuildingaPHEVchargingnetworkinCalifornia[ 34 ],andsimilarlyaplanofbuilding570chargingstationshasbeenproposedinBritishColumbia,Canada[ 51 ].Accordingly,anagent-baseddecisionsupportsystemforaPHEVcharginginfrastructuredeploymentisproposedin[ 71 ],optimaldeploymentofpublicchargingstationsforPHEVshasbeenstudiedin[ 33 ],andinfrastructure-planningforPHEVswithbatteryswappingisproposedin[ 49 ].Onesignicantadvantageofconstructingchargingstations,particularly,batteryswappingstations,istomakeaggregationsofPHEVcharging/dischargingactivitiespossible.Thatis,alargenumberofindividualPHEVscanbeclusteredforcharging/dischargingsimultaneously.Marketparticipantsandelectricityusersaimtomaximizetotalprot(orminimizetotalcost)bymanagingPHEVcharging/dischargingactivitiesfordailyusageandattendingtheauxiliaryservice,real-time,or/andday-aheadmarkets. In[ 64 ],ownersofPHEVsareassumedpricetakers.Intheproposedapproach,forecastedfutureelectricityprices,whicharedeterministic,areutilizedinthecalculationfortheprot.Twodynamicprogrammingalgorithmsareproposedtondtheeconomicallyoptimalsolutionforthevehicleownerforthecaseswithandwithoutparticipatinginancillaryservicemarkets. In[ 2 ] , vehicle-to-gridservice iscoordinatedwiththermalandwindplantsformarketparticipantstoachieveoptimalenergytradingfortheday-aheadmarket.Intheproposedapproach,thewindpoweroutputs,thespotmarketenergyprices,andtheimbalancepricesareassumeduncertain.Atwo-stagestochastic programming modelisderivedandsolvedbycommercialoptimizationsolverCPLEX. 18

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In[ 45 ],theauthorsconsiderdailyenergycostminimizationproblemsofvehicleownersundertime-of-useelectricitypricing frameworks .Thepriceforeachtimeperiodisassumeddeterministicandtheenergydemandateachtimeperiodisrandom.ThePHEVmobilityismodeledasaMarkovchainwiththestatetransitionprobabilitiesdescribingthestatetransitionsforthePHEVs(home,work,andcommute).Astochasticdynamicprogrammingalgorithmisdevelopedtosolvetheproblem. Noteherethat,fromthemodelingperspective,theoptimalscheduleofcharging/dischargingactivitiesconceptforPHEVsaresimilartotheonesstudiedforgeneralelectricitystoragedevices(e.g.,pumped-storagehydro).Forinstance,anoptimalbiddingstrategyisstudiedforelectricitystorageinacompetitiveelectricitymarketin[ 48 ]. Fortheproposedapproach,theelectricitypricesareassumeddeterministicandanonlinearoptimizationmodelisproposedtoformulatetheproblem.Themodelissolvedfortheunconstrainedcaserstbyrelaxingthecapacityconstraintsandthentheoptimizationtimeintervalsareadjustedwhentheupperreservoircapacitylimitsareenforced. Astudyformaximizingtheprotoftheownerofwindpowerandpumped-storagehydroisperformedin[ 18 ].Intheproposedapproach,thewindpowerisassumedrandomandsamplingapproachesareutilizedtogeneratescenarios.Foreachscenario,alinearprogrammingmodel,withthetimehorizontobediscretized,isdevelopedtosolvetheproblem. In[ 3 ],acombinedbiddingstrategyisdevelopedforjointwindandpumpedstorageplantsinapool-basedelectricitymarket.Intheproposedapproach,thewindpowerisassumeduncertainandastochasticmixed-integerlinearprogrammingmodelisproposedtoformulatetheproblem.In[ 17 ],amodelforelectricitystorageisdevelopedthattreatsthestorageproblemasanoptimalswitchingproblemundercontinuoussetting,wherethepricesareassumedaMarkovcontinuous-timestochasticprocessandadiffusionmodelisproposedtomaximizetheexpectedprot.Thedecisionvariables includeoperatingregimesofthestoragefacility (injection,storage,andwithdrawal) andswitchingtimesbetweenrellingthereservoir. In[ 41 ],optimalenergycommitmentswithstorageand 19

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intermittentsupplyhavebeenstudied.Inthisresearch,anoptimalcommitmentpolicyisderivedundertheassumptionthatwindenergyisuniformlydistributedand,furthermore,thestationarydistributionoftheelectricitystoragelevelcorrespondingtotheoptimalpolicyisobtained.Inthisresearch,wesolvetheproblemfromamarketparticipant'sperspective.Noticingthatalltheaboveresearchworksconsideroptimalcontrolsforindividualownerstoparticipateinasingleelectricitymarketanddonot theoreticallyanalyze the market impact.Inourapproach,weconsiderdecisionsforanaggregatorwiththeconsiderationofmarketimpactandparticipatinginboththeday-aheadandreal-timemarkets.Wewilluseindividualownercharging/dischargingactivitiesasabenchmarktoevaluatetheperformanceofourproposedapproach.Recentworksin[ 65 ]and[ 44 ]arerelatedtoourresearch ,inwhichcharging/dischargingactivitiesforagroupofPHEVsarestudied. In[ 65 ],optimalschedulingofPHEVs,insteadofanindividualvehicle,isconsidered ,withthepurposeofreducingdependenciesonsmallexpensiveunitsintheexistingpowersystems.The proposedapproachisforaregulatedmarket andtheobjectiveistominimizethetotalgenerationcost,whileensuringtheloadstobesatised. Noelectricitypricesareinvolvedandmarket-impactfactorsarenotconsideredinthemodel. In[ 44 ],theauthorsassume thepricesaredeterministicforeachtimeperiodandtheaggregatorforagroupofPHEVsisaprice-taker.Alinearprogrammingmodel,oraquadraticprogrammingmodelwhenmarketimpactsareconsidered,isstudiedwiththeobjectiveofminimizingtotaloperationcosts. However,theydonotprovidetheoretical proofs oranalyzetheoptimalpolicystructure. Inourapproach,wecontributetotheliteraturebysolvingtheoptimalelectricitystorageproblem,undertheassumptionthattheelectricitypricesareuncertain,forbothrisk-neutralandrisk-averseaggregatorswhoconsidermarketimpactandparticipateinboththereal-timeandday-aheadmarkets.Inparticular,weanalyzetheoptimalpolicystructureandexploreinsightsintotheproblem. 20

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Theremainingpartofthispaperisorganizedasfollows.InSection 2.2 ,westudyoptimalstoragecontrolpoliciesforPHEVsparticipatinginthereal-timemarket only .Wealsoexploretheinsightsofrisk-averseplayersversusrisk-neutralplayersandcomparetheirperformance.InSection 2.3 ,weextendourstudytoconsiderboththeday-aheadandreal-timemarketsforanaggregator.Finally,inSection 2.4 ,wereportexperimentalresults,andmoreimportantly,explorefurtherinsights. 2.2Real-TimeMarketInthissection,PHEVsareconsideredonlyparticipatinginthereal-timemarket.Underthecurrentderegulatedmarketframework,thereal-timeLMPisupdatedevery15minutesformostmarketsbyrunningthesecurity-constrainedeconomicdispatchproblem. Accordingly,wediscretizethewholeoperationaltimeintervalinto~Ntimeperiods(e.g.,timeperiods1to~N) ,withthelengthofeachtimeperiodtobe15minutes.Inaddition,toparticipateinthereal-timemarket,marketparticipantshavetosubmittheir offers mperiods(4to6,dependingondifferentmarkets)ahead.Inotherwords,attimeperiodn)]TJ /F3 11.955 Tf 12.94 0 Td[(m,weknowthehistorical exogenous real-timeelectricity prices ~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m=(p1;:::;pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m),wherepirepresentsthe exogenous pricefortimeperiodi,andwemakethecharging/dischargingamountofferfortimeperiodn. Wesubmitthisoffer(i.e.,howmuchtocharge/discharge)asaprice-taker(alsonamedself-schedulingatMISOandCAISO). Thatis,we onlyofferelectricityamounts,intermsofbuyingandselling,andtheofferisguaranteedtobetaken.Thismeansthereisnobiddingpriceintheoffer(orthebiddingpriceisverylowforsellingelectricityandveryhighforbuyingelectricity).Followingthismarketruleandourofferingstrategy,attimeperiodn)]TJ /F3 11.955 Tf 12.51 0 Td[(m,thestartingelectricitystoragelevelattimeperiodn,denotedasxncanbecomputedbasedonthecharging/dischargingschedulesintimeperiodsn)]TJ /F3 11.955 Tf 12.54 0 Td[(mton)]TJ /F4 11.955 Tf 12.54 0 Td[(1. Inaddition,therearenocharging/dischargingactivitiesfortherstmtimeperiods(i.e.,time periods 1tom).Fornotationbrevity,intheremainingpartofthispaper,itissufcienttouse[1;N]asthe 21

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operationaltimeinterval,correspondingto[m+1;~N]whereN=~N)]TJ /F3 11.955 Tf 12.06 0 Td[(m,truncatedfrom[1;~N]. Inthecharging/dischargingprocess,twoaspectsmustbeconsidered.First,thereisconversionlossfromstorage.Theconversionofpowerintostoredelectricityandbacktothepowergridisimperfect.Theelectricityamountconvertedfrompowerintostorageduringthedischargingprocessisonlyaportionoftheoriginalelectricityamountspentonchargingthebattery.Leturepresenttheefciencyofconvertingthepowerelectricityintopotentialenergyinthestorageduringthechargingprocess,anddrepresenttheefciencyofconvertingthepotentialenergyinthestorageintothepowerelectivity.Theoverallenergyconversioncoefcient,e.g.,ud,isaround0:6-0:8basedonthedescriptionin[ 68 ].Second,thecharging/dischargingrateisbounded.WeletCubetheupperlimitforthechargingrateandCdbetheupperlimitforthedischargingrateinasingletimeperiod. Finally,wedenotethetotalbatterycapacityownedbytheaggregatortobeC+ClwithClrepresentingtheminimumstoragelevelrequiredtoprotectthebattery andnancialdiscountratetobe. Fornotationbrevity,weusetheinterval[0;C],insteadof[Cl;Cl+C],torepresenttheallowablecharging/discharingrangeofthebattery. Inourapproach,the exogenous electricitypricecouldbeMarkovianorofamoregeneraltype. Ourdecisionishowmuchtocharge/dischargeineachperiod n, whichisequivalenttothedecisionon theending storagelevelinperiodn,denotedasyn, n2[1;N], soastomaximize thetotaldiscountedexpectedprot. 2.2.1Risk-NeutralAggregatorWerststudyarisk-neutralaggregatorcase.Forthiscase,weexploreanite-horizonmodeltomaximizetheaggregator'stotaldiscountedexpectedprotandsolvetheproblemusingstochasticdynamicprogramming.Let Vn(xn;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m) bethemaximumexpectedtotalprotfromperiodntotheendofthehorizon(e.g.,periodN).Werstinitializethevaluefunction VN+1(xN+1;~pN+1)]TJ /F5 7.97 Tf 6.59 0 Td[(m) to 22

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bethesalvagevalueoftheelectricityattheendoftheoperationaltimehorizon,whichisassumedtobeaconcaveandincreasingfunctioninxN+1(e.g., VN+1(xN+1;~pN+1)]TJ /F5 7.97 Tf 6.58 0 Td[(m)=lnxN+1 forasufcientlysmall>0).Foreachparticulartimeperiodn,thechargingamountisequaltotheendingstoragelevelminustheinitialonefortheperiod(i.e.,yn)]TJ /F3 11.955 Tf 11.74 0 Td[(xnifyn)]TJ /F3 11.955 Tf 11.74 0 Td[(xn0).Similarly,thedischargingamountisequaltoxn)]TJ /F3 11.955 Tf 11.54 0 Td[(yn,ifyn)]TJ /F3 11.955 Tf 11.53 0 Td[(xn<0.Therefore,ingeneral,thechargingamountisequalto[yn)]TJ /F3 11.955 Tf 12.26 0 Td[(xn]+andthedischargingamountisequalto[yn)]TJ /F3 11.955 Tf 12.26 0 Td[(xn])]TJ /F1 11.955 Tf 7.09 -4.34 Td[(,wherez)]TJ /F4 11.955 Tf 10.41 -4.34 Td[(=maxf)]TJ /F3 11.955 Tf 15.27 0 Td[(z;0gandz+=maxfz;0g.Inaddition,toconsiderthemarketimpact,weassumethe exogenous real-timeelectricitypriceisnon-negativeandtheinuenceofcharging/dischargingactivitiesonthereal-timeelectricitypriceisbndollarsperunitatperiodn.Forinstance,thereal-timeelectricitypriceisassumedtoincrease(decrease)bndollarsifoneadditionalelectricityunitischarged(discharged).Thisvaluecanbeconsideredastheslopeofthesupplycurveattheclearingpriceforthereal-timemarket. Inaddition,weassumethatthecontroller'sactiondoesnotaffecttheevolutionoftheexogenousprice. Basedonthisassumption,themyopicprotfunctionforthecurrentperiodcanbedescribedasfollows: Ln(yn)]TJ /F3 11.955 Tf 11.96 0 Td[(xn;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)=Epnj~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m[pn])]TJ /F3 11.955 Tf 11.96 0 Td[(bnd[yn)]TJ /F3 11.955 Tf 11.96 0 Td[(xn])]TJ /F4 11.955 Tf 9.74 -4.93 Td[(+bn[yn)]TJ /F3 11.955 Tf 11.95 0 Td[(xn]+=u(d[yn)]TJ /F3 11.955 Tf 11.95 0 Td[(xn])]TJ /F2 11.955 Tf 9.75 -4.93 Td[()]TJ /F4 11.955 Tf 11.95 0 Td[([yn)]TJ /F3 11.955 Tf 11.96 0 Td[(xn]+=u): (2) Accordingly,foragivenbeginningstoragelevelxn,weletymyopn(xn;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)representthemaximizeroftheabove function .ThenthebackwardBellmanequationforthedynamicsofthesystemcanbedescribedasfollows: Vn(xn;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)=maxyn2A(xn)fLn(yn)]TJ /F3 11.955 Tf 11.96 0 Td[(xn;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)+E pn)]TJ /F10 5.978 Tf 5.75 0 Td[(m+1 j~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m[Vn+1(yn;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m+1)]g;(2)wheretheactionspace A(xn)=fyn:(xn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cd)+ynminfxn+Cu;Cgg(2) 23

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describesthefeasiblesetofthetargetstoragelevelwhentheinitialstoragelevelisxn. Noteherethedecisiononynismadeattimeperiodn)]TJ /F3 11.955 Tf 11.95 0 Td[(m. Finally,theobjectiveistomaximizethetotalexpecteddiscountedprotforthegivenoperationaltimeinterval(i.e.,thevalue V1(x1;~p1)]TJ /F5 7.97 Tf 6.58 0 Td[(m)) .Beforedescribingtheoptimalcontrolpolicyfortheproblem,werstexplorethepropertiesofthevaluefunctionsasfollows: Proposition2.1. Forarisk-neutralaggregatorwithmarket-impactconsideration,wecanobservethefollowingproperties: (a) Themyopicoptimalsolutionisymyopn(xn;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)=maxfxn)]TJ /F3 11.955 Tf 9.3 0 Td[(Cd;xn)]TJ /F3 11.955 Tf 9.29 0 Td[(Epnj~pn)]TJ /F10 5.978 Tf 5.75 0 Td[(m[pn]=(2bnd);0g,andthemyopicprotfunctionLn(z;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)isstrictlyconcaveinz. (b) ThejointvaluefunctionJn(xn;yn;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m):=Ln(yn)]TJ /F3 11.955 Tf 9.37 0 Td[(xn;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)+E pn)]TJ /F10 5.978 Tf 5.75 0 Td[(m+1 j~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m[Vn+1(yn;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m+1)]isjointlyconcavein(xn;yn).Foranygivenxn,Jn(xn;yn;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)isstrictlyconcaveinynandaccordingly,thereexistsaunique optimalendingelectricitystoragelevel yn(xn;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)foranygivenxn. (c) ThevaluefunctionVn(xn;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)increasesandisconcaveinxn. Proof. TheproofisgiveninAppendix A.1 . Basedontheaboveproposition,wecanderivetheoptimalpolicyasfollows: Theorem2.1. Foranypriceevolutionmodel,thereexistsauniqueoptimaltargetstoragelevelxn= yn)]TJ /F7 7.97 Tf 6.59 0 Td[(1(yn)]TJ /F7 7.97 Tf 6.59 0 Td[(2(:::y1(x1;~p1)]TJ /F5 7.97 Tf 6.59 0 Td[(m);:::;~pn)]TJ /F7 7.97 Tf 6.59 0 Td[(2)]TJ /F5 7.97 Tf 6.59 0 Td[(m);~pn)]TJ /F7 7.97 Tf 6.59 0 Td[(1)]TJ /F5 7.97 Tf 6.59 0 Td[(m) ,correspondingtoanygiveninitialstoragelevel x1 andanyavailablepriceinformation~pn)]TJ /F7 7.97 Tf 6.58 0 Td[(1)]TJ /F5 7.97 Tf 6.59 0 Td[(mforn=2;:::;N+1 . Proof. TheproofisgiveninAppendix A.2 . Moreover,basedontheconcavityandmonotonicityanalysisdescribedinProposition 2.1 ,wecanfurtherobtainthefollowingstructuralresults. Proposition2.2. Forariskneutral aggregatorwithmarket-impactconsideration,wecanobserve that 24

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(a) Theoptimalendingelectricitystoragelevelyn(xn;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)isanon-decreasingfunc-tionoftheinitialelectricitystoragelevelofthecurrentperiodxn,i.e.,@yn=@xn0.Besides,yn(xn;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)ymyopn(xn;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m). (b) Theoptimalcharging/dischargingamountyn(xn;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m))]TJ /F3 11.955 Tf 12.55 0 Td[(xnisanon-increasingfunctionoftheinitialelectricitystoragelevelofthecurrentperiodxn. (c) Ifelectricitypricesareindependentamongperiods,thenyn(xn;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)isanon-increasingfunctionofEpnj~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m[pn],whichisequaltoE[pn]. Proof. TheproofisgiveninAppendix A.3 . Proposition 2.2 providesaniceoptimalpolicystructurefortheproblem.Proposition 2.2 (a)tellsusthatthehighertheelectricitystoragelevelatthebeginningofoneperiod,thehighertheelectricitystoragelevelweshouldkeepattheendoftheperiod.Inaddition,foreachtimeperiodn,wealwayshavetheoptimalendingelectricitystoragelevelnosmallerthantheoneobtainedbythemyopicapproach.Proposition 2.2 (b)indicatesthatthehighertheinitialelectricitystoragelevelwehave,themoreweshoulddischargeorthelessweshouldcharge.Italsoshowsthatifitisoptimaltodischargeatacertaininitialelectricitystoragelevel,thenitisoptimaltodischargeatahigherinitialelectricitystoragelevel.Similarly,ifitisoptimaltochargeatacertaininitialelectricitystoragelevel,itisoptimaltochargeatalowerinitialelectricitystoragelevel.Soforeachexpectedpricelevel,thereexisttwothresholdss(~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)andS(~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m).Iftheinitialelectricitystoragelevelislowerthans(~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m),thenitisoptimaltocharge;iftheinitialelectricitystoragelevelishigherthanS(~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m),thenitisoptimaltodischarge;iftheinitialelectricitystoragelevelisbetweens(~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)andS(~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m),thenitisoptimaltodonothing.TheoptimalpolicystructureforthisgeneralpriceevolutionisshownontheleftinFigure 2-1 .Proposition 2.2 (c)indicatesthatifpricesareindependentamongperiods,thenthehighertheexpectedcurrentprice,themoreelectricityweshoulddischargeorthelessweshouldcharge.Accordingly,wehavethemonotonepropertyandoptimalpolicystructureforthisprice-independentcase,whichisshownontherightinFigure 2-1 . 25

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Figure2-1. OptimalPolicyStructure(GeneralPricevs.IndependentPrice) 2.2.2Risk-AverseAggregatorIntheprevioussubsection,weassumetheelectricitystorageaggregatorisrisk-neutral.However,inarealisticsituation,peoplearetypicallyrisk-averse(i.e., theycaremoreaboutlossthanbenet ).Inthissubsection,weconsiderthecasewheretheaggregatorhasrisk-averseattitudeanduseanexponentialutilityfunctiontocapturetherisk-averseattitude.Inadditiontothenotationdescribedintheprevioussubsection,weaddanadditionalnotationtorepresentthecoefcientofriskaversion.Underthissetting,weletVn(xn;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)bethemaximumutilityfromperiodntotheendoftheoperationaltimeinterval,andVN+1(xN+1;~pN+1)]TJ /F5 7.97 Tf 6.59 0 Td[(m)betheincreasinglyconcavesalvagevalueoftheelectricityatthelastperiod,asdescribedintheprevioussubsection.Foreachtimeperiodn,weletL0n(yn)]TJ /F3 11.955 Tf 12.88 0 Td[(xn;pn)denotethemyopicprotfunctionwhenthepriceforthecurrentperiodispnandymyopn(xn;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m;)representitsmaximizercorrespondingtoagivenxn,where L0n(z;pn)=[pn)]TJ /F3 11.955 Tf 11.95 0 Td[(bdz)]TJ /F4 11.955 Tf 9.74 -4.93 Td[(+bz+=u](dz)]TJ /F2 11.955 Tf 9.74 -4.93 Td[()]TJ /F3 11.955 Tf 11.96 0 Td[(z+=u): (2) Now wecan denethemyopicutilityfunctionLn(z;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)=Epnj~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m[1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(L0n(z;pn)]: 26

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Accordingly ,thedynamicsofthesystemcanbedescribedasfollows: Vn(xn;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)=maxyn2A(xn)fLn(yn)]TJ /F3 11.955 Tf 11.96 0 Td[(xn;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)+Epn)]TJ /F10 5.978 Tf 5.76 0 Td[(m+1j~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m[Vn+1(yn;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m+1)]g;(2)where A(xn)=fyn:(xn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cd)+ynminfxn+Cu;Cgg(2)isthefeasiblesetwhentheinitialelectricitystorageisxn.Thenalobjectiveoftheproblemistomaximizethetotaldiscountedexpectedutilityofoperatingthesystem V1(x1;~p1)]TJ /F5 7.97 Tf 6.58 0 Td[(m) .Similartothestudyintheprevioussubsection,beforedescribingtheoptimalcontrolpolicyfortheproblem,werstexplorethepropertiesofthevaluefunctionsasfollows: Proposition2.3. Forarisk-averseaggregatorwithmarket-impactconsideration,wecanobservethat (a) ThemyopicutilityfunctionLn(z;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)isstrictlyconcaveinz. (b) ThejointvaluefunctionJn(xn;yn;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)=Ln(yn)]TJ /F3 11.955 Tf 9.29 0 Td[(xn;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)+Epn)]TJ /F10 5.978 Tf 5.76 0 Td[(m+1j~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m[Vn+1(yn;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1)]isjointlyconcavein(xn;yn).Foranygivenxn,Jn(xn;yn;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)isstrictlyconcaveinyn,and,accordingly,thereexistsaunique optimalendingelectricitystoragelevel yn(xn;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)foranygivenxn. (c) ThevaluefunctionVn(xn;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)increasesandisconcaveinxn. Proof. TheproofisgiveninAppendix A.4 . Basedontheaboveproposition,wecanshowtheuniquenessoftheoptimalcontrolpolicyfortherisk-averseaggregatorcase. Theorem2.2. Forarisk-averseaggregatorwithmarket-impactconsideration,thereex-istsauniqueoptimaltargetstoragelevelxn= yn)]TJ /F7 7.97 Tf 6.58 0 Td[(1(yn)]TJ /F7 7.97 Tf 6.59 0 Td[(2(:::y1(x1;~p1)]TJ /F5 7.97 Tf 6.58 0 Td[(m);:::;~pn)]TJ /F7 7.97 Tf 6.58 0 Td[(2)]TJ /F5 7.97 Tf 6.58 0 Td[(m);~pn)]TJ /F7 7.97 Tf 6.59 0 Td[(1)]TJ /F5 7.97 Tf 6.59 0 Td[(m) ,correspondingtoanygiveninitialstoragelevel x1 andanyavailablepriceinformation~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(mforn=2;:::;N+1 . Proof. TheproofisgiveninAppendix A.5 . 27

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BasedonProposition 2.3 andTheorem 2.2 ,wecanfurtherexploretherisk-aversecontrolpolicyasfollows: Proposition2.4. Forarisk-averseaggregatorwithmarket-impactconsideration,wecanobserve: (a) The dischargingamount xn)]TJ /F3 11.955 Tf 11.95 0 Td[(ymyopn(xn;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m;) non-increases in. (b) Theoptimalendingelectricitystoragelevelyn(xn;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)isanon-decreasingfunc-tionoftheinitialelectricitystoragelevelofthecurrentperiodxn,i.e.,@yn=@xn0. (c) Theoptimalcharging/dischargingamountyn(xn;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m))]TJ /F3 11.955 Tf 12.55 0 Td[(xnisanon-increasingfunctionoftheinitialelectricitystoragelevelofthecurrentperiodxn. Proof. TheproofisgiveninAppendix A.6 . FromtheaboveProposition 2.4 ,wecanobservethat,similartotherisk-neutralcase,wealsohaveathresholdpolicycorrespondingtoeachexpectedpricelevel.Thatis,basedonProposition 2.4 (b)and(c),thereexiststwothresholdss(~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)andS(~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)forwhichitisoptimalto discharge whentheinitialelectricitystoragelevelishigherthanS(~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m),to charge whentheinitialelectricitystoragelevelislowerthans(~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m),andtodonothingwhentheinitialelectricitystoragelevelisinbetween.Moreover,basedonProposition 2.4 (a),weobservethat,forthemyopicpolicy,wewilldischargelessiftheaggregatorgetsmoreconservative(e.g.,increases).Accordingly,forthemulti-periodcase,theaggregatortendstocharge/dischargelessasincreases.Lateron,weexploretheinsightsanddiscusstheimpactofrisk-averseattitudeontheoptimalpolicythroughnumericalstudiesinSection 2.4.1.3 . 2.3BothDay-AheadandReal-TimeMarketsIntheprevioussection,weonlyconsiderthecharging/dischargingactivitiesforthereal-timemarket. With thenumberofPHEVsincreasinginthecomingyears,itisrationalforaggregatorstoparticipateinboththereal-timeandday-aheadmarkets,whichcanhelpbringextraprot.However,itismuchmorechallengingtomakeoptimaldecisionsforaggregatorsunderthetwo-marketsettings.Day-aheadandreal-timemarketframeworksforenergytradingworkindifferentways.Asdescribedintheprevious 28

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Figure2-2. ProcesstoSubmitDay-AheadandReal-TimeOffers section,inparticipatinginthereal-timemarket,the offerincluding charging/discharging amounts needstobesubmitted mtimeperiodsahead.However,fortheday-aheadmarket,correspondingtoeachoperatingday(e.g.,dayn)formostU.S.energymarkets,ISOsrequiremarketparticipantstosubmittheiroffersbeforenoonthedaybefore(e.g.,dayn–1). Sometimeintheafternoon(dependingondifferentmarkets), theISOsnishtheirday-aheadunitcommitmentrunsandannounceday-aheadLMPsforeachoperatinghourondayn. Withoutlossofgenerality,weassumetheday-aheadLMPs become availableonetimeperiodaftertheoffersaresubmitted. Therefore,forthecurrentpractice,inordertoparticipateinbothmarkets,anaggregatorhastosubmitthecharging/dischargingamountday-aheadoffersforeachparticularoperatinghouronanoperatingday(e.g.,dayn)atnoonthedaybefore(e.g.,dayn–1)andcontinuesubmittingcharging/dischargingamountreal-timeoffers m periodsbeforeeachoperatingtimeperiod.Theprocessestosubmitday-aheadandreal-timeoffersareshowninFigure 2-2 .Insteadofusingonlyoneindexntorepresentthetimeperiods,weuse(n;k)asthestatespaceindexinthissectionfortheconvenienceoflaterdevelopmentofdynamicequations,wherenrepresentsthenthdateandkindicatesthekthperiodon thenth date.Accordingly,weusepn;kandfn;ktodenotethe exogenous electricitypricesinthekthperiodonthenthdateinthereal-timeandday-aheadmarkets,respectively.Wealso 29

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letqn;kandq0n;kdenotethedischargingamountsofelectricity( or thechargingamountswhennegative)inthekthperiodonthenth date intheday-aheadandreal-timemarkets,respectively.Finally,weassumethereareKtimeperiodsintotalinoneday. 2.3.1NotationandAssumptionWemainlystudytherisk-neutralcaseinthissubsection.Basedonthemarketoperationdescribedintheprevioussubsection,inthedecisionprocess,weneedtodecidetheday-aheadcharging/dischargingamount sforalloperatinghoursthenextday beforetheday-aheadLMPsareavailable.Similartothereal-timemarketonlycase,thecharging/dischargingactivitiesaffecttheday-aheadLMPsaswell.Ascomparedtootherthermalgenerationsources,theportionofthePHEVs'electricitystorageenergyisrelativelysmall.Weassumetheinuenceofcharging/discharging amounts ontheday-aheadelectricitypriceisadollarsperunit.Correspondingtoeachparticulardayn,weassumetheoperationaltimeintervalisfrom12amthecurrentdayto12amthenextday.Toconsidertheactionspaceforeachstate(n;k),wediscusstwocases: (i) Thestatesinwhichaggregatorssubmitboththeday-aheadandreal-time offers .Supposeaggregatorssubmit offers fortheday-aheadmarket`timeperiodsbeforetheoperationaltimeintervalstarts(12am)forthenextday.Inaddition, notingherethat, asdescribed intheprevioussection,aggregatorswillsubmit offers forthereal-timemarketm (m<`) timeperiodsahead. Then,attimeperiodK)]TJ /F3 11.955 Tf 10.84 0 Td[(`foreachparticularday,theaggregatorwillsubmitthe charging/discharging amounts fortheday-aheadmarketonthe next dayandthecharging/dischargingamount for timeperiodK)]TJ /F3 11.955 Tf 11.96 0 Td[(`+mforthereal-timemarketforthecurrentday. (ii) Thestatesinwhichaggregatorscanonlysubmitoffersforthereal-timemarket.Allstatesoneachparticulardayareinthiscategoryexcepttheonedescribedin(i).Fornotationbrevity,werstintroducethefollowingvectorsbeforewedescribetheBellmanequation: ~pn;k=[p1;1;:::;pn;k],~fn=[f1;1;:::;fn;K],~qn=[q1;1;:::;qn;K],and~qn;k:K=[qn;k;:::;qn;K]. Nowweconsidervaluefunctionexpressions.Becausethedecisionsoncharging/dischargingamounts for thereal-timemarketaremademtimeperiodsahead,correspondingtothe 30

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valuefunctionsforthestates(n;k)withkK)]TJ /F3 11.955 Tf 9.73 0 Td[(`+m,allnecessaryavailableinformationforthedecisionmadeforthisperiodincludesxn;k,~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m,~fn,and~qn;k:K.Accordingly,wecandescribethevaluefunction,i.e.,representingtheoptimalvalue(e.g.,maximumprot)fromthekthperiodondayntotheendoftheoperationaltimeinterval,intheformVn;k(xn;k;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn;~qn;k:K).Noteherethatthisvaluefunctionholdsthesameformforthestate(n;k)withk=K)]TJ /F3 11.955 Tf 12.3 0 Td[(`+m,forwhichwe describe theinformationavailableforthedecisionsmadeattimeperiodk)]TJ /F3 11.955 Tf 11.97 0 Td[(`,forboth the day-aheadmarketfordayn+1and the real-timemarketfortimeperiodk)]TJ /F3 11.955 Tf 12.27 0 Td[(`+mfordayn.Ontheotherhand,correspondingtothevaluefunctionsforthestates(n;k)withkK)]TJ /F3 11.955 Tf 12.03 0 Td[(`+m+1,~fn+1and~qn+1become available whenthe decisionsaretobe madefor thesestates .Accordingly,wecan describe thevaluefunctionintheformVn;k(xn;k;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m;~fn+1;~qn;k:K;~qn+1;1:K)forthesestates. Similartothereal-timemarketonlycase,weinitializetheelectricitysalvagevalueforthelastperiod VN;K+1() tobeanincreasinglyconcavefunction.Fornotationbrevity,welet Vn;)]TJ /F5 7.97 Tf 6.58 0 Td[(`()=Vn)]TJ /F7 7.97 Tf 6.59 0 Td[(1;K)]TJ /F5 7.97 Tf 6.59 0 Td[(`()for1`K)]TJ /F4 11.955 Tf 11.96 0 Td[(1:(2)Inourapproach,weallowboth exogenous day-aheadandreal-timepricestofollowanypriceevolutionprocess andassumethatthecontroller'sactiondoesn'taffecttheevolutionoftheexogenousprices .Inaddition,wemakethefollowingmildassumptions.Forinstance,welet[p n;k; pn;k]denotethesupportoftherandomvariablepn;kwithp n;k>0and pn;k<+1.Weassumetheday-aheaddischargingamount q n;kqn;kqn;k ,inwhichq n;kisassumedtobe)]TJ /F3 11.955 Tf 9.3 0 Td[(p n;k=(2bn;k)andqn;kisanarbitrarylargenumber.Underthissetting, weassumetheaggregatorwillnotbuyenoughelectricityfromtheday-aheadmarkettoreducethereal-timepricesignicantly(e.g.,byhalfofthelowest 31

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real-timeprice) ,becauseitismorereasonabletobuytheelectricityfromthereal-timemarketwhenthereal-timepriceisdecreasedsignicantly.1 2.3.2RevenueFunctionforEachPeriodnCorrespondingtoeach state (n;k),weletxn;kandyn;krepresentthebeginningandendingelectricitystoragelevels.Inaddition,wedenotezn;k=yn;k)]TJ /F3 11.955 Tf 12.38 0 Td[(xn;k.Accordingly,wehavez+n;k=maxfzn;k;0grepresentingthetotalnetchargingamountandz)]TJ /F5 7.97 Tf -.54 -8.28 Td[(n;k=maxf)]TJ /F3 11.955 Tf 15.27 0 Td[(zn;k;0grepresentingthetotalnetdischargingamount,combiningtheday-aheadandreal-timecharging/dischargingquantities.Theelectricitystoragebalanceequationcanbe described asfollows: qn;k+q0n;k=dz)]TJ /F5 7.97 Tf -.53 -8.28 Td[(n;k)]TJ /F3 11.955 Tf 11.95 0 Td[(z+n;k=u:(2) Foreachtimeperiod(n;k),because thecharging/dischargingamountforthe day-ahead marketqn;kisknownwhenwemakethe offer forthereal-timemarket,thecharging/dischargingamountforthereal-timemarketq0n;kcanbereplacedbydz)]TJ /F5 7.97 Tf -.53 -8.27 Td[(n;k)]TJ /F3 11.955 Tf 12.62 0 Td[(z+n;k=u)]TJ /F3 11.955 Tf 12.61 0 Td[(qn;kinourformulations.Then,theday-aheadprotvalueforthecurrentperiodisqn;k(fn;k)]TJ /F3 11.955 Tf 11.21 0 Td[(an;kqn;k)withtheconsiderationofmarketimpact.Thereal-timeprotvalueforthecurrentperiodisequalto (dz)]TJ /F5 7.97 Tf -.53 -8.28 Td[(n;k)]TJ /F3 11.955 Tf 11.96 0 Td[(z+n;k=u)]TJ /F3 11.955 Tf 11.96 0 Td[(qn;k)(pn;k)]TJ /F3 11.955 Tf 11.96 0 Td[(bn;k(dz)]TJ /F5 7.97 Tf -.53 -8.28 Td[(n;k)]TJ /F3 11.955 Tf 11.96 0 Td[(z+n;k=u)]TJ /F3 11.955 Tf 11.96 0 Td[(qn;k)):(2) 1Thelowerlimitforthe99%condenceintervalofthereal-timepricefrom9amto11pmisaround$20=MWhinPJM,basedonthedatafromJanuary2008toDecember2012.Atnight,thereal-timepricecouldbelower.Butitisreasonabletoassumethat,duringthistimeinterval,wecanjustbuyelectricityfromthereal-timemarket,insteadoftheday-aheadmarket,sincetheday-aheadpriceisgenerallyhigherintheseperiods. 32

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Accordingly,themyopicprotfunctionfor state (n;k)includesboththeday-aheadandreal-time prots andcanbe described asfollows: Ln;k(zn;k;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m; ~fn ;qn;k)=qn;k(fn;k)]TJ /F3 11.955 Tf 11.95 0 Td[(an;kqn;k)+Epn;kj(~pn;k)]TJ /F10 5.978 Tf 5.76 0 Td[(m; ~fn ))]TJ /F3 11.955 Tf 10.46 -9.68 Td[(dz)]TJ /F5 7.97 Tf -.53 -8.28 Td[(n;k)]TJ /F3 11.955 Tf 11.95 0 Td[(z+n;k=u)]TJ /F3 11.955 Tf 11.95 0 Td[(qn;k)]TJ /F3 11.955 Tf 12.95 -9.68 Td[(pn;k+bn;kz+n;k=u)]TJ /F3 11.955 Tf 11.95 0 Td[(bn;kdz)]TJ /F5 7.97 Tf -.53 -8.28 Td[(n;k+bn;kqn;k: (2) 2.3.3BellmanEquationsandMainResults Foreachtimeperiod (n;k),theactionspaceA(xn;k)representsthefeasiblesetfordecisionmakingwhentheinitialinventoryisxn;k.Underthetwo-marketsetting,thefeasibleactionsetis A(xn;k)=fyn;k:(xn;k)]TJ /F3 11.955 Tf 11.95 0 Td[(Cd)+yn;kminfxn;k+Cu;Cgg:(2)Thedynamicsofthesystemcanberepresentedunder thefollowing threecategories: (1) Thevaluefunctionscorrespondingtostates(n;k)with 1kK)]TJ /F3 11.955 Tf 11.96 0 Td[(`+m)]TJ /F4 11.955 Tf 11.96 0 Td[(1:Vn;k(xn;k;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m;~fn;~qn;k:K)=maxyn;k2A(xn;k)Jn;k(xn;k;yn;k;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn;~qn;k:K);wherethejointvaluefunctionJn;k(xn;k;yn;k;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn;~qn;k:K)=Ln;k(yn;k)]TJ /F3 11.955 Tf 11.96 0 Td[(xn;k;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m; ~fn ;qn;k)+Epn;k)]TJ /F10 5.978 Tf 5.76 0 Td[(m+1j(~pn;k)]TJ /F10 5.978 Tf 5.76 0 Td[(m; ~fn )[Vn;k+1(yn;k;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1;~fn;~qn;k+1:K)]: Forthesestates,thecorresponding timeperiodsfordecidingreal-timeoffers arebefore period K)]TJ /F3 11.955 Tf 11.96 0 Td[(`,whentheday-aheadoffersaresubmitted. (2) Thevaluefunctioncorrespondingtostate(n;k) withk=K)]TJ /F3 11.955 Tf 11.95 0 Td[(`+m:Vn;k(xn;k;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn;~qn;k:K)=maxyn;k2A(xn;k);~qn+1;1:KJn;k(xn;k;yn;k;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m; ~fn ;~qn;k:K;~qn+1;1:K);wherethejointvaluefunctionJn;k(xn;k;yn;k;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m; ~fn ;~qn;k:K;~qn+1;1:K)=Ln;k(yn;k)]TJ /F3 11.955 Tf 11.96 0 Td[(xn;k;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m; ~fn ;qn;k)+E(pn;k)]TJ /F10 5.978 Tf 5.76 0 Td[(m+1; ~fn+1 )j(~pn;k)]TJ /F10 5.978 Tf 5.75 0 Td[(m;~fn)[Vn;k+1(yn;k;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m+1;~fn+1;~qn;k+1:K;~qn+1;1:K)]: Forthiscase, thedecisions forboththeday-aheadandreal-timeoffers aremadeattimeperiodK)]TJ /F3 11.955 Tf 12.58 0 Td[(`,withtheavailableinformation(xn;k;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m;~fn;~qn;k:K).One 33

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timeperiodlater,theday-aheadLMPs(~fn+1)andofferamounts(~qn+1;1:K)fordayn+1areavailable,besidesyn;kandpn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1,which lead totheexpressionVn;k+1(yn;k;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1;~fn+1;~qn;k+1:K;~qn+1;1:K). (3) Thevaluefunctioncorrespondingtostate(n;k)with K)]TJ /F3 11.955 Tf 11.95 0 Td[(`+m+1kK:Vn;k(xn;k;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m;~fn+1;~qn;k:K;~qn+1;1:K)=maxyn;k2A(xn;k)Jn;k(xn;k;yn;k;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m;~fn+1;~qn;k:K;~qn+1;1:K);wherethejointvaluefunctionJn;k(xn;k;yn;k;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn+1;~qn;k:K;~qn+1;1:K)=Ln;k(yn;k)]TJ /F3 11.955 Tf 11.96 0 Td[(xn;k;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m; ~fn+1 ;qn;k)+Epn;k)]TJ /F10 5.978 Tf 5.75 0 Td[(m+1j(~pn;k)]TJ /F10 5.978 Tf 5.75 0 Td[(m; ~fn+1 )[Vn;k+1(yn;k;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1;~fn+1;~qn;k+1:K;~qn+1;1:K)]: Forthesestates,theday-aheadLMPsandofferamountsfordayn+1areavailableandthe onlydecisionisthe real-time offer . Finally,forthelastday(i.e.,dayN),therearenoday-aheadofferstobedetermined. TheobjectiveistomaximizethetotaldiscountedexpectedprotofoperatingthesystemV1;1(x1;1;~p1;1)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~f1;~q1), where~q1isassumedtobe~0 . Fromtheabove Bellman equations,wecanobservethatascomparedtothereal-timemarketonlycase,forthebothmarketscase,wehaveadditionalone-timedecisionintermsofday-aheadofferamountsoneachparticularday.Althoughthisdecisionhappensonlyatonetimeperiodperday,itcanbeobservedthatthedimensionoftheproblemincreasessignicantly.Inthefollowing,weexplorestructurepropertiestohelpreducecomputationaltimes. Proposition2.5. Fortherisk-neutralaggregatorsparticipatinginboththeday-aheadandreal-timemarkets correspondingtoeachoperatingdayn,1nN ,wecanobservethefollowingproperties foranyfeasible policy : (a) Ln;k(zn;k;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m; ~fn ;qn;k)isstrictlyconcavein(zn;k;qn;k)for1kK)]TJ /F3 11.955 Tf 12.05 0 Td[(`+m, andLn;k(zn;k;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m; ~fn+1 ;qn;k)isstrictlyconcavein(zn;k;qn;k)forK)]TJ /F3 11.955 Tf 10.61 0 Td[(`+m+1kK. (b) Jn;k(xn;k;yn;k;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn;~qn;k:K)isstrictlyconcavein(xn;k;yn;k;~qn;k:K)for1kK)]TJ /F3 11.955 Tf 12.76 0 Td[(`+m,andJn;k(xn;k;yn;k;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn+1;~qn;k:K;~qn+1;1:K)isstrictlyconcavein(xn;k;yn;k;~qn;k:K;~qn+1;1:K)forK)]TJ /F3 11.955 Tf 11.95 0 Td[(`+m+1kK. 34

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(c) Vn;k(xn;k;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m;~fn;~qn;k:K)isstrictlyconcavein(xn;k;~qn;k:K)for1kK)]TJ /F3 11.955 Tf 12.09 0 Td[(`+m,andVn;k(xn;k;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn+1;~qn;k:K;~qn+1;1:K)isstrictlyconcavein(xn;k;~qn;k:K;~qn+1;1:K)forK)]TJ /F3 11.955 Tf 11.95 0 Td[(`+m+1kK. (d) Vn;k(xn;k;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m;~fn;~qn;k:K)increasesinxn;kfor1kK)]TJ /F3 11.955 Tf 13.33 0 Td[(`+m,andVn;k(xn;k;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m;~fn+1;~qn;k:K;~qn+1;1:K)increasesinxn;kforK)]TJ /F3 11.955 Tf 11.95 0 Td[(`+m+1kK. Proof. TheproofisgiveninAppendix A.7 . Theorem2.3. Correspondingtoeachoperatingdayn, 1nN ,thereexistauniqueoptimaltargetelectricitystoragelevelyn;kandauniqueoptimalday-aheadmarkettradingamountqn; k , correspondingtoanygiveninitialstoragelevel andanyavailablepriceinformation,foreachtimeperiodk:1kK . Proof. TheproofisgiveninAppendix A.7 . Followingthesimilarapproachasdescribedfortherisk-aversereal-timemarketonlycaseandtherisk-neutralbothmarketcase,wecanobtainasimilarconclusionforarisk-averseaggregatorparticipatinginbothmarkets.Forinstance,wehavethesamestateandactionspacesastheaboverisk-neutralcase.Wecanalsoprovetheconcavityofthemyopicexpectedutilityfunctionandthejointvaluefunctionunder thesame assumptions.Accordingly,correspondingtoeachoperatingdayn,thereexistauniqueoptimaltargetelectricitystoragelevelyn;kandauniqueoptimalday-aheadmarkettradingamountqn+1;r foreachtimeperiodk:1kK forgivenavailablepriceinformation .Weomitthedetaileddescriptionforbrevity. 2.3.4AlgorithmsforParticipatinginBoththeDay-AheadandReal-TimeMarkets Inthissubsection,wedescribeanalgorithmtosolvetheoptimalcontrolproblemforparticipatinginboththereal-timeandday-aheadmarkets.Wesolvethediscreteproblem(i.e.,discretizethestatespace)toapproximatethecontinuousproblemdescribedintheprevioussubsections.Todiscretizethestatespace,weevenlydividethestoragecharging/dischargingrangebyMIunits,thereal-timepricerangebyMPunits,andday-aheadofferamountrangebyQunits.Weassumetheday-aheadprices 35

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independentamongdifferentoperatingdaysandindependentofthereal-timepricesandthereal-timeprices are Markovian.Wealsoassumem=1withoutlossofgenerality. Fromtheanalysisdescribedintheprevioussubsection,wecanobservethatoncetheday-aheadoffersaredecided,theremainingproblemisequivalenttosolvingthereal-timeonlycase.Meanwhile,wenoticethattheday-aheadoffersaredecidedatonetimeunitforeachoperatingdays.Therefore,thewholedynamicprogrammingframeworkconsistsoftwomainparts:(1)decidingthereal-timeofferamountscorrespondingtothegivenday-aheadofferamounts(seeAlgorithm 3 ,whichsolvesCases(1)and(3)inSubsection 2.3.3 and(2)decidingtheday-aheadofferamounts(seeAlgorithm 2 ,whichsolvesCase(2)inSubsection 2.3.3 . Tosolve(1),theconcavityderivedinProposition 2.1 (b)andmonotonicityderivedinProposition 2.2 (a)canhelpspeedupthealgorithm.Morespecically,theconcavitycanhelpderiveabisectionsearchandmonotonicitycanhelpreducethesearchingspace(see,e.g.,line 5 inAlgorithm 3 ). Tosolve(2),wenoticethatforacontinuousjointlystrictlyconcavefunctionf(x1;;xN),ifthereexistsapoint(x1;:::;xN)whereallrst-orderderivativesarezeros,i.e.,@f(x1;:::;xN)=@xi=0fori=1;:::;N,thenthispointistheglobaloptimalpoint.Basedonthisoptimalityconditionandstrictconcavityofthevaluefunctionintheday-aheadofferamounts,provedinProposition 2.5 ,wecandevelopanefcientalgorithm,asdescribedinAlgorithm 2 . Nowwearereadytoexplainthewholeframework,asdescribedinAlgorithm 1 . (i) DayNandtimeperiodsK)]TJ /F3 11.955 Tf 12.38 0 Td[(`+mtoK:Becausetheday-aheadofferamountsaregivenand thereis noneedtomakeanyfurtherday-aheadoffer amount fordayN+1,weonlyneedtouseAlgorithm 3 todecidethereal-timeofferamountsforthesetimeperiods. (ii) TimeperiodsK)]TJ /F3 11.955 Tf 12.56 0 Td[(`+mondayntotimeperiodsK)]TJ /F3 11.955 Tf 12.56 0 Td[(`+m)]TJ /F4 11.955 Tf 12.56 0 Td[(1ondayn+1,forn=N)]TJ /F4 11.955 Tf 12.88 0 Td[(1to1:Followingthedenitiondescribedin( 2 ),theseperiodscorrespondtotimeperiods)]TJ /F3 11.955 Tf 9.3 0 Td[(`+mtoK)]TJ /F3 11.955 Tf 12.6 0 Td[(`+m)]TJ /F4 11.955 Tf 12.6 0 Td[(1ondayn,forn=Nto2.Forthiscase,intimeperiod)]TJ /F3 11.955 Tf 9.3 0 Td[(`+m,weneedtoderivetheoptimalday-aheadofferamountsfordayn.For each every othertimeperiod,weonlyneedtoderive 36

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theoptimalreal-timeofferamountforthisparticulartimeperiod.Algorithm 2 isdesignedtoaccomplishthis,notingthatAlgorithm 3 iscalledinAlgorithm 2 . (iii) Timeperiods1toK)]TJ /F3 11.955 Tf 9.63 0 Td[(`+m)]TJ /F4 11.955 Tf 9.63 0 Td[(1onday1.Forthiscase,theday-aheadofferamountsforday1isinitializedtobezero.WecanalsouseAlgorithm 3 toaccomplishthis,withthedifferentinitialvalueascomparedtoitem(i). Algorithm1: The WholeFramework Output: Optimalday-aheadoffersandendingstoragelevels 1 InitializeLk(q;x;y;p)asdescribedin( 2 )forq=1toQ,x;y=1toMI,andp=1toMP,withfkreplacedbyE[fk]andE[pkjpk)]TJ /F7 7.97 Tf 6.58 0 Td[(1=p]=PMPj=1jPpjfork=1toK,wherePpjisthetransitionprobabilityfrompriceptopricej 2 InitializeVN;K+1(x;p) = ln(x)forallx=1toMIandp=1toMP 3for ~qN;K)]TJ /F5 7.97 Tf 6.59 0 Td[(`+m:K=[1:Q;:::;1:Q] do 4 ObtainVN;K)]TJ /F5 7.97 Tf 6.58 0 Td[(`+m:K(~qN;K)]TJ /F5 7.97 Tf 6.59 0 Td[(`+m:K;:;:)andyN;K)]TJ /F5 7.97 Tf 6.58 0 Td[(`+m:K(~qN;K)]TJ /F5 7.97 Tf 6.58 0 Td[(`+m:K;:;:)fromAlgorithm 3 withinputs~qN;K)]TJ /F5 7.97 Tf 6.58 0 Td[(`+m:KandVN;K+1(:;:) 5for n=Nto2 do 6for ~qn;)]TJ /F5 7.97 Tf 6.59 0 Td[(`+m:0=[1:Q;:::;1:Q] do 7 Obtain ~qn;1:K(~qn;)]TJ /F5 7.97 Tf 6.59 0 Td[(`+m:0;:;:) ,Vn;)]TJ /F5 7.97 Tf 6.59 0 Td[(`+m(~qn;)]TJ /F5 7.97 Tf 6.58 0 Td[(`+m:0;:;:), Y n;)]TJ /F5 7.97 Tf 6.59 0 Td[(`+m(~qn;)]TJ /F5 7.97 Tf 6.59 0 Td[(`+m:0;:;:),Vn;)]TJ /F5 7.97 Tf 6.59 0 Td[(`+m+1:K)]TJ /F5 7.97 Tf 6.59 0 Td[(`+m)]TJ /F7 7.97 Tf 6.59 0 Td[(1(~qn;)]TJ /F5 7.97 Tf 6.59 0 Td[(`+m:0;:;:;:;:),and Y n;)]TJ /F5 7.97 Tf 6.58 0 Td[(`+m+1:K)]TJ /F5 7.97 Tf 6.59 0 Td[(`+m)]TJ /F7 7.97 Tf 6.59 0 Td[(1(~qn;)]TJ /F5 7.97 Tf 6.59 0 Td[(`+m:0;:;:;:;:)1 fromAlgorithm 2 withinputs~qn;)]TJ /F5 7.97 Tf 6.58 0 Td[(`+m:0andVn;K)]TJ /F5 7.97 Tf 6.58 0 Td[(`+m(:;:;:) 8 ObtainV1;1:K)]TJ /F5 7.97 Tf 6.58 0 Td[(`+m)]TJ /F7 7.97 Tf 6.59 0 Td[(1(:;:)and Y 1;1:K)]TJ /F5 7.97 Tf 6.59 0 Td[(`+m)]TJ /F7 7.97 Tf 6.59 0 Td[(1(:;:)fromAlgorithm 3 withzeroday-aheadofferamountsinputsandV2;)]TJ /F5 7.97 Tf 6.58 0 Td[(`+m(:;:;:) 9return ~q2:N;1:K(:;:;:) and Y 1:N;1:K(:;:::;:) 101 Inthispart,weuse(xn;)]TJ /F5 7.97 Tf 6.59 0 Td[(`+m;pn;)]TJ /F5 7.97 Tf 6.59 0 Td[(`)insteadofqn;1:KasthevariablesforCases(3)and(1)inSubsection 2.3.3 ,becausetheoptimalqn;1:Kisdecidedby(xn;)]TJ /F5 7.97 Tf 6.58 0 Td[(`+m;pn;)]TJ /F5 7.97 Tf 6.58 0 Td[(`)andqn;)]TJ /F5 7.97 Tf 6.59 0 Td[(`+m:KbasedonCase(2)inSubsection 2.3.3 . 2.3.5Discussion Thecomputationcomplexityofouralgorithmdependsonhowmanyiterationsneededtondtheoptimalday-aheadofferamountsinAlgorithm 2 .Fromourexperience,itusuallytakesnomorethan3iterationsunderourcurrentnumericalsettings,whichisspeciedinSubsection 2.4.2.1 .Inthiscase,thecomputationcomplexityisO(Q`)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1log2(Q)K2MI2MP2log2(MI)N).Fortheparkinglotofacompanywhoseworkingtimeisfrom9to16,iftheISOrequiresmarketparticipantstosubmitday-aheadoffersbefore14andm=2,thenthecomputationcomplexityisO(Qlog2(Q)K2MI2MP2log2(MI)N),whichisnotlarge.However,iftheISO 37

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Algorithm2: DeriveOptimalDay-AheadCharging/DischargingAmounts Input: Day-aheadoffers~q)]TJ /F5 7.97 Tf 6.58 0 Td[(`+m:0andVK)]TJ /F5 7.97 Tf 6.59 0 Td[(`+m(:;:;:) Output: Optimalday-aheadandreal-timeoffersandcorrespondingprots 1for x=1toMIandp=1toMP do 2 Initialize~q1:K=(0;0;:::;0)and~q01:K=()]TJ /F4 11.955 Tf 9.3 0 Td[(1;)]TJ /F4 11.955 Tf 9.3 0 Td[(1;:::;)]TJ /F4 11.955 Tf 9.3 0 Td[(1) 3while ~q01:K6=~q1:K do 4 Set ~q01:K=~q1:K and ~q01:K=~q1:K 5for k = 1 toK do 6 Usethebisectionsearchmethodtondq1k=argmaxqkJ(~q11:k)]TJ /F7 7.97 Tf 6.59 0 Td[(1;qk;~q0k+1:K;x;p),whereJ(~q11:k)]TJ /F7 7.97 Tf 6.59 0 Td[(1;qk;~q0k+1:K; x;p ) =v)]TJ /F5 7.97 Tf 6.59 0 Td[(`+m(x;p) iscomputedthroughAlgorithm 3 withinputs(~q)]TJ /F5 7.97 Tf 6.58 0 Td[(`+m:0;~q11:k)]TJ /F7 7.97 Tf 6.58 0 Td[(1;qk;~q0k+1:K)]TJ /F5 7.97 Tf 6.59 0 Td[(`+m)]TJ /F7 7.97 Tf 6.58 0 Td[(1)andVK)]TJ /F5 7.97 Tf 6.59 0 Td[(`+m(~q0K)]TJ /F5 7.97 Tf 6.59 0 Td[(`+m:K;:;:)ifk
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requiresmarketparticipantstosubmitday-aheadoffersbefore12andm=1,thenthecomputationcomplexityisO(Q4log2(Q)K2MI2MP2log2(MI)N),whichtakestimetocompute. Ingeneral,thecomputationcomplexityisO(Qrlog2(Q)K2MI2MP2log2(MI)N),whereristhenumberofthedecisionperiodsfrom24)]TJ /F3 11.955 Tf 12.14 0 Td[(`+mto24.Forexample,fortheparkinglotofacommunitywithoperationtimeofPHEVsfrom0to6,thecomputationcomplexityisO(log2(Q)K2MI2MP2log2(MI)N). 2.4FurtherInsightsExplorationandDiscussionsInthissection,wenumericallystudytheoptimalcontrolproblem.Ournumericalresultsnotonlyconrmourtheoreticalresults,butalsorevealnewinsights. 2.4.1CaseStudiesforParticipatinginReal-TimeMarketOnlyInthissection,weperformcasestudiesforthePHEVaggregatorsparticipatinginthereal-timemarket thatincluderisk-neutralandrisk-averseattitudes,and compare theseapproacheswithindividualcharging/discharging approaches andamongthemselves. Werstcomparetheoptimalpoliciesamongtheseapproachesformarketparticipants.Then,wecompareindividualandaggregatorbehaviorsandexplore theinsightsonthebenets providedbyconsideringmarketimpactinthemodelfortheaggregators,ascomparedtotheindividualcharging/discharingactivitieswithoutconsideringmarketimpact. Finally, weevaluatetheseimpacts betweenrisk-neutralandrisk-averseaggregators. 2.4.1.1OptimalPolicyComparisonWecomparerisk-neutralandrisk-averseaggregatorcasemodels,usinganindividualcontrollercaseasabenchmark. 39

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Todeveloptheprice-dependencymodel,werstassumethelengthofeachtimeperiodtobeonehour.BasedonthehistoricaldataonthePJMmarket,2foreachday,operatinghoursbetween11–21 constitute ahigh-demandregime,andthe remaininghours oftheday constitute alow-demandregime.Withineachhigh-orlow-demandregime,weassumethe exogenousreal-time pricefollowsaGeometricBrownianmotion.Forinstance,thepriceevolvesaspt+4t=pte()]TJ /F5 7.97 Tf 6.59 0 Td[(2=2)4t+W4t,whereW4tN(0;(4t)2). Inthisexperiment,wechoose4t=1todiscretizethetimeinterval .Inthehigh-demandregime,high=0:0666,andhigh=0:3327;inthelow-demandregime,low=0:1806,andlow=0:5765.Duringtheoperatinghour10,thereisanupwardjump withsize 20(i.e., p11=(p10+20)e(high)]TJ /F5 7.97 Tf 6.59 0 Td[(2high=2)+highW1 ).Duringtheoperatinghour21,thereisadownwardjump withsize 30(i.e., p22=(p21)]TJ /F4 11.955 Tf 11.95 0 Td[(30)e(low)]TJ /F5 7.97 Tf 6.59 0 Td[(2low=2)+lowW1 ).Theoperationaltimeintervalincludes10daysintotal.Inaddition,(high;high),(low;low);andthetwojumpsizesareobtainedbasedonthePJMreal-time LMP datasetduringtheweekdaysinAugust2007andAugust2008.3Accordingly,fromtheevolutionoftheprice,weknowthatW1=(log(pt+1))]TJ /F4 11.955 Tf 12.5 0 Td[(log(pt))]TJ /F4 11.955 Tf 12.49 0 Td[(()]TJ /F3 11.955 Tf 12.49 0 Td[(2=2))=N(0;1),whichisastandardnormaldistribution.Thus,fori=1;:::;MP,thetransition probability canbeapproximatedasfollows:Pij=8>>>>>>><>>>>>>>:((log(1))]TJ /F4 11.955 Tf 11.95 0 Td[(log(i))]TJ /F4 11.955 Tf 11.95 0 Td[(()]TJ /F3 11.955 Tf 11.95 0 Td[(2=2))=)j=1((log(j))]TJ /F4 11.955 Tf 11.96 0 Td[(log(i))]TJ /F4 11.955 Tf 11.96 0 Td[(()]TJ /F3 11.955 Tf 11.95 0 Td[(2=2))=))]TJ /F4 11.955 Tf 9.3 0 Td[(((log(j)]TJ /F4 11.955 Tf 11.95 0 Td[(1))]TJ /F4 11.955 Tf 11.96 0 Td[(log(i))]TJ /F4 11.955 Tf 11.96 0 Td[(()]TJ /F3 11.955 Tf 11.96 0 Td[(2=2))=)2jMP)]TJ /F4 11.955 Tf 11.95 0 Td[(11)]TJ /F4 11.955 Tf 11.96 0 Td[(((log(MP)]TJ /F4 11.955 Tf 11.96 0 Td[(1))]TJ /F4 11.955 Tf 11.95 0 Td[(log(i))]TJ /F4 11.955 Tf 11.95 0 Td[(()]TJ /F3 11.955 Tf 11.95 0 Td[(2=2))=)j=MP; (2)where(x)isthestandardnormaldistributionfunctionandMPrepresentsthemaximumreal-timeprice. 2 HistoricalPECGENReal-TimeLMP(XLS)02.18.2009 at http://www.pjm.com/markets-and-operations/energy/real-time.aspx 3http://www.pjm.com/markets-and-operations/energy/real-time.aspx 40

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Figure2-3. OptimalTrade-Down-ToLevelofElectricityStorage Inaddition,inour experiments,charginganddischargingefcienciesareallsetat0:9. The price isnormalizedtobeintheinterval (0;1]and this intervalisdividedby MPunits(MP=100inthisexperiment) todiscretizethepricevalue. Similarly theelectricitystoragecapacityisnormalizedtobe1andisdividedby MIunits(MI=101inthisexperiment) . Weassumethecontrollershavetomakethe real-timeoffer decisiononeperiodahead(i.e.,m=1) .Finally,themarketimpactforthereal-timemarketissettobe0:05 (i.e.,b=0:05) andthecoefcientofabsoluteriskaversion issettobe0:1(i.e.,=0:1). Theresultsareshownin Figure 2-3 .AsweprovedinProposition 2.2 ,theoptimaltrade-down-tolevelofelectricitystorageincreasesintheinitialelectricitystorage. 2.4.1.2MarketImpactComparisonsbetweenIndividualandAggregatorActivitiesTocomparetheinuenceofthemarketimpact sbetweenindividualandaggregatoractivities, we testtheperformancesforbothapproaches .For theindividualcase ,thereareKindividualPHEVs,andeachperformscharging/dischargingactivitiesseparately,withoutconsideringmarketimpact.WeassumethecapacityofeachPHEVisequalto0:1electricityunitunderournormalizationscheme.For theaggregatorcase ,weconsideranaggregatorwhoownsKPHEVsandperformsenergytradingwiththeconsiderationofmarketimpact. 41

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Wecanrstexploretheinsightsandperformthecomparisonforasingleperiodcase. Weassume allPHEVsarefullychargedattheverybeginning(i.e.,x1=0:1K) andthedischargingspeedlimitissufcientlylarge. For theindividualcase ,theKindividualPHEVunitswilldischargealltheirelectricitystoragetomaximizetheirprot.Althoughtheimpactofeachindividualactionisnegligible,thesumcouldnotbeignored.TheexpectedpriceaftertheiractionsisE[p1])]TJ /F4 11.955 Tf 12.75 0 Td[(0:1b1dK. Fortheaggregatorcase, ymyop(x1;~p)]TJ /F5 7.97 Tf 6.59 0 Td[(m)=maxf 0:1K )]TJ /F3 11.955 Tf 12.42 0 Td[(E[p1]=(2b1d);0g,basedonProposition 2.1 (a),makingtheexpectedpriceaftertheactionbecomemaxfE[p1])]TJ /F4 11.955 Tf 13.12 0 Td[(0:1b1dK;E[p1]=2g.ThissimpleresultshowsthatifKissufcientlylarge,theindividualactivitiesreducethereal-timepricemuch more than theaggregatoractivitydoes.Similarly,whenchargingisneeded(e.g.,forthelatermulti-periodcase),wecanobservethattheindividualchargingactivitieswillincreasethereal-timepricemuchmorethantheaggregatoractivitydoes. For a multi-periodcase(wesetN=240), followingtheobservationfromtheabovesingleperiodcaseanalysis,if therearetoomanyindividualcontrollerscrowdedinthereal-timemarket,theiractionswillincreasethereal-timepricevolatility.SupposeeachPHEVhasthesameelectricitystoragelevel,then,iftheexpectedreal-timepriceislowerthanthechargingthreshold,eachindividualcontrollerstartstoperformchargingactivities.Theiractionssuddenlyincreasetheloadsignicantlyandmakethereal-timepricesurge.Meanwhile,acontroller'sdecisionhastobemadecertainperiods beforethereal-timepriceisrealized .Therefore,thecontrollerstendtopaymoremoneythantheyhaveexpected forcharging .Similarly,whenthereal-timepriceishighenoughtocausetheexpectedreal-timepricetoexceedthedischargingthreshold,eachcontrollerstartstodischarge,expectingtoearnaprot.Situationslikethismightcausethereal-timepricetouctuateintensely.Therefore,theaggregatedcontrolwiththeconsiderationofmarketimpactcansmooththereal-timepricebetterthanindividualcontrols. 42

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Figure2-4. ComparisonofExpectedPriceafterActionwhenK=100 Nowwereportthenumericalresults.We adopt the similarsettings asdescribedinSubsection 2.4.1.1 and rstreportthereal-time price changesfortherst time period. TheresultsareshowninFigure 2-4 fortheK=100caseandFigure 2-5 fortheK=300case. Inthesetwogures,the x -axisrepresentsthereal-timepriceavailable on therstperiod,andthe y -axisrepresentstheexpectedreal-timepriceaftertherstperiod'saction. From thesegures ,itcanbeobservedthattheindividualchargingbehaviorscausetheelectricitypricesuddenlyjump-upwhenthechargingthresholdishit.Similarly,itcanbeobservedthattheindividualdischargingbehaviorsmaketheelectricitypricesuddenlyjump-downwhenthedischargingthresholdishit.Allofthesechangescauseincreasedpriceuctuationsascomparedtoonerisk-neutral/risk-averseaggregatorcases.Inaddition,asthenumberofindividualPHEVsincreases,thevolatilitycausedbyindividualPHEVsenlarges.Forinstance, comparingFigures 2-4 and 2-5 ,whenthenumberofindividualPHEVsincreasesfrom100to300,thepriceuctuationfortheindividualcharging/dischargingcasesrisessignicantly.However,theelectricitypricesdonotuctuateasmuchfortherisk-neutralandrisk-averseaggregatorcases. Forthenumericalstudyforthemulti-periodcase,wecomparetheresults byconsideringdifferentstoragecapacities,e.g.,K=100,K=300,andK=1000.For 43

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Figure2-5. ComparisonofExpectedPriceafterActionwhenK=300 eachstoragecapacitysetting,wecomparetheperformancesof different approachesbysimulatingthereal-timepricerealizationsforthese240timeperiods.Wetake1000samples(denotedass=1;::;1000).Foreachsamples,wecalculatethemeanvalueofthereal-timepricesforthese240timeperiodsanddenoteitasps=P240n=1psn=240,wherepsnrepresentsthereal-timepriceafteractionforperiod n .Finally,wereportthesamplemeanandsamplevariancevaluesforallreal-timepricesamongallscenarios,withsamplemeandenotedasp=1=1000P1000s=1psandsamplevariancedenotedasVar=P1000s=1P240n=1(psn)]TJ /F4 11.955 Tf 13.56 0 Td[(p)2=(2401000)]TJ /F4 11.955 Tf 12.57 0 Td[(1). Fromtheresults,wendthatthemoreindividualcontrollers,themoreelectricitystorageintotaltheyuse,andsothehigheraveragepriceafteraction, due toenergylossduringtheenergyconversionprocess.Withtheaggregatedoptimalcontrolbyanaggregator,theelectricitystoragecouldreducepricevolatilitysignicantly.Forexample,ascomparedtotheoriginalelectricityprices,theonerisk-neutralaggregatorcaseincreasestheaveragepriceby0:57%anddecreasesthevarianceby9:1%whenthe numberofPHEVs is100.Whenthe number increasesto300,thecorrespondingaveragepriceisincreasedby1:3%,andthevarianceisdecreasedby18:6%.Thesetwovaluesincreaseevenlargerto2:9%and31:4%whenthe numberofPHEVs reaches1000. 44

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Ingeneral,whenthetotal numberofPHEVs isnottoo large (sayK=100),boththeoneaggregatedcontrollerandindividualcontrollerapproachesreducethevariance.However,whenthenumberofPHEVsissufcientlylarge,individualcontrollerapproachcouldcausetheelectricitypricetouctuatemorerapidlyandincreasesthepricevolatility. Thereforeitmightnotbe optimaltoalloweachPHEVcontrollertocharge/dischargeindependently. 2.4.1.3ComparisonofRisk-NeutralandRisk-AverseAggregators Inthissubsection,wehighlighthowtherisk-aversefactoraffectsthesystemperformancebasedonthecomputationalresultsobtainedintheprevious subsection .First,wenoticethat basedonProposition 2.4 (a),xn)]TJ /F3 11.955 Tf 11.23 0 Td[(ymyopn(xn;)decreasesin,whichmeansthemorerisk-aversetheaggregatoris,thelessthepersonsellstothereal-timemarketinmyopiccases.Soforaone-periodcase,therisk-reverseaggregatortendstouselesselectricitystoragecomparedtotherisk-neutralcase.Foranite multi-period case,weobservethat,as the coefcientofabsoluteriskaversionincreases,theaveragepricedecreases.Itindicatesthatarisk-averseaggregatorchargesanddischargesless ,because theaveragepricewill increase ifmoreelectricitystorageanaggreagoruses,duetoenergylossintheenergyconversionprocess.Tosummarize,comparedtoindividualcontrollers,an aggregator withtheconsiderationofmarketimpacttendstouselesselectricitystorage,andtherisk-averseattitudefurtherstrengthensthistrend. 2.4.2BothDay-AheadandReal-TimeMarketsInthissubsection,weperformnumericalstudieson an aggregator'sofferingstrategies when participatinginboth theday-aheadandreal-time markets.Wealsocomparetheperformancesofparticipatinginbothmarketsversusparticipatingin the real-timemarketonly. 45

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2.4.2.1ExperimentalSettingandAlgorithms Inthissubsection,weconsidertheoperationaltimeintervaltobetwodays.Inaddition,ontherstday,theaggregatoronly submits thecharging/discharging amount offersfortheday-aheadmarket on thesecondday,and on thesecondday,theaggregator participatesinthereal-timemarket . Theoperationaltimeinterval on thesecondday contains 8hours,fromoperatinghours9to16(e.g.,operationsforaparkinglotofacompany),andaccordinglythereareN=8tradingperiodsinthiscasestudy.Charginganddischargingefcienciesarebothsettobe0:9,themarketimpactfortheday-aheadmarketissettobe0:01,andthemarketimpactforthereal-timemarketissettobe0:05.Theday-aheadandreal-timepricesarenormalizedintheinterval(0;1]andtheintervalisdiscretizedby20units (i.e.,MP=20) .Weutilizeday-aheadpricesduringtheweekdaysinthePJMday-aheadmarketinAugust2007andAugust20084tohelpgenerate theexpected day-aheadprices (i.e.,takingtheaveragevalueoftheday-aheadpricesinthegiventimeinterval).Inaddition,weassumethattheday-aheadpricesareindependentamongdifferentoperatingdaysandindependentofthereal-timeprices. Forthereal-timemarket,weassumethepriceevolvesaspt+4t=pte()]TJ /F5 7.97 Tf 6.59 0 Td[(2=2)4t+W4t,whereW4tN(0;(4t)2).Inthisexperiment,weset4t=1,=0:8239and=1:2408,wheretheparameters(;)areobtainedbasedonthePJMreal-timeelectricitypricesduringtheweekdaysinAugust2007andAugust2008.5Electricitystoragecapacityis normalized tobe1andisdiscretizedby 21 units (i.e.,MI=21) .Charginganddischargingspeedlimitsarebothsettobe veunits electricitystorage perperiod . 4 HistoricalPECGENDay-AheadLMP(XLS)02.18.2009 at http://www.pjm.com/markets-and-operations/energy/day-ahead.aspx 5 HistoricalPECGENReal-TimeLMP(XLS)02.18.2009 at http://www.pjm.com/markets-and-operations/energy/real-time.aspx 46

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Inouractionspace,fortheday-aheadmarket,toreducecomputationalburden,weconsider7possibletradingquantities,discharging0:45C,0:3C,0:15C,0,)]TJ /F4 11.955 Tf 9.3 0 Td[(0:15C,)]TJ /F4 11.955 Tf 9.3 0 Td[(0:3C,or)]TJ /F4 11.955 Tf 9.3 0 Td[(0:45C,whereCistheelectricitystoragecapacity.Meanwhile,wenoticethatforacontinuousjointly strictly concavefunctionf(x1;;xN),ifthereexistsapoint(x1;:::;xN)whereallrst-orderderivativesarezeros,i.e.,@f(x1;:::;xN)=@xi=0fori=1;:::;N,thenthispointistheglobaloptimalpoint. BasedonthisoptimalityconditionandstrictconcavityprovedinProposition 2.5 ,wecandevelopanefcientstochasticdynamicprogrammingalgorithm, asdescribedinAppendix 2.3.4 . Ouralgorithmismuchmoreefcientthantheenumerationbackwarddynamicprogrammingapproach. Toobtaintheoptimalpolicy,our algorithmnishesin3:5minuteswhiletheenumerationalgorithmusesmorethan4:5hourswhenN=4.WhenN=8,ouralgorithmnishesin14minutes,whiletheenumerationalgorithm couldnotnishwithinthetimelimit,whichissettobeoneday. 6 2.4.2.2ComparisonofParticipatinginReal-TimeMarketOnlyversusinBothMarketsInthissubsection,weshowtheadvantageofconsideringbothmarketsinsteadofparticipatinginthereal-timemarketonly.Theexperimentalsettinginthissubsectionisthesameasthatinthe previous subsection.Underthissetting,wederivetheoptimaltradingstrategiesforparticipatinginbothmarketsandparticipatinginthereal-timemarketonlycases,respectively.WealsoutilizethePJMreal-timepricesduringtheweekdaysinAugust2007and2008tocomputehowmuchprotthesestrategiesprovide.Oneachparticularday,theaggregatortradesfrom9amto5pm,thenormal 6 The experimentswerecoded inMatlabatIntelCore(TM)DuoCPU2.93GHzwith4GBmemory. 47

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Figure2-6. ProtofParticipatinginReal-TimeMarketOnlyandinBothMarkets workingtimeinmostcompanies.Theprotofparticipatingonlyinthereal-timemarketandinbothmarketsoneachdayaredescribedinFigure 2-6 .FromFigure 2-6 ,wecanobserveamong44 instanceswitheachinstancerepresentingoneparticularday ,participatinginbothmarketsgeneratesmoreprotin40 instances ,andalittlelessprotin4 instances thanparticipatinginthereal-timemarketonlydoes.In general ,theaverageprotofparticipatinginthereal-timemarketonlycaseis 0:242 ,whichismuchlowerthan that ofparticipatinginbothmarkets ,whichis 0:400 .In sum ,participatinginbothmarketsismuchmoreprotable. 2.5ConcludingRemarksInthischapter,throughconcavityanalysis,weprovedtheexistenceoftheoptimalpolicyandanalyzedtheoptimalpolicystructure.Furthermore,basedonthederivedtheoreticalresultsandnumericalstudiesperformed,wediscussedhowtocontrolPHEVsoptimally,foresawhowPHEVswouldaffectthemarket,andprovidedsuggestionsforownersofPHEVsonhowtousePHEVsbetter.Inparticular,weexploredthefollowingtwoinsights. 48

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First,throughnumericalstudies,weshowedthat,withtheconsiderationofmarketimpact,anaggregatoruseslesselectricitystorageandarisk-averseaggregatorusesevenless.Ontheotherhand,toomanyindividualactivitiesmightincreasereal-timepricevolatilityandfurthermakereal-timepriceuctuateintenselyinthecurrentmarketmechanism.Soallowingoneortwoaggregators,withtheconsiderationofmarketimpact,tocontroltheelectricitystorageisbenecialtothesystem.Second,whentheelectricitystoragecapacityissufcientlylarge,participatingonlyinthereal-timemarketis neither optimalforaggregators norfor thesystemoperator.Largeelectricitystorageownersshouldparticipateinboththeday-aheadandreal-timemarkets. 49

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CHAPTER3OPTIMALCONTROLOFBROWNIANINVENTORYWITHBATCHORDERING 3.1MotivationandLiteratureReviewIn most inventory control systems, an inventorycontroller'sorderamountmustbeanintegermultipleofaxedquantityQ.Forexample,thesuppliermightusestandardpackingsthatcontainaxednumberofunits,oraretailerordersafulltruckloadtoreceiveaquantitydiscount.In[ 72 ],Veinottrst studied thiskindofbatchorderinginventorycontrolproblem.Assumingunsatiseddemandcanbebackloggedandwithsomeassumptionsontheexpectedholdingandshortage costs ,heprovesthat a (R;Q)policyisoptimalunderdiscountedcostcriterion. In[ 79 ],ZhengandChenprovideanoptimizationalgorithmandasensitivityanalysisof(R;nQ)policy. In[ 22 ],ChengeneralizesVeinott'sresultstomulti-echelonsettingsunderthelong-runaverage cost criterionandweakensVeinott'sassumption. Recently,in[ 66 ],Shang etal. proposemechanismstocoordinateaserialsupplychainwherematerialsareshippedaccordingto(R;nQ)policy. In[ 21 ],ChaoandZhouextendChen'sworkto the multi-echelonserialsystemwithbothbatchorderingandxedreplenishmentintervals.Theyalsoprovideanefcientalgorithmtocomputetheparameters.Alltheabovepapersexploretheoptimalinventorypolicyforbatchorderingunder thediscretetimesetting . Thereare also papersthatstudythepolicyundercontinuous time setting,suchas[ 4 ],[ 13 ],and[ 69 ].However,thesepapersonlyanalyzetheperformanceof(R;Q)policywithoutprovingthe(R;Q)policyisoptimal. Inthispaper,westudythebatchorderinginventorycontrolproblemunder a continuousreviewsystem wherethe demandfollows a Brownianmotion.Instochasticcontrolarea,in[ 8 ],BatherrststudiestheBrownianinventorycontrolproblem.Heassumesthatthedemandprocessfollows aBrownianmotion andtheinventorycanbeadjustedinstantaneouslywithaxedsetupcostandproportional 50

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adjustmentcost.Underaconvexholdingandpenaltycostandthelong-runaveragecostcriterion,hesuggeststhatan(s;S)policyisoptimal.However,he didnot provetheexistenceanduniquenessofsandS.In[ 70 ],Sulemextendsthemodeltodiscountedcost cases withlinearholdingandpenalty costs .Sheprovidesconditionstoguaranteetheexistenceanduniquenessof the parameterssandS.In[ 20 ],ChaoextendsBather'smodeltodiscountedcostproblemswithandwithoutdemandbacklogging.In[ 7 ],Bar-IlanandSulem extend Bather'smodeltodiscountedcostproblemswithconstantleadtime.Theyuse a quasi-variationalinequality(QVI)approachtoexplicitlycharacterizetheoptimalpolicywhentheholdingandshortagecostsarelinear.In[ 9 ],BenkheroufextendsBather'smodeltodiscountedcostproblemswithgeneralconvexholdingandshortage costs .Recently,in[ 77 ],WuandChaostudythelong-runaveragecostproblemwithconvexholdingandshortage costs andniteproductioncapacity.Theyuseatwo-dimensionalBrownianmotionprocesstomodelthecumulativeproductionanddemand.Forthebackloggingmodel,theyprove that theoptimalpolicyisan(s;S)policy.Forthelostsalesmodel,theyshowthattheoptimalpolicyiseitheran(s;S)policyor neverentering thebusiness.We contribute tothe literatureintwoways: (i) weprovethat(R;Q)policyremainsoptimalundercontinuousreview setting whendemandfollowsaBrownianmotion,and(ii)wesolvethestochasticinventory(impulse)controlproblemwhentheordering(transaction)amountshavetobeintegertimesof a xed quantity . 3.2TheModelDescriptionandAssumptionsInthissection,wedescribeourproblemandtheassumptions.Considerasingleiteminventorycontrolproblemunderacontinuouslyreviewsetting.WeassumethattheunsatiseddemandisbackloggedandthecumulativedemandD(t)untiltimetfollowsadriftedBrownianMotion. 51

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(EvolutionofInventory)ThedemandfollowaBrownianmotion(BM),i.e., D(t)=D0t+B(t);(3)wherethedriftD0representsthedemandrateand>0representsstandardvariance,andthenoisetermB(t)isastandardBrownianmotiononthelteredprobabilityspace(;F;(Ft)t0;P).Thecontrollercanplaceanordertoincreasetheinventoryatanytime.However,theorderhastobebatchsize.(PurchaseCostsandOrderQuantity)EachorderhastobeintegraltimesofQunitsandtheunitpurchasecostcisstrictlypositive.Thereisaholdingandpenaltycosth(x)whentheinventorylevelisx.Wemakethefollowingreasonableassumptions:(HoldingCostsandPenaltyCosts)Theholdingandpenaltycostsfunctionh(x)satisesthat(a)h(x)isconvex,(b)h(x)2C2(R)exceptatonepointa,(c)h0(x)0forxaandh0(x)0forxa,(d)limx!h0(x)<)]TJ /F3 11.955 Tf 9.3 0 Td[(rcandlimx!+1h(x)=+1,and(e)h(x)ispolynomiallygrowthinx.Thecontroller'sdecisionsareasequenceofFt)]TJ /F1 11.955 Tf 9.3 0 Td[(stoppingtimes,1;2;:::,andcorrespondingFk)]TJ /F1 11.955 Tf 12.62 0 Td[(measurableintegertradingquantitiesvk,k=1;2;:::,denotedasw=((1;v1);(2;v2);:::);wherekk+1,vk=lkQ,andlk2N,k=1;2;:::.WedenotethesetofallsuchtransactiondecisionsasW.LetN(t)=maxfn:ntgbethenumberofordersplaceduntiltimetandY(t)=PN(t)n=0vnbethecumulativeamountofordersuntiltimet.DeneZ(t)=x)]TJ /F3 11.955 Tf 12.12 0 Td[(D(t)+Y(t),whereZ(0)]TJ /F4 11.955 Tf 9.3 0 Td[()=xistheinitialinventorylevel.Let=)]TJ /F3 11.955 Tf 9.3 0 Td[(D0.Fort2[n;n+1)forn=0;1;2;:::,from( 3 ),wehavetheevolutionofinventoryfollows dZ(t)=dt+dB(t):(3) 52

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. . inventory . x . Q . x+Q . Q . x+2Q . Q . x+3Q . y . (x;y)=3 Figure3-1. Illustrationof(x;y) Thestateinoursettingistheinventorylevelx.Giventhestatex,thecontroller'sobjectivefunctionisgivenbytheexpecteddiscountedcostsfromtheorderingactivityw2W,i.e., Vw(x):=Ex241Z0e)]TJ /F5 7.97 Tf 6.58 0 Td[(rth(Z(t))dt+1Xk=1e)]TJ /F5 7.97 Tf 6.58 0 Td[(rkcvk35;(3)whereristhediscountrate.Thedecisionproblemistondw2WtominimizeJw(x).Thatis,wewanttosolve V(x)=infw2WVw(x):(3) 3.3MainTheoremandExistenceofParametersInthissection,westatetheoptimalvaluefunctionandthecontrolpolicyoftheinventorycontrolproblemdescribedinthelastsection. Dene ( x;y )=mink2 Z+ :x+kQ>y;(3) whereZ+representsthesetofnon-negativeintegers . ( x;y )represents theminimumintegeral times ofQneededtoincreaseinventoryfromxtostrictlyabovey.AnexampleisgiveninFig 3-1 .Lettheexpecteddiscountedholdingcostwithoutordering H(x)=EZ10e)]TJ /F5 7.97 Tf 6.59 0 Td[(rth(x)]TJ /F3 11.955 Tf 11.96 0 Td[(D(t))dt;(3)wherexistheinitialinventorylevel. Theorem3.1. Theoptimalvaluefunctionis 53

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V(x)=infw2WVw(x)=8><>:H(x+( x;R )Q))]TJ /F3 11.955 Tf 11.96 0 Td[(Ke)]TJ /F5 7.97 Tf 6.59 0 Td[(2(x+( x;R )Q)+c( x;R )QifxR;H(x))]TJ /F3 11.955 Tf 11.96 0 Td[(Ke)]TJ /F5 7.97 Tf 6.59 0 Td[(2xifx>R;(3)where2=h+p 2+2r2i=2>0,RistheuniquepointsuchthathH0(x+Q))]TJ /F3 11.955 Tf 11.96 0 Td[(H0(x)i=2+H(x+Q))]TJ /F3 11.955 Tf 11.95 0 Td[(H(x)+cQ=0;K=[H(R+Q))]TJ /F3 11.955 Tf 11.96 0 Td[(H(R)+c]=e)]TJ /F5 7.97 Tf 6.58 0 Td[(2(R+Q))]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2R>0; H(x)=e1xZ1xh(y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1ydy+e)]TJ /F5 7.97 Tf 6.59 0 Td[(2xZxh(y)e2ydy=p 2+2r2;(3) and1=h)]TJ /F3 11.955 Tf 9.3 0 Td[(+p 2+2r2i=2. Accordingly,theoptimalorderstrategyisathresholdorderpolicy.Morespecically,thereexistsathresholdRandwhentheinventorylevelxisnohigherthanR,weorderthemiminumintegral times ofQamountssuchthattheinventoryincreases to strictlyaboveR,i.e.,theoptimalorderquantityis( x;R )Q.Fig 3-2 explainsthe(R;Q)policy.Intheremainingpartofthispaper,weprovethefunctiondescribedin( 3 )istheoptimalvaluefunction.WerstprovidelowerboundtheoremsinSection 3.4 .Thenweprovethefunctiondescribedin( 3 )satisestheconsitionsoflowerboundtheoremsandsoistheminimumcostfunctioninSections 3.5 and 3.6 . 3.4LowerBoundTheoremsThissectionprovidesamethodfordeterminingtheoptimalsolution.ThesetwolowerboundtheoremsrequiretheknowledgeofV(x)andthecontinuationregion,i.e.,theregionwherethecontrollerdoesnotorder.Formanyproblems,however,thesecanbeguessed,andthenthelowerboundtheoremsareinvoked. Denetheoperatorforthestochasticdifferentialequation( 3 )as 54

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. . . . . 0 . 0:2 . 0:4 . 0:6 . 0:8 . 1 . 1:2 . 1:4 . 1:6 . 1:8 . 2 . )]TJ /F4 11.955 Tf 9.3 0 Td[(2 . 0 . 2 . R . R+Q . t . Inventory . (R;Q)policy Figure3-2. Illustrationof(R;Q)policy )]TJ /F3 11.955 Tf 7.32 0 Td[(V(x):=V0(x)+1 22V00(x):(3)Westateageneralized Ito's lemma. Lemma3.1. (ExtendedIto'sformula) SupposethatV(x)2C1(R)andV00(x)2C(R)exceptforcountablepoints.Wehavee)]TJ /F5 7.97 Tf 6.59 0 Td[(rTV(Z(T))=V(Z(0)]TJ /F4 11.955 Tf 9.3 0 Td[())+ZT0e)]TJ /F5 7.97 Tf 6.59 0 Td[(rtAV(Z(t))dt+ZT0e)]TJ /F5 7.97 Tf 6.59 0 Td[(rtV0(Z(t))dB(t) (3)+N(T)Xn=1e)]TJ /F5 7.97 Tf 6.59 0 Td[(rn[V(Z(n)))]TJ /F3 11.955 Tf 11.96 0 Td[(V(Z(n)]TJ /F4 11.955 Tf 9.3 0 Td[())];where AV(x):=)]TJ /F3 11.955 Tf 9.3 0 Td[(rV(x)+)]TJ /F3 11.955 Tf 26.29 0 Td[(V(x): (3) Proof. TheproofisgiveninAppendix B.1 . 55

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Furthermore,fornotationbrevityandconvinenceofofanalyzingourgenerallowerboundandboundachievedtheorems,wedene theswitchoperator LV(x)= inf v2fV(x+v)+(x;v)g;(3)whereisthefeasibleorderregionand(x;v)istheorderingcostwhentheinventoryisxandtheorderquantityisv. Theorem3.2. (Lowerboundforoptimalvaluefunction)Fornon-negativecostfunctitonsh(x)and(x;v),supposethatthereexistsaV(x)suchthat 1. V(x)2C1(R)andV00(x)2C(R)exceptforcountablepoints, 2. V(x)LV(x)forallx2R, 3. AV(x)+h(x)0foralmostallx2R, 4. ExR10e)]TJ /F5 7.97 Tf 6.59 0 Td[(rth(x)]TJ /F3 11.955 Tf 11.96 0 Td[(D(t))dt<1foranyx2Randlimx!V(x)=h(x)<1.Then,V(x)infw2WVw(x),whereVw(x)=EZ10e)]TJ /F5 7.97 Tf 6.59 0 Td[(rth(Z(t))dt+E"1Xi=1)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Z)]TJ /F3 11.955 Tf 5.48 -9.68 Td[()]TJ /F5 7.97 Tf -1.34 -8.02 Td[(i;vi#istheobjectivefunctioncorrespondingtoanyfeasiblepolicyw=((1;v1);(2;v2);:::)2W. Proof. TheproofisgiveninAppendix B.2 . Theorem3.3. (Lowerboundachieved)SupposeV(x)satisestheconditionsofTheorem 3.2 andsuchthat 5. AV(x)+h(x)=0forx2C:=fx:V(x)^k)]TJ /F7 7.97 Tf 6.59 0 Td[(1: Z(t)=2C g,and ^qk isdenedsuchthat LV(Z(^-k))=V(Z(^-k)+ ^qk )+(Z(^-k); ^qk ) if ^k <1and ^qk :=0if ^k =1,fork=1;2;:::. 56

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ThenV(x)=V ^w (x)=infw2WVw(x) and^w:=f^1;^2;:::;^q1;^q2;:::gisanoptimalorderingpolicy. Proof. TheproofisgiveninAppendix B.3 . Thelowerboundtheoremsweprovidedhereareforgeneral impulsecontrol problems.In ourproblemsetting ,thefeasibleorderregionis:=fv>0:v=lQ;l2Ng, accordingly theswitchoperator asdescribedin( 3 )isspeciedas LV(x)= inf l2Z+ fV(x+lQ)+clQg;(3)andtheorderingcostis (x;v)=cv;(3)wherexistheinventorylevelandvistheorderquantity. Inremainingpart,weconsider( 3 )and( 3 )forourproblemsetting. Inaddition,notethat x2RnC,x2fx:V(x)=LV(x)g (3) basedonCondition(ii)andthedenitionofCinCondition(v) .ThesetC:=fx:V(x)Rg. Inthesetwotheorems 3.2 and 3.3 ,basedonthedenitionofLV,thisactuallyimplysnoorderregionforourproblem. 3.5OptimalPolicyParametersInthissection,weshowthecontinueandsmoothnessofV(x)describedin( 3 ),asrequiredinTheorem 3.2 ,canuniquelydecidetheparametersRandK. 57

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First,from( 3 ),wenotethatV(x)iscontinueandsmoothinxifandonlyifV(x)iscontinueandsmoothatx=R)]TJ /F3 11.955 Tf 12.03 0 Td[(mQ,(i.e.,V(x)iscontinuousatx=R)]TJ /F3 11.955 Tf 12.03 0 Td[(mQanditsleftandrightderivativesareequalatx=R)]TJ /F3 11.955 Tf 11.95 0 Td[(mQ),form=0;1;:::.Second,weknowthat ( R)]TJ /F3 11.955 Tf 11.95 0 Td[(mQ;R )=(m+1)form=1;2;::::(3)basedonthedenitionof( x;y )in( 3 ).From( 3 )and( 3 ),wehavethatV(x)iscontinueandsmoothinxifandonlyifH(R)]TJ /F3 11.955 Tf 11.96 0 Td[(mQ+(m+1)Q))]TJ /F3 11.955 Tf 11.95 0 Td[(Ke)]TJ /F5 7.97 Tf 6.59 0 Td[(2(R)]TJ /F5 7.97 Tf 6.59 0 Td[(mQ+(m+1)Q)+c(m+1)Q=H(R)]TJ /F3 11.955 Tf 11.96 0 Td[(mQ+mQ))]TJ /F3 11.955 Tf 11.95 0 Td[(Ke)]TJ /F5 7.97 Tf 6.58 0 Td[(2(R)]TJ /F5 7.97 Tf 6.59 0 Td[(mQ+mQ)+cmQ;whichisequivalentto H(R+Q))]TJ /F3 11.955 Tf 11.96 0 Td[(Ke)]TJ /F5 7.97 Tf 6.59 0 Td[(2(R+Q)+cQ=H(R))]TJ /F3 11.955 Tf 11.95 0 Td[(Ke)]TJ /F5 7.97 Tf 6.58 0 Td[(2R;(3)andH0(R)]TJ /F3 11.955 Tf 11.95 0 Td[(mQ+(m+1)Q)+K2e)]TJ /F5 7.97 Tf 6.59 0 Td[(2(R)]TJ /F5 7.97 Tf 6.59 0 Td[(mQ+(m+1)Q)=H0(R)]TJ /F3 11.955 Tf 11.95 0 Td[(mQ+mQ)+K2e)]TJ /F5 7.97 Tf 6.59 0 Td[(2(R)]TJ /F5 7.97 Tf 6.59 0 Td[(mQ+mQ);whichisequivalentto H0(R+Q)+K2e)]TJ /F5 7.97 Tf 6.58 0 Td[(2(R+Q)=H0(R)+K2e)]TJ /F5 7.97 Tf 6.59 0 Td[(2R:(3)Third,byadding( 3 )times2and( 3 ),we havethefollowingequation: H0(R)=2+H(R)+cR=H0(R+Q)=2+H(R+Q)+c(R+Q):(3) 58

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Now ,weshowthattheequationH0(x)=2+H(x)+cx=H0(x+Q)=2+H(x+Q)+c(x+Q)hasauniquesolution .AndthenRshouldbethecorrespondingsolution . Toprovethis,welet f(x):=H0(x)=2+H(x)+cx:(3) Weonlyneedtoshow f(x)=f(x+Q)hasauniquesolution.Before proving this,werstprovethatH(x)denedin( 3 )has an explicitexpression asshowninthefollowingLemma. Lemma3.2. FortheexpectedholdingcostfunctionH(x)denedin( 3 ), asshownin( 3 ) ,wehaveH(x)=e1xZ1xh(y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1ydy+e)]TJ /F5 7.97 Tf 6.59 0 Td[(2xZxh(y)e2ydy=p 2+2r2; andH00(x)>0:(3) Proof. TheproofisgiveninAppendix B.5 . Proposition3.1. ThereexistsauniqueRsuchthat f(x))]TJ /F3 11.955 Tf 11.96 0 Td[(f(x+Q)8>>>><>>>>:>0forxR(3) Proof. TheproofisgiveninAppendix B.6 . Finally, byreorganizingtermsof ( 3 ),wehave K=hH0(R+Q))]TJ /F3 11.955 Tf 11.96 0 Td[(H0(R)i=2)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2R)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2(R+Q)>0;(3)wheretheinequalityholdsbecauseofH00(x)>0basedon( 3 ). 59

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3.6VericationofOptimalValueFunctionandControlPolicyInthissection,weverifythatourguessedvaluefunctionandorderpolicyareoptimalbyprovingtheysatisfytheconditionsinTheorems 3.2 and 3.3 .SinceCondition(i)ofTheorem 3.2 hasbeensatisedbythechoiceofRandKprovedinSection 3.5 ,inordertoshowthatthevaluefunctionandcontrolpolicyareoptimal,itsufcestoverifythatthefollowingconditionsaresatised:(ii)V(x)=LV(x)forxR andLV(x)=V(x+(x;R)Q)+c(x;R)Q ,(iii)V(x)R,(iv)AV(x)+h(x)0forxRandx6=R)]TJ /F3 11.955 Tf 12.43 0 Td[(lQforl=1;2;:::,(v)AV(x)+h(x)=0forx>R,and(vi)limx!V(x)=h(x)<1. NoteherethatweactuallyguessC:=fx:V(x)RgasdiscussedinRemark 3.4 .IfweverifyConditions(ii)and(iii),weverifyConditions2and6inTheorems 3.2 and 3.3 . Inaddition,Conditions(iv)and(v)guaranteeConditions3and5,andCondition(vi)guaranteesCondition4inTheorems 3.2 and 3.3 . Beforewestart,weprovesomepreliminaryresults. Lemma3.3. Forg(x;l) denedas g(x;l)=H(x+lQ))]TJ /F3 11.955 Tf 11.96 0 Td[(Ke)]TJ /F5 7.97 Tf 6.59 0 Td[(2(x+lQ)+lcQ;(3)whereH(x)isdenedin( 3 ) andl2Z+ ,wehave g(x;l+1)>g(x;l)forl( x;R );(3)whereRisdenedinLemma 3.1 and( x;y )isdenedin( 3 ). Proof. TheproofisgiveninAppendix B.7 . NowweverifyCondition(ii)in thefollowingp roposition. Proposition3.2. ForthefunctionV(x)denedin( 3 )andtheoperatorLdenedin( 3 ),wehaveLV(x) =V(x+(x;R)Q)+c(x;R)Q =V(x)forxR,whereRisdecidedin ( 3 ). 60

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Proof. NoticethatforxR,basedon( 3 ),wecanletLV(x)= inf fL1V(x);L2V(x)g;where L1V(x)=inf l2Z+:l>0; l<( x;R )fV(x+lQ)+clQg(3) representstheminimumvaluebyorderingthequantityafterwhichtheinventorystillfallsintheorderregion, and L2V(x)=inf l2Z+: l( x;R )fV(x+lQ)+clQg(3) representstheminimumvaluebyorderingthequantityafterwhichtheinventoryfallsintheno-order(continuation)region. WewillproveLV(x)=V(x)byproving L1V(x)=V(x)forxR(3)and L2V(x)=V(x)forxR:(3)Notethat basedon ( 3 )and( 3 ),wehaveV(x+lQ)=8>>>><>>>>:H(x+lQ))]TJ /F3 11.955 Tf 11.95 0 Td[(Ke)]TJ /F5 7.97 Tf 6.59 0 Td[(2(x+lQ)ifl( x;R )H(x+lQ+( x+lQ;R )))]TJ /F3 11.955 Tf 11.95 0 Td[(Ke)]TJ /F5 7.97 Tf 6.58 0 Td[(2(x+lQ+( x+lQ;R ))+c( x+lQ;R )Qifl<( x;R ):ForxRandl<( x;R ),wehavex+( x;R )Q=x+lQ+( x+lQ;R )Qbasedonthedenitionof( x;y )in( 3 ).Sotheaboveequationisequivalentto 61

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V(x+lQ)=8><>:H(x+lQ))]TJ /F3 11.955 Tf 11.95 0 Td[(Ke)]TJ /F5 7.97 Tf 6.59 0 Td[(2(x+lQ)ifl( x;R )H(x+( x;R ) Q ))]TJ /F3 11.955 Tf 11.96 0 Td[(Ke)]TJ /F5 7.97 Tf 6.58 0 Td[(2(x+( x;R ) Q )+c(( x;R ))]TJ /F3 11.955 Tf 11.96 0 Td[(l)Qifl<( x;R ):(3) Nowweprove( 3 ). ForxR,wehave L1V(x)=infl<( x;R )fV(x+lQ)+clQg=infl<( x;R )H(x+( x;R ) Q ))]TJ /F3 11.955 Tf 11.95 0 Td[(Ke)]TJ /F5 7.97 Tf 6.58 0 Td[(2(x+( x;R ) Q )+c(( x;R ))]TJ /F3 11.955 Tf 11.95 0 Td[(l)Q+clQ=H(x+( x;R ) Q ))]TJ /F3 11.955 Tf 11.95 0 Td[(Ke)]TJ /F5 7.97 Tf 6.58 0 Td[(2(x+( x;R ) Q )+c( x;R )Q=V(x);(3)wheretherstequalityholdsbecauseof( 3 ),thesecondequalityholdsbecauseof( 3 ),andthelastequalityholdsbecauseof( 3 ). Nowweprove( 3 ). ForxR,wehave L2V(x)=infl( x;R )fV(x+lQ)+clQg=infl( x;R )H(x+lQ))]TJ /F3 11.955 Tf 11.96 0 Td[(Ke)]TJ /F5 7.97 Tf 6.59 0 Td[(2(x+lQ)+clQ=infl( x;R )g(x;l):=g(x;( x;R ))=H(x+( x;R )))]TJ /F3 11.955 Tf 11.95 0 Td[(Ke)]TJ /F5 7.97 Tf 6.59 0 Td[(2(x+( x;R ))+c( x;R )Q=V(x);(3)wheretherstequalityholdsbecauseof( 3 ),thesecondequalityholdsbecauseof( 3 ),thethirdequalityholdsbecauseof( 3 ),thefourthequalityholdsbecauseof( 3 ),thefthequalityholdsbecauseof( 3 )and( 3 ),andthelastequalityholdsbecauseof( 3 ).Thus,basedon( 3 ),( 3 ),and( 3 ),wehave L V(x)=inffL1V(x);L2V(x)g=V(x) andLV(x)=V(x+(x;R)Q)+c(x;R)Q forxR. Next,weproveCondition(iii)issatisedinProposition 3.3 . 62

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Proposition3.3. ForthefunctionV(x)denedin( 3 )andtheoperatorLdenedin( 3 ),wehaveLV(x)>V(x)forx>R,whereRisdecidedinLemma 3.1 . Proof. Forx>R,wehaveLV(x)=inf l2Z+ fV(x+lQ)+clQg=inf l2Z+ H(x+lQ))]TJ /F3 11.955 Tf 11.96 0 Td[(Ke)]TJ /F5 7.97 Tf 6.59 0 Td[(2(x+lQ)+clQ=inf l2Z+ g(x;l)=g(x;1)>g(x;0)=H(x))]TJ /F3 11.955 Tf 11.96 0 Td[(Ke)]TJ /F5 7.97 Tf 6.59 0 Td[(2x=V(x);wheretherstequalityholdsbecauseof ( 3 ) ,thesecondequalityholdsbecauseof( 3 ),thethirdequalityholdsbecauseof( 3 ),thefourthequalityandthe following inequalityholdbecauseof( 3 ),thefthequalityholdsbecauseof( 3 ),andthelastequalityholdsbecauseof( 3 ). Furthermore,weproveCondition(vi)issatiedinthefollowingproposition. Proposition3.4. ForthefunctionV(x)denedin( 3 )andtheoperatorAdenedin( 3 ),wehaveAV(x)+h(x)0forxRandx6=R)]TJ /F3 11.955 Tf 12.19 0 Td[(lQforl=1;2;:::,whereRisdecidedinLemma 3.1 . Proof. Werstprovethefollowingclaimthatwewilluselater: )]TJ /F3 11.955 Tf 11.96 0 Td[(r[H(x+lQ))]TJ /F3 11.955 Tf 11.95 0 Td[(H(x)])]TJ /F3 11.955 Tf 11.96 0 Td[(rclQrhH0(x+Q))]TJ /F3 11.955 Tf 11.96 0 Td[(H0(x)i=2forxR:(3)Theaboveinequalityis equivalent toH0(x+lQ))]TJ /F3 11.955 Tf 11.95 0 Td[(H0(x)=2+[H(x+lQ))]TJ /F3 11.955 Tf 11.95 0 Td[(H(x)]+clQ0forxR bydividingrfrombothsidesandreorganizingterms ,whichis 63

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equilalenttof(x+ l Q))]TJ /F3 11.955 Tf 11.98 0 Td[(f(x)0forxR,whichholdsbasedon( B ) inLemma B.5 inAppendix B.7 andthefactthate)]TJ /F5 7.97 Tf 6.59 0 Td[(2x)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2Q>0 . Nowweprovetheclaimholds. Forx2(R)]TJ /F3 11.955 Tf 11.95 0 Td[(lQ;R)]TJ /F4 11.955 Tf 11.95 0 Td[((l)]TJ /F4 11.955 Tf 11.96 0 Td[(1)Q)for each l=1;2;:::,AV(x)+h(x) =A)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(H(x+(x;R)Q))]TJ /F3 11.955 Tf 11.96 0 Td[(Ke)]TJ /F5 7.97 Tf 6.59 0 Td[(2(x+(x;R)Q)+c(x;R)Q+h(x) =A)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(H(x+lQ))]TJ /F3 11.955 Tf 11.96 0 Td[(Ke)]TJ /F5 7.97 Tf 6.58 0 Td[(2(x+lQ)+clQ+h(x) =A)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(H(x+lQ))]TJ /F3 11.955 Tf 11.96 0 Td[(Ke)]TJ /F5 7.97 Tf 6.58 0 Td[(2(x+lQ)+clQ)-222(AH(x)=A(H(x+lQ))]TJ /F3 11.955 Tf 11.96 0 Td[(H(x))+A(clQ)=)]TJ /F3 11.955 Tf 9.3 0 Td[(r[H(x+lQ))]TJ /F3 11.955 Tf 11.95 0 Td[(H(x)]+H0(x+lQ))]TJ /F3 11.955 Tf 11.96 0 Td[(H0(x)+(2=2)H00(x+lQ))]TJ /F3 11.955 Tf 11.95 0 Td[(H00(x))]TJ /F3 11.955 Tf 11.95 0 Td[(rclQrH0(x+lQ))]TJ /F3 11.955 Tf 11.96 0 Td[(H0(x)=2+H0(x+lQ))]TJ /F3 11.955 Tf 11.96 0 Td[(H0(x)+(2=2)H00(x+lQ))]TJ /F3 11.955 Tf 11.95 0 Td[(H00(x) =(r=2+)H0(x+lQ))]TJ /F3 11.955 Tf 11.95 0 Td[(H0(x)+(2=2)H00(x+lQ))]TJ /F3 11.955 Tf 11.95 0 Td[(H00(x) =(2=2)2H0(x+Q))]TJ /F3 11.955 Tf 11.95 0 Td[(H0(x)+(2=2)H00(x+lQ))]TJ /F3 11.955 Tf 11.95 0 Td[(H00(x) = (2=2)2 f0(x+lQ))]TJ /F3 11.955 Tf 11.96 0 Td[(f0(x)0;wheretherstequalityholdsbecauseof( 3 ), thesecondequalityholdsbecause( x;R )=lforx2(R)]TJ /F3 11.955 Tf 12.41 0 Td[(lQ;R)]TJ /F4 11.955 Tf 12.41 0 Td[((l)]TJ /F4 11.955 Tf 12.41 0 Td[(1)Q],thethird equalityholdsbecauseAH(x)+h(x)=0basedonLemma B.2 inAppendix B.4 ,the fourth equalityholdsbecauseofLemma B.3 ,the fth equalityholdsbecauseof ( 3 )and ( 3 ),therstinequalityholdsbecauseof( 3 ), theseventhequalityholdsfollowedbythefactthatr=2+=(2=2)2,whichfollowsfrom)]TJ /F3 11.955 Tf 9.3 0 Td[(r+()]TJ /F3 11.955 Tf 9.3 0 Td[(2)+(2=2)22=0basedonLemma B.1 ,thelastequalityholdsbecauseof( 3 ),andthelastinequalityholdsbecausef0(x+lQ))]TJ /F3 11.955 Tf 12.14 0 Td[(f0(x)=f00()lQ0forsome2[x;x+lQ]duetof00(:)0asprovedin( B ). Finally,weproveConditions(v)and(vi)aresatiedinthefollowingproposition. 64

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Proposition3.5. ForthefunctionV(x)denedin( 3 )andtheoperatorAdenedin( 3 ),wehaveAV(x)+h(x)=0forx>R,whereRisdecidedinLemma 3.1 .Wealsohavelimx!V(x)=h(x)<1. Proof. From( 3 )inourguessedprocess,wehaveV(x)=H(x))]TJ /F3 11.955 Tf 12.32 0 Td[(Ke)]TJ /F5 7.97 Tf 6.59 0 Td[(2xforx>R.FromLemma B.3 inAppendix B.4 ,wehaveA)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Ke)]TJ /F5 7.97 Tf 6.58 0 Td[(2x=0.FromLemma B.2 inAppendix B.4 ,wehave AH(x) +h(x)=0.Sowehave AV(x)+h(x) = AH(x))-222(A)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Ke)]TJ /F5 7.97 Tf 6.58 0 Td[(2x+h(x) = AH(x)+h(x) =0:Next,weprove(vi)limx!V(x)=h(x)<1 separately.Forthecasex! ,wehavelimx!V(x)=h(x)=limx!H(x+( x;R )Q))]TJ /F3 11.955 Tf 11.95 0 Td[(Ke)]TJ /F5 7.97 Tf 6.58 0 Td[(2(x+( x;R )Q)+c( x;R )Q=h(x)=limx!c( x;R )Q=h(x)=limx!c()]TJ /F3 11.955 Tf 9.3 0 Td[(x)=h(x)=limx!)]TJ /F3 11.955 Tf 9.29 0 Td[(c=h0(x)2[0;1=r];wheretherstequalityholdsbecauseof( 3 ),thesecondequalityholdsbecauselimx!h(x)=1basedon(d)ofAssumption 3.2 andH(x+( x;R )Q))]TJ /F3 11.955 Tf 9.3 0 Td[(Ke)]TJ /F5 7.97 Tf 6.59 0 Td[(2(x+( x;R )Q)haslowerboundminH(y))]TJ /F3 11.955 Tf 11.95 0 Td[(Ke)]TJ /F5 7.97 Tf 6.59 0 Td[(2y:y2[R;R+Q]andupperboundmaxH(y))]TJ /F3 11.955 Tf 11.95 0 Td[(Ke)]TJ /F5 7.97 Tf 6.58 0 Td[(2y:y2[R;R+Q]basedonthedenitionof( x;R )in( 3 ),thethirdequalityholdsbecauseofthedenitionof( x;R )in( 3 ),thelastpartholdsbecauselimx!h0(x)<)]TJ /F3 11.955 Tf 9.3 0 Td[(rcbasedon(d)ofAssumption 3.2 . 65

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Forthecasex!+1,basedon (e)ofAssumption 3.2 ,( 3 )and( 3 ),wehavelimx!+1V(x)=h(x)=limx!+1H(x)=h(x)<1. 3.7ConcludingRemarksInthischapter,wediscussthestochasticinventorycontrolproblemwithBrowniandemandandbatchordering.Weprovideclosedformexpressionforthevaluefunctionandweprovethattheoptimalpolicyisa(R;Q)policy. 66

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CHAPTER4BROWNIANINVENTORYCONTROLWITHBATCHCONTROLQUANTITIES 4.1MotivationandLiteratureReviewInthischapter,westudytheBrownianinventorycontrolproblem where theinventorycanbeincreased or decreasedinbatchsizes.Therearepapers discussing theoptimalcontrolpolicies where theinventorycanbecontrolledbothupwardanddownward. Forinstance,in [ 63 ],Richardstudiestheimpulsecontrolproblemofageneralone-dimensionalhomogeneousdiffusionprocess.Heconsidersboth the innitehorizon problem withdiscountedcostobjectivesand the nitehorizonproblems.Basedontheassumptionoftheexistenceofsolutionstoagroupofinequalities,heprovidessufcientconditionsfortheoptimalvaluefunctionandoptimalpolicy.In[ 24 ],ConstantinidesandRichardstudythediscountedcostcashmanagementproblemwithlinearholdingandpenaltycosts.Theyprovethattheoptimalpolicyisa(d;D;U;u)policy.Underthe lostsales settingwheretheinventoryisrestrictedtobenonnegative,Harrisonetal.[ 32 ]discussthediscountedcostproblemwithlinearholdingcostandprovethata(0;D;U;u)policyisoptimal.Withthesamesetting,Ormecietal.[ 59 ]explorethelong-runaveragecostproblemandshowthata(0;D;U;u)policyremainsoptimal.Theyalsogeneralizetheresulttothecasewherebothpossibleadjustmentquantityandinventorycapacityhavelimits.Backto the backloggingsetting,DaiandYao[ 25 , 26 ]solvetheproblemwithgeneralconvexholdingandpenalty costs underlong-runaveragecostcriterionanddiscountedcostcriterionrespectively.Theyprovethat(d;D;U;u)policiesareoptimalforbothcases.To thebestof ourknowledge,thereisno existingworksolving theBrownianinventorycontrolproblemwhenthetransactionamountshavetobeintegertimesof a xed quantity andtheinventorycanbeadjustedbothupwardanddownward.Thischapter tries toll in thisgap. 67

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4.2TheModelDescriptionandAssumptionsInthissection,wedescribeourproblemandtheassumptions.Considerasingleiteminventorycontrolproblemunderacontinuouslyreviewsetting.WeassumethattheunsatiseddemandisbackloggedandthecumulativedemandD(t)untiltimetfollowsadriftedBrownianMotion.(EvolutionofInventory)ThedemandfollowaBrownianmotion(BM),i.e., D(t)= d0 t+B(t);(4)wherethedrift d0 representsthedemandrateand>0representsstandardvariance,andthenoisetermB(t)isastandardBrownianmotiononthelteredprobabilityspace(;F;(Ft)t0;P).Thecontrollercanplaceanordertoincreaseordecreasetheinventoryatanytime.However,theorderhastobebatchsize.(PurchaseCostsandOrderQuantity)EachorderhastobeintegraltimesofQunitswithQ>0andtheunitcostofincreasinginventoryis cu >0andtheunitcostofdecreasinginventoryis cd >0.Thereisaholdingandpenaltycosth(x)whentheinventorylevelisx.Wemakethefollowingreasonableassumptions:(PenaltyCosts)Theholdingandpenaltycostsfunctionh(x)satisesthat(a)h(x)isconvex,(b)h(x)2C2(R)exceptatonepointa,(c) h0 (x)<0forx0forx>a,(d)limx! h0 (x)<)]TJ /F3 11.955 Tf 9.3 0 Td[(r cu , limx!+1 h0 (x)>rcd ,andlimx!h(x)=+1,and(e)h(x)ispolynomiallygrowthinx.Thecontroller'sdecisionsareasequenceofFt)]TJ /F1 11.955 Tf 9.3 0 Td[(stoppingtimes,1;2;:::,andcorrespondingFk)]TJ /F1 11.955 Tf 12.62 0 Td[(measurableintegertradingquantitiesvk,k=1;2;:::,denotedasw=((1;v1);(2;v2);:::);wherekk+1,vk=lkQ,andlk2 Znf0g ,k=1;2;:::.WedenotethesetofallsuchtransactiondecisionsasW.LetN(t)=maxfn:ntgbethenumberofcontrolsuntiltimetand 68

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Yu(t)=PN(t)n=0[vn]+ bethecumulativeincreasedamountuntiltimet, Yd(t)=PN(t)n=0[vn])]TJ /F1 11.955 Tf -451.02 -29.72 Td[(bethecumulativedecreasedamountuntiltimet .DeneZ(t)=X(t)+ Yu(t))]TJ /F3 11.955 Tf 11.95 0 Td[(Yd(t) ,whereZ(0)]TJ /F4 11.955 Tf 9.29 0 Td[()=xistheinitialstate.Let=)]TJ /F3 11.955 Tf 9.3 0 Td[(d0.Fort2[n;n+1)forn=0;1;2;:::,from( 4 ),wehavetheevolutionofinventoryfollows dZ(t)=dt+dB(t):(4)Thestateinoursettingisinventorylevelx.Giventhestatex,thecontroller'sobjectivefunctionisgivenbytheexpecteddiscountedcostsfromtheorderingactivityw2W,i.e., Jw(x):=Ex241Z0e)]TJ /F5 7.97 Tf 6.59 0 Td[(rth(Z(t))dt+1Xk=1e)]TJ /F5 7.97 Tf 6.59 0 Td[(rkcu[vk]++1Xk=1e)]TJ /F5 7.97 Tf 6.59 0 Td[(rkcd[vk])]TJ /F9 11.955 Tf 7.09 18.22 Td[(35;(4)whereristhediscountrate.Thedecisionproblemistondw2WtominimizeJw(x).Thatis,wewanttosolve V(x)=infw2WJw(x):(4) 4.3MainTheoremInthissection,westatetheoptimalvaluefunctionandthecontrolpolicyoftheinventorycontrolproblemdescribedinthelastsection.Dene u(y;x)=mink2Z+:x+kQ>y:(4) d(y;x)=mink2Z+:x+kQ
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. . inventory . x . Q . x+Q . Q . x+2Q . Q . x+3Q . y . u(y;x)=3 Figure4-1. Illustrationofu(y;x) Lettheexpectedholdingcostwithoutordering w(x)=ExZ10e)]TJ /F5 7.97 Tf 6.58 0 Td[(rth(x)]TJ /F3 11.955 Tf 11.95 0 Td[(D(t))dt;(4)wherexistheinitialinventorylevel. Theorem4.1. Theoptimalvaluefunctionis V(x)=infw2WVw(x)=8>>>><>>>>:(x+u(S;x)Q;A;B)+cuu(S;x)QforxS(x;A;B)forS0,2=h+p 2+2r2i=2>0,S,D,A,andBaredecidedinTheorem 4.4 .Accordingly,theoptimalorderstrategyistowaitwhentheinventorylevelxisbetweenSandD,toincreaseinventorysuchthattheinventorylevelxstrictlyaboveSandnolargerthanS+QiftheinventorylevelfallsonorbelowS,i.e.,theoptimalorderquantityisn(x),anddecreaseinventorysuchthattheinventorylevelxstrictlybelowD 70

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. . . . . 0 . 0:2 . 0:4 . 0:6 . 0:8 . 1 . 1:2 . 1:4 . 1:6 . 1:8 . 2 . )]TJ /F4 11.955 Tf 9.3 0 Td[(2 . )]TJ /F4 11.955 Tf 9.3 0 Td[(1 . 0 . 1 . 2 . 3 . S . D . t . Inventory . (S;D;Q)policy Figure4-2. Illustrationof(S;D;Q)policy andnolowerthanD)]TJ /F3 11.955 Tf 12.38 0 Td[(QiftheinventorylevelfallsonoraboveD.Figure 4-2 explainsthe(S;D;Q)policy.Intheremainingpartofthispaper,weprovethefunctiondescribedin( 4 )istheoptimalvaluefunction.WerstprovidelowerboundtheoremsinSection 4.4 .Thenweprovethefunctiondescribedin( 4 )satisestheconsitionsoflowerboundtheoremsandsoistheminimumcostfunctioninSections 4.5 and 4.6 . 4.4LowerBoundTheoremsThissectionprovidesamethodfordeterminingtheoptimalsolution.ThesetwolowerboundtheoremsrequiretheknowledgeofV(x)andthecontinuationregion,i.e.,theregionwherethecontrollerdoesnotorder.Formanyproblems,however,thesecanbeguessed,andthenthelowerboundtheoremsareinvoked. Denetheoperatorforthestochasticdifferentialequation( 4 )as )]TJ /F3 11.955 Tf 7.32 0 Td[(V(x):= V0 (x)+1 22 V00 (x):(4)WestateageneralizedIto'slemma. 71

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Lemma4.1. (ExtendedIto'sformula) SupposethatV(x)2C1(R)and V00 (x)2C(R)exceptforcountablepoints.Wehavee)]TJ /F5 7.97 Tf 6.59 0 Td[(rTV(Z(T))=V(Z(0)]TJ /F4 11.955 Tf 9.29 0 Td[())+ZT0e)]TJ /F5 7.97 Tf 6.59 0 Td[(rtAV(Z(t))dt+ZT0e)]TJ /F5 7.97 Tf 6.59 0 Td[(rt V0 (Z(t))dB(t) (4)+N(T)Xn=1e)]TJ /F5 7.97 Tf 6.59 0 Td[(rn[V(Z(n)))]TJ /F3 11.955 Tf 11.96 0 Td[(V(Z(n)]TJ /F4 11.955 Tf 9.29 0 Td[())];where AV(x):=)]TJ /F3 11.955 Tf 9.3 0 Td[(rV(x)+)]TJ /F3 11.955 Tf 26.29 0 Td[(V(x): (4) Proof. TheproofisgiveninAppendix B.1 . Furthermore,fornotationbrevityandconvinenceofofanalyzingourgenerallowerboundandboundachievedtheorems,wedene theswitchoperator LV(x)= inf v2fV(x+v)+(x;v)g;(4)whereisthefeasibleorderregionand(x;v)istheorderingcostwhentheinventoryisxandtheorderquantityisv. Theorem4.2. (Lowerboundforoptimalvaluefunction)Fornon-negativecostfunctitonsp(x)and(x;v),supposethatthereexistsaV(x)suchthat 1. V(x)2C1(R)and V00 (x)2C(R)exceptforcountablepoints, 2. V(x)LV(x)forallx2R, 3. AV(x)+h(x)0foralmostallx2R, 4. ExR10e)]TJ /F5 7.97 Tf 6.59 0 Td[(rth(x)]TJ /F3 11.955 Tf 11.96 0 Td[(D(t))dt<1foranyx2Randlimx!V(x)=h(x)<1.Then,V(x)infw2WVw(x),whereVw(x)=EZ10e)]TJ /F5 7.97 Tf 6.59 0 Td[(rth(Z(t))dt+E"1Xi=1)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Z)]TJ /F3 11.955 Tf 5.48 -9.68 Td[()]TJ /F5 7.97 Tf -1.34 -8.01 Td[(i;vi# 72

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istheobjectivefunctioncorrespondingtoanyfeasiblepolicyw=((1;v1);(2;v2);:::)2W. Proof. TheproofisgiveninAppendix B.2 . Theorem4.3. (Lowerboundachieved)SupposeV(x)satisestheconditionsofTheorem 4.2 andsuchthat 5. AV(x)+h(x)=0forx2C:=fx:V(x)^k)]TJ /F7 7.97 Tf 6.59 0 Td[(1: Z(t)=2C g,and ^qk isdenedsuchthat LV(Z(^-k))=V(Z(^-k)+ ^qk )+(Z(^-k); ^qk ) if ^k <1and ^qk :=0if ^k =1,fork=1;2;:::.ThenV(x)=V ^w (x)=infw2WVw(x) and^w:=f^1;^2;:::;^q1;^q2;:::gisanoptimalorderingpolicy. Proof. TheproofisgiveninAppendix B.3 . Notehere thelowerboundtheoremsweprovidedaboveareforgeneral impulsecontrol problems.In ourproblemsetting ,thefeasibleorderregionis:=fv=lQ:l2Z;l6=0g; accordingly theswitchoperator asdescribedin( 4 )isspeciedas LV(x)=minminl2NfV(x+lQ)+culQg;minl2NfV(x)]TJ /F3 11.955 Tf 11.96 0 Td[(lQ)+cdlQg;(4)andtheorderingcostis (x;v)=8><>:cuvifv0cdvifv<0;(4) 73

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wherexistheinventorylevelandvistheorderquantity. Intheremainingpart,weconsider( 4 )and( 4 )forourproblemsetting. Inaddition,notethat x2RnC,x2fx:V(x)=LV(x)g (4) basedonCondition(ii)andthedenitionofCinCondition(v) .ThesetC:=fx:V(x)
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S)]TJ /F3 11.955 Tf 11.95 0 Td[(mQ+u(S;S)]TJ /F3 11.955 Tf 11.95 0 Td[(mQ+0)Q=S)]TJ /F3 11.955 Tf 11.95 0 Td[(mQ+mQ=S;(4) D+nQ)]TJ /F3 11.955 Tf 11.96 0 Td[(d(D;D+nQ)Q=D+nQ)]TJ /F4 11.955 Tf 11.95 0 Td[((n+1)Q=D)]TJ /F3 11.955 Tf 11.95 0 Td[(Q;(4)and D+nQ)]TJ /F3 11.955 Tf 11.95 0 Td[(d(D;D+nQ)]TJ /F4 11.955 Tf 11.95 0 Td[(0)Q=D+nQ)]TJ /F3 11.955 Tf 11.96 0 Td[(nQ=D:(4)From( 4 )and( 4 )( 4 ),wehavethatV(x)iscontinueandsmoothinxifandonlyif w(S+Q)+Ae1(S+Q)+Be)]TJ /F5 7.97 Tf 6.59 0 Td[(2(S+Q)+cu(m+1)Q=w(S)+Ae1S+Be)]TJ /F5 7.97 Tf 6.59 0 Td[(2S+cumQ;(4) w0 (S+Q)+1Ae1(S+Q))]TJ /F3 11.955 Tf 11.96 0 Td[(2Be)]TJ /F5 7.97 Tf 6.59 0 Td[(2(S+Q)= w0 (S)+1Ae1S)]TJ /F3 11.955 Tf 11.96 0 Td[(2Be)]TJ /F5 7.97 Tf 6.58 0 Td[(2S;(4) w(D)]TJ /F3 11.955 Tf 11.96 0 Td[(Q)+Ae1(D)]TJ /F5 7.97 Tf 6.58 0 Td[(Q)+Be)]TJ /F5 7.97 Tf 6.59 0 Td[(2(D)]TJ /F5 7.97 Tf 6.58 0 Td[(Q)+cd(n+1)Q=w(D)+Ae1D+Be)]TJ /F5 7.97 Tf 6.58 0 Td[(2D+cdnQ;(4)and w0 (D)]TJ /F3 11.955 Tf 11.96 0 Td[(Q)+1Ae1(D)]TJ /F5 7.97 Tf 6.59 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(2Be)]TJ /F5 7.97 Tf 6.59 0 Td[(2(D)]TJ /F5 7.97 Tf 6.58 0 Td[(Q)= w0 (D)+1Ae1D)]TJ /F3 11.955 Tf 11.95 0 Td[(2Be)]TJ /F5 7.97 Tf 6.58 0 Td[(2D:(4)Notethat( 4 )and( 4 )areequivalentto w(S+Q)+Ae1(S+Q)+Be)]TJ /F5 7.97 Tf 6.59 0 Td[(2(S+Q)+cuQ=w(S)+Ae1S+Be)]TJ /F5 7.97 Tf 6.58 0 Td[(2S(4) 75

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and w(D)]TJ /F3 11.955 Tf 11.95 0 Td[(Q)+Ae1(D)]TJ /F5 7.97 Tf 6.59 0 Td[(Q)+Be)]TJ /F5 7.97 Tf 6.58 0 Td[(2(D)]TJ /F5 7.97 Tf 6.59 0 Td[(Q)+cdQ=w(D)+Ae1D+Be)]TJ /F5 7.97 Tf 6.59 0 Td[(2D:(4)Thus,A,B,SandDshouldbechosensuchthat( 4 ),( 4 ),( 4 ),and( 4 )hold.Toprovesuchfourparametersexist,werstdeneseveralfunctionsandstudytheirpropertiesinthefollowingsubsection. 4.5.1PreliminaryresultsInthissubsection,wedenesomefunctionsandexploretheirpropertiesthatwillbeusedtoprovetheexistenceofparametersandverifyourguessedfunctionisoptimal.First,let f(x):= w0 (x)=2+w(x):(4) F(x):=f(x))]TJ ET BT /F3 11.955 Tf 251.55 -329.86 Td[(f0 (x)=1:(4) g(x):=w(x))]TJ ET BT /F3 11.955 Tf 249.97 -384.15 Td[(w0 (x)=1:(4) G(x):= g0 (x)=2+g(x):(4) Lemma4.2. Fortheexpectedholdingcostfunctionw(x)denedin( 4 ), asshownin( 4 ) ,wehave w(x)=e1xZ1xh(y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1ydy+e)]TJ /F5 7.97 Tf 6.59 0 Td[(2xZxh(y)e2ydy=p 2+2r22C2(R)(4) and w00 (x)>0:(4) Proof. TheproofisgiveninAppendix C.2 . 76

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Lemma4.3. (i)Forf(x)denedin( 4 ),wehave f00 (x)0;(4)andthereexistsazfsuchthattherst-orderderivativeoff(x) f0 (x)= w00 (x)=2+ w0 (x)8>>>><>>>>:<0x0x>zf:(4)(ii)Forg(x)denedin( 4 ),wehave g00 (x)0;(4)andthereexistsazgsuchthattherst-orderderivativeofg(x) g0 (x)= w00 (x)=2+ w0 (x)8>>>><>>>>:<0x0x>zg:(iii)ForF(x)denedin( 4 ),wehaveF(x)=h(x)=r.ForG(x)denedin( 4 ),wehaveG(x)=F(x). Proof. TheproofisgiveninAppendix C.2 . Next,dene Ad(x)=[2=(1+2)][f(x)]TJ /F3 11.955 Tf 11.96 0 Td[(Q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(x)+cdQ]=)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(e1x)]TJ /F3 11.955 Tf 11.95 0 Td[(e1(x)]TJ /F5 7.97 Tf 6.59 0 Td[(Q)(4) Au(x)=)]TJ /F4 11.955 Tf 11.29 0 Td[([2=(1+2)][f(x+Q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(x)+cuQ]=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(e1(x+Q))]TJ /F3 11.955 Tf 11.95 0 Td[(e1x(4) 77

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. . x . D1 . D0 Figure4-3. IllustrationofAd(x) Bd(x)=[1=(1+2)][g(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(g(x)+cdQ]=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(e)]TJ /F5 7.97 Tf 6.58 0 Td[(2x)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F5 7.97 Tf 6.58 0 Td[(2(x)]TJ /F5 7.97 Tf 6.59 0 Td[(Q)(4) andBu(x)=)]TJ /F4 11.955 Tf 11.29 0 Td[([1=(1+2)][g(x+Q))]TJ /F3 11.955 Tf 11.95 0 Td[(g(x)+cuQ]=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2(x+Q))]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2x:(4)Westatesomepropertiesofthesefunctionsinthefollowinglemmas. Lemma4.4. ThereexistsaD0suchthat Ad(x)8>>>><>>>>:>0ifxD0(4)andthereexistsaD1>D0suchthat A0d (x)=[12=(1+2)][F(x))]TJ /F3 11.955 Tf 11.95 0 Td[(F(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(cdQ]=e1x)]TJ /F3 11.955 Tf 11.96 0 Td[(e1(x)]TJ /F5 7.97 Tf 6.59 0 Td[(Q)8>>>><>>>>:<0ifx0ifx>D1:(4) Proof. TheproofisgiveninAppendix C.3 . Figure 4-3 givesanillustrationofAd(x). Lemma4.5. ThereexistsaS0suchthat 78

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Au(x)8>>>><>>>>:>0ifxS0(4)andthereexistsaS1>S0suchthat A0u (x)=[12=(1+2)][F(x+Q))]TJ /F3 11.955 Tf 11.95 0 Td[(F(x)+cuQ]=e1(x+Q))]TJ /F3 11.955 Tf 11.95 0 Td[(e1x8>>>><>>>>:<0ifx0ifx>S1:(4) Proof. TheproofisgiveninAppendix C.3 . Au(x)hassimilarshapewiththatofAd(x). Lemma4.6. ForD1decidedinLemma 4.4 ,thereexistsaD2>D1suchthat Bd(x)8>>>><>>>>:<0ifx0ifx>D2and B0d (x)8>>>><>>>>:<0ifx0ifx>D1:(4)Inaddition,wehave B0d (x)= A0d (x)e1x)]TJ /F3 11.955 Tf 11.95 0 Td[(e1(x)]TJ /F5 7.97 Tf 6.58 0 Td[(Q)=e)]TJ /F5 7.97 Tf 6.59 0 Td[(2(x)]TJ /F5 7.97 Tf 6.59 0 Td[(Q))]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F5 7.97 Tf 6.58 0 Td[(2x:(4) Proof. TheproofisgiveninAppendix C.3 . Figure 4-4 givesanillustrationofBd(x). Lemma4.7. ForS1decidedinLemma 4.5 ,thereexistsaS2>S1suchthat Bu(x)8>>>><>>>>:<0ifx0ifx>S2and B0u (x)8>>>><>>>>:<0ifx0ifx>S1:(4) 79

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. . x . D1 . D2 Figure4-4. IllustrationofBd(x) Inaddition,wehave B0u (x)= A0u (x)e1(x+Q))]TJ /F3 11.955 Tf 11.95 0 Td[(e1x=e)]TJ /F5 7.97 Tf 6.59 0 Td[(2x)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F5 7.97 Tf 6.58 0 Td[(2(x+Q):(4) Proof. TheproofisgiveninAppendix C.3 . NotethatBu(x)hassimilarshapewiththatofBd(x). Lemma4.8. ForD0decidedinLemma 4.4 ,S0decidedinLemma 4.5 ,Bd(x)denedin( 4 ),andBu(x)denedin( 4 ),wehave Bd(D0)Bu(S0):(4) Proof. TheproofisgiveninAppendix C.7 . 4.5.2ReducedequationsInthissubsection,wesimplifythefourequationsthatcontainA,B,S,andDtotwoequationsthatonlycontainSandD.Werstwritethefourequationsin( 4 )tothefollowingfourequivalentequations: A)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(1Qe1D)]TJ /F3 11.955 Tf 11.95 0 Td[(B)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(e2Q)]TJ /F4 11.955 Tf 11.96 0 Td[(1e)]TJ /F5 7.97 Tf 6.58 0 Td[(2D=w(D)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(w(D)+cdQ;(4) 1A)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(1Qe1D+2B)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(e2Q)]TJ /F4 11.955 Tf 11.95 0 Td[(1e)]TJ /F5 7.97 Tf 6.59 0 Td[(2D= w0 (D)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ ET BT /F3 11.955 Tf 355 -641.75 Td[(w0 (D);(4) 80

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)]TJ /F3 11.955 Tf 11.96 0 Td[(A)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(e1Q)]TJ /F4 11.955 Tf 11.95 0 Td[(1e1S+B)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F5 7.97 Tf 6.58 0 Td[(2Qe)]TJ /F5 7.97 Tf 6.59 0 Td[(2S=w(S+Q))]TJ /F3 11.955 Tf 11.96 0 Td[(w(S)+cuQ;(4)and )]TJ /F3 11.955 Tf 11.95 0 Td[(1A)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(e1Q)]TJ /F4 11.955 Tf 11.96 0 Td[(1e1S)]TJ /F3 11.955 Tf 11.96 0 Td[(2B)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F5 7.97 Tf 6.58 0 Td[(2Qe)]TJ /F5 7.97 Tf 6.59 0 Td[(2S= w0 (S+Q))]TJ ET BT /F3 11.955 Tf 360.72 -143.45 Td[(w0 (S):(4)Next,wesimplifythefourequationsandgettwoequationsthatonlycontainSandD.WebeginwithcharacterizingAandBbasedonSandD.From( 4 ),wehave B=)]TJ /F3 11.955 Tf 9.3 0 Td[(A)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(1Qe1D1=2)]TJ /F4 11.955 Tf 11.95 0 Td[([ w0 (D))]TJ ET BT /F3 11.955 Tf 243.58 -298.86 Td[(w0 (D)]TJ /F3 11.955 Tf 11.96 0 Td[(Q)]=2e2D=)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(e2Q)]TJ /F4 11.955 Tf 11.96 0 Td[(1:(4)Substituting( 4 )into( 4 ),wehave A=[2=(1+2)]fw(D)]TJ /F3 11.955 Tf 11.96 0 Td[(Q))]TJ /F3 11.955 Tf 11.95 0 Td[(w(D)+cdQ)]TJ /F4 11.955 Tf 11.96 0 Td[([ w0 (D))]TJ ET BT /F3 11.955 Tf 328.42 -388.31 Td[(w0 (D)]TJ /F3 11.955 Tf 11.95 0 Td[(Q)]=2ge)]TJ /F5 7.97 Tf 6.58 0 Td[(1D=)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(1Q=[2=(1+2)][f(D)]TJ /F3 11.955 Tf 11.96 0 Td[(Q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(D)+cdQ]e)]TJ /F5 7.97 Tf 6.59 0 Td[(1D=)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F5 7.97 Tf 6.58 0 Td[(1Q=Ad(D);(4)wherethesecondequalityholdsbecauseof( 4 )andthelastequalityholdsbecauseof( 4 ).Substituting( 4 )into( 4 ),wehaveanotherexpressionofBasfollows 81

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B=)]TJ /F4 11.955 Tf 11.29 0 Td[([1=(1+2)]fw(D)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(w(D)+cdQ+[ w0 (D))]TJ ET BT /F3 11.955 Tf 348.28 -47.82 Td[(w0 (D)]TJ /F3 11.955 Tf 11.96 0 Td[(Q)]=1ge2D=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(e2Q)]TJ /F4 11.955 Tf 11.96 0 Td[(1=)]TJ /F4 11.955 Tf 11.29 0 Td[([1=(1+2)][g(D)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(g(D)+cdQ]=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2(D)]TJ /F5 7.97 Tf 6.59 0 Td[(Q))]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2D=Bd(D); (4)wherethesecondequalityholdsbecauseof( 4 )andthelastequalityholdsbecauseof( 4 ).Nowwestate anotherwaytocharacterAandBbasedonSandD. From( 4 ),wehave B=)]TJ /F3 11.955 Tf 9.3 0 Td[(A)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(e1Q)]TJ /F4 11.955 Tf 11.95 0 Td[(1e1S1=2)]TJ /F4 11.955 Tf 11.95 0 Td[([ w0 (S+Q))]TJ ET BT /F3 11.955 Tf 260.78 -295.87 Td[(w0 (S)]=2e2S=)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2Q:(4)Substituting( 4 )into( 4 ),wehave A=)]TJ /F4 11.955 Tf 11.29 0 Td[([2=(1+2)]fw(S+Q))]TJ /F3 11.955 Tf 11.95 0 Td[(w(S)+cuQ+[ w0 (S+Q))]TJ ET BT /F3 11.955 Tf 355.78 -385.32 Td[(w0 (S)]=2ge)]TJ /F5 7.97 Tf 6.58 0 Td[(1S=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(e1Q)]TJ /F4 11.955 Tf 11.95 0 Td[(1=)]TJ /F4 11.955 Tf 11.29 0 Td[([2=(1+2)][f(S+Q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(S)+cuQ]e)]TJ /F5 7.97 Tf 6.58 0 Td[(1S=)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(e1Q)]TJ /F4 11.955 Tf 11.95 0 Td[(1=Au(S);(4)wherethesecondequalityholdsbecauseof( 4 )andthelastequalityholdsbecauseof( 4 ).Substituting( 4 )into( 4 ),wehaveanotherexpressionofBasfollows 82

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B=[1=(1+2)]fw(S+Q))]TJ /F3 11.955 Tf 11.95 0 Td[(w(S)+cuQ)]TJ /F4 11.955 Tf 11.95 0 Td[([ w0 (S+Q))]TJ ET BT /F3 11.955 Tf 364.54 -47.82 Td[(w0 (S)]=1ge2S=)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2Q=[1=(1+2)][g(S+Q))]TJ /F3 11.955 Tf 11.95 0 Td[(g(S)+cuQ]=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2S)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2(S+Q)=Bu(S); (4)wherethesecondequalityholdsbecauseof( 4 )andthelastequalityholdsbecauseof( 4 ).SoifweknowthevaluesofSandD,wecangetthevaluesofAandBbasedon( 4 )and( 4 )or( 4 )and( 4 ).Finally,weusetheaboverelationshipstoreducetheoriginalfourequationstotwoequationsthatonlycontainSandD.From( 4 )and( 4 ),wehave Ad(D)=Au(S):(4)From( 4 )and( 4 ),wehave Bd(D)=Bu(S)(4)NowweonlyneedtoprovethatthereexistSandDsuchthat( 4 )and( 4 )hold. 4.5.3ExistenceofSandD Noteherethatallcombinations(S;D)satisfying( 4 )formalineinthetwodimensionalspaceforSandD. Forillustration,weuse1torepresenttheline,i.e.,1=f(z;(z)):Ad(z)=Au((z))g. ToshowtheexistenceofSandDsatisfying( 4 )and( 4 ),itissufcienttoprovethatthereexistsauniquepair(S;D)21suchthatBd(D))]TJ /F3 11.955 Tf 11.95 0 Td[(Bu(S)=0. 83

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. . x . S1 . S0 . D1 . D0 . D2 . (z) . Au((z)) . Ad(z) . z . Au(x) . Ad(x) Figure4-5. Illustrationof(z) WerstprovethecasewhenAu(S1)Ad(D1),whereD1andS1aredenedinLemmas 4.4 and 4.5 . Dene(z)andH(z) FromLemmas 4.4 and 4.5 andAu(S1)Ad(D1),wehavethatfor eachz2[D1;D2] ,thereexistsaunique(z)2(S0;S1]suchthatAd(z)=Au((z)).Figure 4-5 givesanillustrationof(z). Thus,bydening H(z):=Bd(z))]TJ /F3 11.955 Tf 11.96 0 Td[(Bu((z));(4)weonlyneedtoprovethatthereexistsauniqueD2(D1;D2)suchthatH(D)=0. Accordinglywe letS=(D).Fromthedenitionof(z) asshowninFig 4-5 ,wehave S=(D)2(S0;S1]:(4) WerstshowthepropertiesofH(z)forz2[D1;D2].Theclaimwillbeproved. Beforewestart,werstprovetworesultsthatwillbeusedlater. Takingthederivativeof bothsidesof Ad(z)=Au((z))withrespectivetoz,wehave 0 (z)= A0d (z)= A0u ((z)):(4)Inaddition,from( 4 )and( 4 ),wehave 84

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Ad(x)=[2=(1+2)][f(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(x)+cdQ]=)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(e1x)]TJ /F3 11.955 Tf 11.96 0 Td[(e1(x)]TJ /F5 7.97 Tf 6.59 0 Td[(Q)>[2=(1+2)][f(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(x))]TJ /F3 11.955 Tf 11.96 0 Td[(cuQ]=)]TJ /F3 11.955 Tf 5.47 -9.68 Td[(e1x)]TJ /F3 11.955 Tf 11.96 0 Td[(e1(x)]TJ /F5 7.97 Tf 6.59 0 Td[(Q)=Au(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q):FromAu((z))=Ad(z)>Au(z)]TJ /F3 11.955 Tf 11.96 0 Td[(Q),Lemma 4.5 ,and(z)2(S0;S1],wehave (z)0forD10,and(c)H(D1)<0 Werstprove H0 (z)>0forD10forz>D1basedon( 4 )inLemma 4.4 ,itsufcestoprove e1z)]TJ /F3 11.955 Tf 11.95 0 Td[(e1(z)]TJ /F5 7.97 Tf 6.59 0 Td[(Q)=e)]TJ /F5 7.97 Tf 6.59 0 Td[(2(z)]TJ /F5 7.97 Tf 6.59 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F5 7.97 Tf 6.58 0 Td[(2z)]TJ /F9 11.955 Tf 11.96 9.68 Td[(e1((z)+Q))]TJ /F3 11.955 Tf 11.95 0 Td[(e1(z)=e)]TJ /F5 7.97 Tf 6.59 0 Td[(2(z))]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F5 7.97 Tf 6.58 0 Td[(2((z)+Q)>0: Thiscanbeshownasfollows: 85

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. . x . S1 . S0 . D1 . D0 . D2 . 1(z) . Au(1(z)) . Ad(z) . z . Au(x) . Ad(x) Figure4-6. Illustrationof1(z) e1z)]TJ /F3 11.955 Tf 11.95 0 Td[(e1(z)]TJ /F5 7.97 Tf 6.58 0 Td[(Q)=e)]TJ /F5 7.97 Tf 6.59 0 Td[(2(z)]TJ /F5 7.97 Tf 6.58 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2z=)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F5 7.97 Tf 6.58 0 Td[(1Qe(1+2)z=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(e2Q)]TJ /F4 11.955 Tf 11.95 0 Td[(1>)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F5 7.97 Tf 6.58 0 Td[(1Qe(1+2)((z)+Q)=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(e2Q)]TJ /F4 11.955 Tf 11.96 0 Td[(1=e1((z)+Q))]TJ /F3 11.955 Tf 11.96 0 Td[(e1(z)=e)]TJ /F5 7.97 Tf 6.59 0 Td[(2(z))]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2((z)+Q);(4)wheretheinequalityholdsbecauseof(z)+Q0.From( 4 ),wehaveH(D2)=Bd(D2))]TJ /F3 11.955 Tf -413.93 -23.91 Td[(Bu((D2))=0)]TJ /F3 11.955 Tf 12.08 0 Td[(Bu((D2))>0,wherethesecondequalityholdsbecauseBd(D2)=0basedonLemma 4.6 andtheinequalityholdsbecauseBu(x)<0forx
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H1(z):=Bd(z))]TJ /F3 11.955 Tf 11.96 0 Td[(Bu(1(z)):(4)From( 4 ),( 4 ),and1(D1)=(D1),wehaveH(D1)=H1(D1).InordertoproveH(D1)<0,itsufcestoproveH1(D1)<0.InthefollowingweproveH1(D1)<0byproving H01 (z)<0forD0zAd(D1),wedenetheline(z;(z))forS0zS2 andprovetheclaimsysmmetrically .Finally,wesummerizetheresultsinthefollowingtheorem. Theorem4.4. ThereexistA<0,B<0,S,andDsuchthatS+Q
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SinceCondition1ofTheorem 4.2 hasbeensatisedbythechoiceofS,D,A,andBprovedinSection 4.5 ,inordertoshowthatthevaluefunctionandcontrolpolicyareoptimal,itsufcestoverifythatthefollowingconditionsaresatised:(ii)V(x)=LV(x)=V(x+u(S;x)Q)+cuu(S;x)QforxSandV(x)=LV(x)=V(x)]TJ /F3 11.955 Tf 12.71 0 Td[(d(D;x)Q)+cdd(D;x)QforxD,(iii) LV(x)>V(x) forS
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(i)qu(S)=0andqu(x)>0forS0forS+Qx
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Lu1V(x):=minl2N;lu(S;x)fV(x+lQ)+culQg(4) represents thecasewhenx+lQS+Q, Lu2V(x):=minl2N;u(S;x)+1lu(D)]TJ /F7 7.97 Tf 6.59 0 Td[(0;x))]TJ /F7 7.97 Tf 6.59 0 Td[(1fV(x+lQ)+culQg(4) represents thecasewhenx+lQbelongstothecontinuationregion,i.e.,S+QV(x),(b)Lu1V(x)=V(x),(c)Lu2V(x)>V(x),and(d)Lu3V(x)>V(x)forxS.(a)LdV(x)>V(x)forxS Notethatx)]TJ /F3 11.955 Tf 11.96 0 Td[(lQSforl2NandxS.Fromthedenitionofu(S;x)in( 4 ),wehaveu(S;x)]TJ /F3 11.955 Tf 11.95 0 Td[(lQ)=u(S;x)+landso x)]TJ /F3 11.955 Tf 11.95 0 Td[(lQ+u(S;x)]TJ /F3 11.955 Tf 11.96 0 Td[(lQ)Q=x+u(S;x)Q:(4)From( 4 ),wehave 90

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V(x)]TJ /F3 11.955 Tf 11.95 0 Td[(lQ)( 4\0008 )=(x)]TJ /F3 11.955 Tf 11.96 0 Td[(lQ+u(S;x)]TJ /F3 11.955 Tf 11.95 0 Td[(lQ)Q)+cuu(S;x)]TJ /F3 11.955 Tf 11.95 0 Td[(lQ)Q( 4\00080 )=(x+u(S;x)Q)+cuu(S;x)Q+culQ( 4\0008 )=V(x)+culQ:(4)Thus, from( 4 ),wehave LdV(x)=minl2NfV(x)]TJ /F3 11.955 Tf 11.96 0 Td[(lQ)+cdlQg( 4\00081 )=minl2NfV(x)+culQ+cdlQg>V(x):(4)(b)Lu1V(x)=V(x)forxS For1lu(S;x)andl2N,fromthedenitionofu(S;x)in( 4 ),wehaveu(S;x+lQ)=u(S;x))]TJ /F3 11.955 Tf 11.95 0 Td[(land accordingly x+lQ+u(S;x+lQ)Q=x+u(S;x)Q:(4)From( 4 ),wehave V(x+lQ)( 4\0008 )=(x+lQ+u(S;x+lQ)Q)+cuu(S;x+lQ)Q( 4\00083 )=(x+u(S;x)Q)+cuu(S;x)Q)]TJ /F3 11.955 Tf 11.95 0 Td[(culQ( 4\0008 )=V(x))]TJ /F3 11.955 Tf 11.96 0 Td[(culQ:(4)Thus, from( 4 ),wehave Lu1V(x)( 4\00077 )=minl2N;lu(S;x)fV(x+lQ)+culQg (4)( 4\00084 )=minl2N;lu(S;x)fV(x))]TJ /F3 11.955 Tf 11.95 0 Td[(culQ+culQg=minl2N;lu(S;x)fV(x)g=V(x): 91

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(c)and(d)Lu2V(x)>V(x)forxSandLu3V(x)>V(x)forxS Thedetailedproof sare givenin C.9 inAppendix.Insum,from( 4 ),( 4 ),( 4 ),( 4 ),( 4 ) inthepreviousargument,and ( C ),( C ),and( C )inAppendix,wehaveLV(x)=minfLdV(x);Lu1V(x);Lu2V(x);Lu3V(x)g=V(x)forxS: 4.6.3LV(x)>V(x)forSV(x)forSV(x)forSV(x)andLdV(x)>V(x)forSV(x)forSV(x)holdssimilarly duetosymmetry . Werstdiscussaspecialcase.Whenx=D)]TJ /F3 11.955 Tf 11.95 0 Td[(Q,wehave x+lQ=D)]TJ /F3 11.955 Tf 11.95 0 Td[(Q+lQDforl2N(4)and d(D;x+lQ)=d(D;D+(l)]TJ /F4 11.955 Tf 11.96 0 Td[(1)Q)=lforl2N;(4) 92

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wherethelastequalityholdsbecauseofthedenitionofd(y;x)in( 4 ).Hence,basedon( 4 ),wehave LuV(x):=minl2NfV(x+lQ)+culQg=minl2Nf(x+lQ)]TJ /F3 11.955 Tf 11.96 0 Td[(d(D;x+lQ)Q)+cdd(D;x+lQ)Q+culQg( 4\00087 )=minl2Nf(x+lQ)]TJ /F3 11.955 Tf 11.95 0 Td[(lQ)+cdlQ+culQg=minl2Nf(x)+cdlQ+culQg=(x)+cdQ+cuQ>(x)=V(x); wherethesecondinequalityholdsbecauseof( 4 )and( 4 ),andthelastequalityholdsbecauseof( 4 )andthefactthatx=D)]TJ /F3 11.955 Tf 12.43 0 Td[(Q2(S;D)basedonD)]TJ /F3 11.955 Tf 12.43 0 Td[(Q>Sin( 4.4 ). Next,wediscussthegeneralcasewherex2(S;D)andx6=D)]TJ /F3 11.955 Tf 12.29 0 Td[(Q. Notethatforx2(S;D)]TJ /F3 11.955 Tf 11.95 0 Td[(Q), following( 4 ) ,we canrepresent LuV(x)=minfLu1V(x);Lu2V(x)g;(4)where Lu1V(x):=minl2N;lu(D;x))]TJ /F7 7.97 Tf 6.59 0 Td[(1fV(x+lQ)+culQg;(4) andLu2V(x):=minl2N;lu(D;x)fV(x+lQ)+culQg:(4)Andforx2(D)]TJ /F3 11.955 Tf 11.95 0 Td[(Q;D), following( 4 ) ,we canrepresent 93

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LV(x)=Lu2V(x):(4) becauseu(D;x)=1forx2(D)]TJ /F3 11.955 Tf 11.95 0 Td[(Q;D)basedonthedenitionofu(y;x)in( 4 ). Then,i nthefollowing, dueto( 4 )( 4 ),weonlyneedto prove (i) Lu1V(x)>V(x)forSV(x)forS0;(4)wheretheinequalityholdsbecause of( 4 )and qu(y)>0forS
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Lu1V(x)=minl2N;lu(D;x))]TJ /F7 7.97 Tf 6.59 0 Td[(1fV(x+lQ)+culQg( 4\00095 )>V(x): For(ii), weproveLu2V(x)>V(x)forS0;2>0;(4) 12=2r=2and(1=2)]TJ /F4 11.955 Tf 11.96 0 Td[(1=1)=(1)]TJ /F3 11.955 Tf 11.96 0 Td[(2)=(12)=)]TJ /F3 11.955 Tf 9.3 0 Td[(=r:(4) Proof. Theproofisobvious. Lemma4.11. Forthefunctionse1xande)]TJ /F5 7.97 Tf 6.59 0 Td[(2x,where1and2aredescribedin( 4 ),wehaveA)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(e1x=0andA)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(e)]TJ /F5 7.97 Tf 6.58 0 Td[(2x=0withAdenedin( 4 ). Proof. From( 4 ),wehaveA)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2x=e)]TJ /F5 7.97 Tf 6.58 0 Td[(2x)]TJ /F3 11.955 Tf 9.3 0 Td[(r)]TJ /F3 11.955 Tf 11.95 0 Td[(2+1 2222=e)]TJ /F5 7.97 Tf 6.58 0 Td[(2xf0()]TJ /F3 11.955 Tf 9.3 0 Td[(2)=0,wherethesecondequalityfollows( 4 )andthelastequalityholdsbecausef0()]TJ /F3 11.955 Tf 9.3 0 Td[(2)= 95

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0basedonLemma 4.10 .Similarly,wehaveA)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(e1x=e1x)]TJ /F3 11.955 Tf 9.3 0 Td[(r+1+1 2221=e1xf0(1)=0. Lemma4.12. Forthefunctionw(x)=ExR10e)]TJ /F5 7.97 Tf 6.59 0 Td[(rth(x)]TJ /F3 11.955 Tf 11.96 0 Td[(D(t))dt,wehaveAw(x)+h(x)=0,whereAisdenedin( 4 ). Proof. Thisconclusionfollowsfrom equation(9)inP age45of[ 31 ]. Nowweprovethemainclaim. Proposition4.3. ForthefunctionV(x)denedin( 4 )andtheoperatorAdenedin( 4 ),wehave AV(x)+h(x)=0forS
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F(x)( 4\00029 )=f(x))]TJ ET BT /F3 11.955 Tf 191.17 -47.82 Td[(f0 (x)=1( 4\00028 )=w(x)+(1=2)]TJ /F4 11.955 Tf 11.96 0 Td[(1=1) w0 (x))]TJ ET BT /F3 11.955 Tf 308.7 -74.71 Td[(w00 (x)=(12)=w(x))]TJ /F4 11.955 Tf 11.95 0 Td[((=r) w0 (x))]TJ /F3 11.955 Tf 11.95 0 Td[(2 w00 (x)=(2r); (4)wherethelastequalityholdsbecause(1=2)]TJ /F4 11.955 Tf 11.95 0 Td[(1=1)=)]TJ /F3 11.955 Tf 9.3 0 Td[(=rand12=2r=2basedon ( 4 )in Lemma 4.10 . Furthermore,following( 4 )andconsideringthefactthat1>0,2>0,andQ>0basedonAssumption 4.2 and( 4 ), wehave F(x+Q))]TJ /F3 11.955 Tf 11.96 0 Td[(F(x)+cuQ8>>>><>>>>:<0ifx0ifx>S1:(4) Nowweareabletoprovethemainclaim. Notethat[1k=1(S)]TJ /F3 11.955 Tf 11.95 0 Td[(kQ;S)]TJ /F4 11.955 Tf 11.96 0 Td[((k)]TJ /F4 11.955 Tf 11.95 0 Td[(1)Q).Inthefollowing,weproveAV(x)+h(x)0forxSbyprovingAV(x)+h(x)0forx2(S)]TJ /F3 11.955 Tf 11.95 0 Td[(kQ;S)]TJ /F4 11.955 Tf 11.96 0 Td[((k)]TJ /F4 11.955 Tf 11.95 0 Td[(1)Q)foreachk=1;2;:::. Forx2(S)]TJ /F3 11.955 Tf 11.95 0 Td[(kQ;S)]TJ /F4 11.955 Tf 11.96 0 Td[((k)]TJ /F4 11.955 Tf 11.95 0 Td[(1)Q)foragivenk2N,wehave 97

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AV(x)+h(x)=A)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(w(x+u(S;x)Q)+Ae1(x+u(S;x)Q)+Be)]TJ /F5 7.97 Tf 6.58 0 Td[(2(x+u(S;x)Q)+cuu(S;x)Q+h(x)=A)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(w(x+kQ)+Ae1(x+kQ)+Be)]TJ /F5 7.97 Tf 6.59 0 Td[(2(x+kQ)+cukQ+h(x)=A)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(w(x+kQ)+Ae1(x+kQ)+Be)]TJ /F5 7.97 Tf 6.59 0 Td[(2(x+kQ)+cukQ)-221(Aw(x)=A(w(x+kQ))]TJ /F3 11.955 Tf 11.96 0 Td[(w(x))+A(cukQ)=)]TJ /F3 11.955 Tf 9.3 0 Td[(r[w(x+kQ))]TJ /F3 11.955 Tf 11.95 0 Td[(w(x)]+[ w0 (x+kQ))]TJ ET BT /F3 11.955 Tf 271.85 -185.09 Td[(w0 (x)]+(2=2)[ w00 (x+kQ))]TJ ET BT /F3 11.955 Tf 189.29 -209 Td[(w00 (x)])]TJ /F3 11.955 Tf 11.95 0 Td[(rcukQ=)]TJ /F3 11.955 Tf 9.3 0 Td[(rfw(x+kQ))]TJ /F4 11.955 Tf 11.96 0 Td[((=r) w0 (x+kQ))]TJ /F3 11.955 Tf 11.96 0 Td[(2 w00 (x+kQ)=(2r)g+rfw(x))]TJ /F4 11.955 Tf 11.96 0 Td[((=r) w0 (x))]TJ ET BT /F3 11.955 Tf 201.17 -256.81 Td[(w00 (x)g)]TJ /F3 11.955 Tf 20.59 0 Td[(rcukQ( 4\000100 )=)]TJ /F3 11.955 Tf 9.29 0 Td[(r[F(x+kQ))]TJ /F3 11.955 Tf 11.95 0 Td[(F(x)+cukQ]=)]TJ /F3 11.955 Tf 9.3 0 Td[(rnPk)]TJ /F7 7.97 Tf 6.59 0 Td[(1i=0[F(x+(i+1)Q))]TJ /F3 11.955 Tf 11.95 0 Td[(F(x+iQ)+cuQ]o0; wheretherstequalityholdsbecauseof( 4 )and( 4 ),thesecondequalityholdsbecauseu(S;x)=kforx2(S)]TJ /F3 11.955 Tf 11.95 0 Td[(kQ;S)]TJ /F4 11.955 Tf 11.96 0 Td[((k)]TJ /F4 11.955 Tf 11.95 0 Td[(1)Q]basedonthedenitionofu(y;x)in( 4 ),thethirdequalityholdsbecauseAw(x)+h(x)=0basedonLemma 4.12 ,thefourthequalityholdsbecauseA)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(e1x=0andA)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(e)]TJ /F5 7.97 Tf 6.58 0 Td[(2x=0basedonLemma 4.11 ,thefthequalityholdsbecauseof( 4 )and( 4 ),andtheinequalityholdsbecauseof( 4 )andx+iQx+(k)]TJ /F4 11.955 Tf 11.95 0 Td[(1)QSS1duetoik)]TJ /F4 11.955 Tf 9.85 0 Td[(1,x2(S)]TJ /F3 11.955 Tf 11.96 0 Td[(kQ;S)]TJ /F4 11.955 Tf 11.95 0 Td[((k)]TJ /F4 11.955 Tf 11.96 0 Td[(1)Q],andSS1basedon( 4 ). 4.6.5limx!V(x)=h(x)<1.WeverifyCondition(vi)inthissubsection.Weonlyneedtoshowlimx!V(x)=h(x)<1,becauselimx!+1V(x)=h(x)<1canbeprovedsimilarlyduetosymmetry. 98

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Toobtainourconclusion,werstshowthat(x+u(S;x)Q)isboundedforallxS.Notethat x+u(S;x)Q2(S;S+Q][S;S+Q]forallxS(4)basedonthedenitionofu(S;x)in( 4 ).Inaddition,basedon( 4 ),wehave(y)=w(y)+Ae1y+Be)]TJ /F5 7.97 Tf 6.59 0 Td[(2yisacontinuousfuncitonbecausew(y)2C2(R)basedon( 4 )inLemma 4.2 .Sinceacontinuousfunctionhasupperandlowerboundsinaclosedinterval,wehave (x+u(S;x)Q)2[ ; ]forxS;(4)where :=minf(y):y2[S;S+Q]gand :=maxf(y):y2[S;S+Q]g.Nowweprovetheclaimthatlimx!V(x)=h(x)isbounded.From( 4 ),wehavelimx!V(x)=h(x)=limx![(x+u(S;x)Q)+cuu(S;x)Q]=h(x)=limx!cuu(S;x)Q=h(x)=limx!cu()]TJ /F3 11.955 Tf 9.29 0 Td[(x)=h(x)=limx!)]TJ /F3 11.955 Tf 9.3 0 Td[(cu= h0 (x)2[0;1=r];wherethesecondequalityholdsbecause(x+u(S;x)Q)isboundedbasedon( 4 )andlimx!h(x)=1basedon(d)ofAssumption 4.2 ,thethirdequalityholdsbecauseofu(S;x)Q)]TJ /F4 11.955 Tf 9.38 0 Td[(()]TJ /F3 11.955 Tf 9.3 0 Td[(x)=u(S;x)Q+xisboundedbasedon( 4 )andlimx!h(x)=1basedon(d)ofAssumption 4.2 ,thefourthequalityholdsbecauseofL'Hospital'sRule,andthelastpartholdsbecauselimx! h0 (x)<)]TJ /F3 11.955 Tf 9.29 0 Td[(rcubasedon(d)ofAssumption 4.2 . 99

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. . supply-demand . . supply-demand . . supply-demand Figure4-7. BatteryControlinElectricityReserveMarket 4.7ApplicationinElectricityReserveMarketInthissection,wediscusshowtoapplytheabovemodeltocontrolelectricbatteryinelectricityreservemarket.OperatingreserveisthegeneratingcapacityavailabletoIndependentSystemOperator(ISO)inashorttimeintervaltomeetdemand,incaseageneratorgoesdown,thereisotherdisruptiontothesupply,orunderpeakload.Thequickreactpropertyofelectricbatterymakeitagoodchoicetoserveasoperatingreserve.Letthenetsupplybesupplyminusdemand.Wetreatthecumulativenetsupplyasourinventory.WeassumethenetsupplyX(t)followsaBrownianmotion,i.e.,X(t)=t+B(t),where>0byplanning.Thismeansmostofthetime,thebatteryischarged.Whenemergencyhappens,thebatterycanserveasshorttermsupplyuntilthetraditionalpowerplantsincreasetheiroutput.Startingatsomeinventorylevel,thesupplyislargerthandemandinmostoftime,sotheinventoryisincreasingmostoftime.However,sincetherearemanyuncertaincircumstances,theinventorycanalsodecreasesometimes.Thus,wewouldnotletthebatteryleavethesystemimmediatelyaftertheyarefullycharged,becausethesebatteriescanserveasabuffer,asinFigure 4-7 .Ononehand,itisnotbenecialtokeeptoomanyfullychargedbatteries.Hence,whentheinventoryissufcientlylarge,i.e.,thecumulativenetsupplyreachestheupperthresholdD,thecontrollercanletabatchoffullychargedbatteriesleavethesystem 100

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. . x . D . . x . D . . x . D)]TJ /F3 11.955 Tf 11.95 0 Td[(Q Figure4-8. UpperThresholdisReached . . x . S . . x . S . . x . S+Q Figure4-9. LowerThresholdisReached andletabatchofemptybatteriesenterthesystem.TheinventorydecreasesbyQafterthiscontrolaction,asinFigure 4.7 .Ontheotherhand,itisriskyifthereisnotenoughbackupoffullbatteriesincaseofemergency.Therefore,whentheinventoryissufcientlylow,i.e.,thecumulativenetsupplyreachesthelowerthresholdS,thecontrollercanletabatchofemptybatteriesleavethesystemandletabatchoffullbatteriesenterthesystem.TheinventoryincreasesbyQafterthiscontrolaction,asinFigure 4.7 .Finally,whenthecontrollerchoosesaholdingandpenaltycostfunction,theexpectedholdingcostwithoutcontrolcanbecomputedbasedon( 4 ).ThentheoptimalchoiceofSandDcanbeobtainedthroughsolving( 4 )( 4 )basedonourresultsinthispaper. 101

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4.8ConcludingRemarksInthischapter,wesolvethestochasticimpulsecontrolproblemwhentheunderlyingprocessevolvesasBrownianmotionsandadjustmentquantitieshavetobeintegerofsomebasequantity.Weprovethattheoptimalcontrolpolicyis(S;D;Q)policy.Specically,whentheunderlyingprocessfallsonorbelowS,weincreasetheminimumofamountssuchthattheunderlyingprocessexceedS;whentheunderlyingprocessfallsonoraboveD,wedecreasetheminimumofamountssuchthattheunderlyingprocessdecreasesbelowD. 102

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CHAPTER5BROWNIANINVENTORYCONTROLWITHPIECE-WISELINEARCONCAVECONTROLCOSTUNDERDISCOUNTEDCOSTCRITERION 5.1MotivationandLiteratureReviewImpulsecontroliswidelyappliedineconomics,nance,andoperationsresearch.Ineconomics,impulsecontrol canbe usedtosolvetheoptimalcontrolofexchangerateproblems( see,e.g., [ 15 ],[ 16 ],[ 37 ],and[ 55 ]).Innance,impulsecontrol canbe usedtosolveassetmanagementandportfoliomanagementproblems( see,e.g., [ 10 ],[ 29 ],[ 43 ],[ 47 ],and[ 54 ]),cashmanagementproblems( see,e.g., [ 5 ],[ 6 ],and[ 23 ]),dividendpolicy( see,e.g., [ 14 ]and[ 35 ]),andinvestmentandconsumptionmanagementproblems( see,e.g., [ 28 ],[ 30 ],[ 46 ],and[ 57 ]).Inoperationsresearch,theimpulsecontrolismainlyusedtosolvetheoptimalinventorycontrolproblems.Inthisarea,in[ 8 ],Batherrststudiesthestochasticinventorycontrolproblemwhentheinventoryevolvesas a Brownianmotionandtheinventorycanbeincreasedinstantaneouslywithaxedsetupcostandlinearcontrolcost.Underthelong-runaveragecostcriterion,hesuggeststhatan(s;S)policyisoptimalwithoutprovingtheexistenceanduniquenessofparameterssandS.In[ 70 ],SulemextendsBather'smodeltothediscountedcostproblemswithlinearholding/penaltycost.SheprovidessufcientconditionstoguaranteetheexistenceanduniquenessofparameterssandS.In[ 20 ],ChaoextendsBather'smodelbydiscussingthecaseswithandwithout considering backlogging.In[ 7 ],Bar-IlanandSulem extend Bather'smodelbyconsideringaconstantleadtime.In[ 9 ],Benkheroufextendsthemodeltodiscountedcostproblemswithageneralconvexholding/shortagecostfunction.Recently,in[ 77 ],WuandChaostudythelong-runaverage-costproblemwithageneralconvexholding/penaltycostfunctionand a two-dimensionalBrownianmotionproductionanddemandsetting.Forthebackloggingcase,theyshowtheoptimalpolicyisan(s;S)policy.Forthelostsalescase,theyprovethattheoptimalcontrolpolicyiseitheran(s;S)policyor neverto control. 103

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Theabovepapersonlyconsiderthecasewheretheinventorycanonlybeincreased.Whentheinventorycanbebothincreasedanddecreasedinstantaneously,problemsbecomemoredifcult.In[ 63 ],Richardrststudiesthiskindofinventorycontrol problems .Basedontheassumptionoftheexistenceofsolutionstoagroupofinequalities,heprovidessufcientconditionsfortheoptimalvaluefunctionandoptimalpolicy.In[ 24 ],ConstantinidesandRichardexplorethediscountedcostcashmanagementproblemwithlinearholding/penaltycost.Theyprovethattheoptimalpolicyisa(d;D;U;u)policy.Underthe lostsales settingwheretheinventoryisrestrictedtobenonnegative,Harrisonetal.[ 32 ]studythediscountedcostproblemwithalinearholdingcostfunctionandshowthata(0;D;U;u)policyisoptimal.Withthesamesettingof lostsales andlinearholdingcost,Ormecietal.[ 59 ]discussthelong-runaveragecostinventorycontrolproblemandprovethata(0;D;U;u)policyremainsoptimal.Recently,DaiandYao[ 25 , 26 ]solvetheproblemwithgeneralconvexholding/penaltycostunderdiscountedcostandlong-runaveragecostcriteriarespectively.Theyprovethat(d;D;U;u)policiesareoptimalforbacklogging cases and(0;D;U;u)policiesareoptimalforlost-salessetting.The(d;D;U;u)policyand(0;D;U;u)policycanbeviewedasbothsides(s;S)policy.Mostexisting workson Brownianinventorycontrolproblemsonlyconsiderlinearcontrol costs .Thereare onlya fewpapersdiscussingnon-linearcontrolcostproblems.In[ 78 ],Yaoetal.studythestochasticinventorycontrolproblemwithconcaveordering costs .Under the long-runaveragecostcriterion,theyshowthattheoptimalcontrolpolicyisdeterminedbyasinglepair(s;S).Intraditionaldiscretetimeinventorycontrol areas ,thepiecewiselinearcontrolcostproblemisalsoknownasquantitydiscountproblem.In[ 60 ]and[ 61 ],Porteusstudiesthediscretetimeinventorycontrolproblemwithaconcaveincreasingorderingcostfunction.Understrong assumptions ofdemandprocess,heproposesthatageneralized(s;S)policy,whichconsistsofasequenceofparameters(si;Si),isoptimal.However, 104

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sincethedemandisalwaysnon-negative,thereisnoadditionalwaitingregionasinourcase.To thebestof ourknowledge,thereisno existingworksolving theimpulsecontrolproblemwhenthecontrolcostispiecewiselinearconcaveunder the discountedcostcriterion,andalltheexistingBrownianinventorycontrolpoliciesare(s;S)and(d;D;U;u)typesof policies withonlyonewaitingregion. Inthisstudy,we contributeto the literaturebylling in thegapand proving anewtypeofoptimal policies which are differentfromtraditional ones . 5.2TheModelDescriptionandAssumptionsInthissection,wedescribeourproblemandtheassumptions.Considerasingleiteminventorycontrolproblemunderacontinuouslyreviewsetting.WeassumethattheunsatiseddemandisbackloggedandthecumulativedemandD(t)untiltimetfollowsadriftedBrownianmotion.(EvolutionofInventory)Thedemandfollowsa drifted Brownianmotion(BM),i.e., D(t)=D0t+B(t);(5)wherethedrifttermD0representsthedemandrateand>0representsstandardvariance,andthenoisetermB(t)isastandardBrownianmotiononthelteredprobabilityspace(;F;(Ft)t0;P).(controlCostsandOrderQuantity)Thecostofcontrolvunitsis (v)=8><>:c0+c1vfor0d;wherec0>0,c1>c2>0,andd>0, asshowninFigure 5-1 . Inaddition ,thereisa holding/penalty costh(x)whentheinventorylevelisx.We assume 105

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. . v . c(v) . d . c0 . c1 . c2 Figure5-1. PiecewiseLinearControlCosts (HoldingCostsandPenaltyCosts)Theholding/penaltycostfunctionh(x)satisesthat(a)h(x)isconvex,(b)h(x)2C2(R)exceptatonepointa,(c) h0 (x)0forxaand h0 (x)0forxa,(d)limx! h0 (x)<)]TJ /F3 11.955 Tf 9.3 0 Td[(rc1andlimx!+1h(x)=+1,and(e)h(x)ispolynomiallygrowthinx, whererrepresentsthediscountrate. The decisions areasequenceofFt)]TJ /F1 11.955 Tf 9.3 0 Td[(stoppingtimes,1;2;:::,and the correspondingFk)]TJ /F1 11.955 Tf 12.62 0 Td[(measurable order quantitiesvk,k=1;2;:::,denotedasw=((1;v1);(2;v2);:::);wherekk+1 andvk0for k=1;2;:::.WedenotethesetofallsuchdecisionsasW.LetN(t)=maxfn:ntgbethenumberofordersplaceduntiltimetandY(t)=PN(t)n=0vnbethecumulativeamountofordersuntiltimet. Nowwehavetheinventorylevel Z(t)=x)]TJ /F3 11.955 Tf 11.95 0 Td[(D(t)+Y(t);(5)whereZ(0)]TJ /F4 11.955 Tf 9.3 0 Td[()=xistheinitialinventorylevel. Fornotationconveniencewelet =)]TJ /F3 11.955 Tf 9.29 0 Td[(D0. Basedon( 5 )and( 5 ) ,wehavetheevolutionofinventoryfollows dZ(t)=8><>:dt+dB(t)fort2[0;1)[[1n=1(n;n+1)vnfort=n;forn=1;2;::: :(5) Accordingly,correspoindingtoagiveninitialinventorylevelxandancontrolpolicyw2W,theobjectivefunctionunderdiscountedcostcriteriacanbewrittenasfollows: 106

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Vw(x):=Ex"Z10e)]TJ /F5 7.97 Tf 6.59 0 Td[(rth(Z(t))dt+1Xk=1e)]TJ /F5 7.97 Tf 6.59 0 Td[(rkc(vk)#:(5) Wewanttoderiveanorderpolicyw2WtominimizeVw(x),whichisshownasfollows: V(x)=infw2WVw(x):(5) 5.3MainResultsInthissection,werstdescribetheoptimalpoliciesforthestochasticinventorycontrolproblemwithpiecewiselinearconcavecontrolcost.Ouroptimalpoliciesdependontwocasesbasedontherelationshipbetweend,c0,c1andc2. Specically,forgivenc0,c1andc2,thereexistsadasdescribedin( 5 )and( 5 ).Case1correspondstoddandCase2correspondstod>>>>>><>>>>>>>:w)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 148.51 -343.58 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d+c2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(xforxz2w(x))]TJ /F3 11.955 Tf 11.96 0 Td[(Ae 1x)]TJ /F3 11.955 Tf 11.95 0 Td[(Be)]TJ ET BT /F5 7.97 Tf 177.36 -367.49 Td[( 2xforz2s1(5)where w(x)=e1xZ1xh(y)e)]TJ /F5 7.97 Tf 6.58 0 Td[(1ydy+e)]TJ /F5 7.97 Tf 6.58 0 Td[(2xZxh(y)e2ydy=p 2+2r2;(5) 1=h)]TJ /F3 11.955 Tf 9.3 0 Td[(+p 2+2r2i=2;2=h+p 2+2r2i=2;(5)s1,S1,andK1areprovidedinTheorem 5.4.1 ,S2isdecidedin( 5 ),andz1,z2,A,andBareprovidedinTheorem 5.6 . 107

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. . x . Increase . toS2 . Wait . V(x) . z2 . z1 . Increase . toS1 . s1 . Wait . S1 . S2 Figure5-2. IllustrationofV(x)whend>d Correspondingly,theoptimalcontrolpolicyistowaitwhentheinventoryisaboves1orbetweenz2andz1,orderS1)]TJ /F3 11.955 Tf 10.86 0 Td[(xunitswithcostc0+c1(S1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)whentheinventoryisonorbelows1butonorabovez1,andorderS2)]TJ /F3 11.955 Tf 12.03 0 Td[(xunitswithcostc0+c1d+c2)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(x)]TJ /F3 11.955 Tf 11.96 0 Td[(dwhentheinventoryisonorbelowz2,asillustratedinFigure 5-2 . Itcanbeprove(see(2)inPage44and(4)inPage45of[ 31 ])thatw(x)hasthefollowingexpression: w(x)=EZ10e)]TJ /F5 7.97 Tf 6.59 0 Td[(rth(x)]TJ /F3 11.955 Tf 11.95 0 Td[(D(t))dt;(5) whoseeconomicmeaningistheexpecteddiscountedholdingcostwithoutcontrolwhenxistheinitialinventorylevel. Proof. TheproofisgiveninSection 5.5.1 . Whenthediscountpointishighenough(dd),thecontrollerorderswithasmalleramountS1)]TJ /F3 11.955 Tf 10.44 0 Td[(xd,takingadvantageofdiscountcostc2,whenthediffusionislowenoughxz2. 108

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. . x . Increase . toS2 . V(x) . z2 . Wait . z1=s1 . IncreasetoS1 . Wait . S1 . S2 Figure5-3. IllustrationofV(x)whend=d Notethatthereisawaitingregionwhentheinventorylevelxisbetweenz2andz1. Theintuitionis,sincethebasicdiffusionmightincreaseordecreasewithoutcontrol,thecontrollercanbenetfromwaitinginsteadofimmediatelyorderingasmallerquantitywithproportionalcostc1orabiggerquantitywithadditionalproportionalcostc2.Throughwaiting,thecontrollercangetmoreinformationaboutthediffusionprocessandreducethecostcausedbymakingwrongdecision. Notethatwhend=d,asillustratedinFigure 5-3 ,thetwowaitingregionscoincideinz1=s1point.Ifdcontinuesdecreaseandd<>:w(x))]TJ /F3 11.955 Tf 11.95 0 Td[(K2e)]TJ /F5 7.97 Tf 6.59 0 Td[(2xifx>s2w(S2))]TJ /F3 11.955 Tf 11.95 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 161.06 -575.78 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d+c2(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(x)ifxs2(5) 109

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. . x . Increase . toS2 . V(x) . s2 . Wait . S2 Figure5-4. IllustrationofV(x)whend
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Theorems 5.3 and 5.4 ,thenV(x)istheoptimalvaluefunctionandwisanoptimalorderpolicy. Beforewedescribingthetwotheorems,werstdene theoperatorforthestochasticdifferentialequation( 5 )as )]TJ /F3 11.955 Tf 7.32 0 Td[(V(x):= V0 (x)+1 22 V00 (x):(5) andwe stateageneralized Ito's lemma asfollows. Lemma5.1. (ExtendedIto'sformula) SupposethatV(x)2C1(R)and V00 (x)2C(R)exceptforcountablepoints.Wehavee)]TJ /F5 7.97 Tf 6.59 0 Td[(rTV(Z(T))=V(Z(0)]TJ /F4 11.955 Tf 9.29 0 Td[())+ZT0e)]TJ /F5 7.97 Tf 6.59 0 Td[(rtAV(Z(t))dt+ZT0e)]TJ /F5 7.97 Tf 6.59 0 Td[(rt V0 (Z(t))dB(t) (5)+N(T)Xn=1e)]TJ /F5 7.97 Tf 6.59 0 Td[(rn[V(Z(n)))]TJ /F3 11.955 Tf 11.96 0 Td[(V(Z(n)]TJ /F4 11.955 Tf 9.29 0 Td[())];where AV(x):=)]TJ /F3 11.955 Tf 9.3 0 Td[(rV(x)+)]TJ /F3 11.955 Tf 26.29 0 Td[(V(x): (5) Proof. TheproofisgiveninAppendix B.1 . Inaddition, fornotationbrevityandconvinenceofofanalyzingourgenerallowerboundandboundachievedtheorems,wedenetheswitchoperator LV(x)= inf v2fV(x+v)+ (v)g;(5)whereisthefeasibleorderquantityregionandc(v)isthecontrolcostwhentheinventoryisxandtheorderquantityisv. Theorem5.3. (Lowerboundforoptimalvaluefunction)Fornon-negativecostfunctitonsh(x)and(x;v),supposethatthereexistsaV(x)suchthat 111

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. . v . c(v) . c0 . c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d . d . c1 . c2 Figure5-5. PiecewiseLinearControlCostsandTwoLinearPieces 1. V(x)2C1(R)and V00 (x)2C(R)exceptforcountablepoints, 2. LV(x)V(x)forallx2R, 3. AV(x)+h(x)0foralmostallx2R, 4. ExR10e)]TJ /F5 7.97 Tf 6.59 0 Td[(rth(x)]TJ /F3 11.955 Tf 11.96 0 Td[(D(t))dt<1foranyx2Randlimx!V(x)=h(x)<1.Then,V(x)infw2WVw(x),whereVw(x)=ER10e)]TJ /F5 7.97 Tf 6.59 0 Td[(rth(Z(t))dt+E[P1i=1 (vi)]istheobjectivefunctioncorrespondingtoanyfeasiblepolicyw=((1;v1);(2;v2);:::)2W. Proof. TheproofisgiveninAppendix B.2 . Theorem5.4. (Lowerboundachieved)SupposeV(x)satisestheconditionsofTheorem 5.3 andsuchthat 5. AV(x)+h(x)=0forx2C:=fx:LV(x)>V(x)g, 6. ^k:=infft>^k)]TJ /F7 7.97 Tf 6.59 0 Td[(1: Z(t)=2C g,and ^qk isdenedsuchthat LV(Z(^-k))=V(Z(^-k)+ ^qk )+ ( ^qk ) if ^k <1and ^qk :=0if ^k =1,fork=1;2;:::.ThenV(x)=V ^w (x)=infw2WVw(x) and^w:=f^1;^2;:::;^q1;^q2;:::gisanoptimalcontrolpolicy. Proof. TheproofisgiveninAppendix B.3 . 112

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Thelowerboundtheoremsweprovidedhereareforgeneral impulsecontrol problems.Inourproblemsetting,thefeasibleorderquantityregionis :=fv:v>0g(5)and,basedonAssumption 5.2 ,thecontrolcostis (v)=8><>:c0+c1vfor0d;(5)wherexistheinventorylevelandvistheorderquantity.FromFigure 5-5 ,wenotethatthe piecewise linearconcavecontrolcost (v)istheminimumoftwolinearcontrolcostswithtwodifferentset-upcosts.Thus,wecanalsorepresent (v)inthefollowingway (v)=minfc0+c1v;c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2vg:(5)Following( 5 ),( 5 ),and( 5 ),wecanrepresent L V(x)=infv>0fV(x+v)+ (v)g (5)=infy>xfV(y)+ (y)]TJ /F3 11.955 Tf 11.95 0 Td[(x)g( 5\00017 )=infy>xfV(y)+minfc0+c1v;c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d+c2vgg=inff L1 V(x); L2 V(x)g;where L1 V(x)=infy>xfV(y)+c0+c1(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x)g;(5) and L2 V(x)=infy>xfV(y)+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x)g;(5) 113

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correspondingtotwolinearpieceswithslopesc1andc2inFigure 5-5 . Intheremainingpart ofthispaper ,weconsider( 5 )and( 5 )forourproblemsetting.ThesetC:=fx:LV(x)>V(x)gisalsoreferredascontinuationregion(see,e.g.[ 58 ]).InTheorems 5.3 and 5.4 ,basedonthedenitionofLV,thecontinuationregionactuallyimplysnoorderregionforourproblem. Inthispaper,correspondingtoCase1inwhichdd,weguessV(x)takesformin( 5 )andthecontinuationregiontakestheformC=fx:z2s1g,andcorrespondingtoCase2inwhichds2g. 5.4PreliminaryResultsInthissection,westatesomepreliminaryresultsaboutthelinear control/control/order costproblem.Theseresultscanhelpustoanalyzethe quantitydiscount/piecewiselinearconcavecontrol problem.FromFigure 5-5 ,wenotethatthepiecewiselinearconcavecontrolcost canbeviewedas thecombinationoftwolinearcontrolcostswithdifferentset-upcosts.SowerstexploretheproblemswithlinearcontrolcostintheSubsection 5.4.1 .TheninSubsection 5.4.2 ,wediscusstherelationship among parametersofthesetwoproblemswithlinearcontrolcosts.InSubsection 5.4.3 ,weprovidesomepropertiesthatweusetoverifythevaluefunctionsareoptimalinSections 5.5.1 and 5.5.3 . 5.4.1Classic(s;S)PolicyandExtensionInthissubsection,weprove lemmasandpropositions thatwillbeusedtocharacterizethesolution. Noteherethatintheexistingliterature,suchas[ 9 ],thetraditional(s;S)isprovedoptimalfortheimpulsecontrolproblemwithlinearcontrolcost.Weproveheretheexistenceofparameters(s;S;K)usingadifferentapproach,inwhichtheexploredpropertieswillbeusedinthelaterproofinSection 5.5 . Beforeweprovetheexistenceofparameters,werstintroducetwolemmastodescribetwofunctions. 114

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Lemma5.2. Fortheexpectedholdingcostfunctionw(x)denedin( 5 ),wehave w(x)2C2(R); w00 (x)>0:(5) and w0 (x)=e1xZ1xh0(y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1ydy+e)]TJ /F5 7.97 Tf 6.59 0 Td[(2xZxh0(y)e2ydy(5) Proof. TheproofisgiveninAppendix B.5 . Lemma5.3. Let f(x):=[ w0 (x)+c]= 2+w(x)+cx;(5)wherew(x)isdescribedin( 5 ),2isdescribedin( 5 ),andc>0.Wehave limx!f(x)=+1;(5) f00 (x)= w000 (x)= 2+ w00 (x)>0forx>>><>>>>:<0x0x>zf;(5) Proof. Theproofisgivenin B.6 . Fromthepropertiesofw(x)and f(x) providedin Lemmas 5.2 and 5.4 ,wecanprovethefollowingpropositionthatcanbeusedtodecidethe threshold parametersintheoptimalcontrolpolicies. 115

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Proposition5.1. Foranygivenpositiveconstantsc,^c0,andr,iflimx! h0 (x)<)]TJ /F3 11.955 Tf 9.3 0 Td[(rc,thenthereexistss,S,andKsuchthat s0;(5) w(s))]TJ /F3 11.955 Tf 11.95 0 Td[(Ke)]TJ ET BT /F5 7.97 Tf 168.58 -133.03 Td[( 2s=w(S))]TJ /F3 11.955 Tf 11.95 0 Td[(Ke)]TJ ET BT /F5 7.97 Tf 260.96 -133.03 Td[( 2S+^c0+c(S)]TJ /F3 11.955 Tf 11.95 0 Td[(s);(5) w0 (S)+K 2e)]TJ ET BT /F5 7.97 Tf 242.9 -187.33 Td[( 2S+c=0;(5) w0 (s)+K 2e)]TJ ET BT /F5 7.97 Tf 242.55 -241.62 Td[( 2s+c=0;(5) w00 (s)= 2+ w0 (s)+c<0;and w00 (S)= 2+ w0 (S)+c>0;wherew(x)isdescribedin( 5 )and 2isdescribedin( 5 ).Inaddition,sinceKisdecidedbyc2and^c0,wewriteKasK(c2;^c0)andwehave lim^c0!+1K(c2;^c0)=0:(5)Furthermore,wehave [ w0 (x)+c]=)]TJ ET BT /F3 11.955 Tf 178.51 -575.29 Td[( 2e)]TJ ET BT /F5 7.97 Tf 202.09 -570.35 Td[( 2x8>>>>>>>>>><>>>>>>>>>>:>)]TJ /F3 11.955 Tf 9.3 0 Td[(Kforx)]TJ /F3 11.955 Tf 9.3 0 Td[(Kforx>S:(5) 116

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Proof. TheproofisgiveninAppendix D.1 . Proposition 5.1 canbeusedtodecidethethresholdsparametersintheoptimalcontrolpolicies.Forexample,iftheset-upcostis^c0andthelinearcontrolcostisc,thentheparameterssandSdecidedinthispropositionarethestart-to-orderandorder-up-tothresholdsin the classic(s;S)policy ,asshowninthefollowingtheorem . Theorem5.5. Whentheset-upcostis^c0>0andthelinearcontrolcostisc>0,thevaluefunctionisV(x)=8><>:w(S))]TJ /F3 11.955 Tf 11.96 0 Td[(Ke)]TJ ET BT /F5 7.97 Tf 215.23 -188.12 Td[( 2S+^c0+c(S)]TJ /F3 11.955 Tf 11.96 0 Td[(x)ifxs;w(x))]TJ /F3 11.955 Tf 11.95 0 Td[(Ke)]TJ ET BT /F5 7.97 Tf 256.97 -212.03 Td[( 2xifx>swhere 2isdescribedin( 5 ),s,S,andKaredecidedinProposition 5.1 .Accordingly,theoptimalorderstrategyistowaittoorderuntilinventorylevelxreachesbelows,thentoordersuchthattheinventorylevelxincreasestoS,i.e.,theoptimalorderquantityisv(x)=S)]TJ /F3 11.955 Tf 11.96 0 Td[(x. FollowingTheorem 5.5 ,wecanimmediatelyobtainthefollowingcorollaryintermsofoptimalpoliciesfortwolinearcontrolcosts(c1andc2)withdifferentset-upcosts(c0vsc0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d)whichwillbereferredfrequentlylateron. Let c1;0=c0andc2;0=c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d:(5)Fori=1;2,iftheset-upcostisci;0andcontrolcostisci,thenthevaluefunctionisVi(x)=8><>:w(Si))]TJ /F3 11.955 Tf 11.96 0 Td[(Kie)]TJ ET BT /F5 7.97 Tf 207.53 -530.48 Td[( 2Si+ci;0+ci(Si)]TJ /F3 11.955 Tf 11.96 0 Td[(x)ifxsiw(x))]TJ /F3 11.955 Tf 11.96 0 Td[(Kie)]TJ ET BT /F5 7.97 Tf 254.94 -554.39 Td[( 2xifx>si;wheresi,Si,andKiaredecidedinProposition 5.1 withci;0replacing^c0andcireplacingc.TheoptimalorderpolicyistoordersuththattheinventoryincreasetoSiwhentheinventorylevelisonorbelowsianddonothingwhentheinventorylevelisabovesi. 117

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Proof. TheprooffollowsimmediatelyfromTheorem 5.5 . 5.4.2DiscussionandClassication Startingfromthissubsection,westarttoanalyzethepiecewiselinearconcavecontrolcostproblem.First,wediscusstheseparationandclassicationofcasesleadingtoTheorems 5.1 and 5.2 ,respectively.Specically,we showthatthereexistsad>0suchthatd0and^c0>0,thereexistsauniquepairof(s;S;K)satisfying( 5 ),( 5 ),and( 5 ).SoKisafunctionofcand^c0andaccordinglywewriteKasK(c;^c0).Next,basedonProposition 5.1 ,wenotethatfortherstlinearpiecewherec=c1and^c0=c0, K1=K(c1;c0)>0doesnotdependond;(5)andforthesecondlinearpiecewherec=c2and^c0=(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d, K2(d)=K(c2;c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d)>0dependsond:(5)Inthefollowing,werstanalyzehowthethreeparameters(s;S;K)decidedinProposition 5.1 dependonthecontrolcostcandtheset-upcost^c0.Thenwediscusstherelationship among K1,K2,andd. Proposition5.2. Forc,s,S,andKdescribedinProposition 5.1 ,wehaves0(c)<0,S0(c)<0,and K0 (c)<0. Proof. TheproofisgiveninAppendix D.2 . Proposition5.3. For^c0,s,S,andKdescribedinProposition 5.1 ,wehaves0(^c0)0,S0(^c0)0,and K0 (^c0)<0. Proof. TheproofisgiveninAppendix D.3 . 118

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. . d . K2(d) . d . 0 . K1 Figure5-6. IllustrationofK1,K2(d),andd NowwearereadytoprovethattheexistenceofdsuchthatK10(5)suchthat K1K1,(ii) K02 (d)<0,and(iii)limd!+1K2(d)0;(5)s3,andS3suchthat 119

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Q2(x):=[ w0 (x)+c2]=)]TJ ET BT /F3 11.955 Tf 197.94 -75.71 Td[( 2e)]TJ ET BT /F5 7.97 Tf 221.51 -70.77 Td[( 2x8>>>>>>>>>><>>>>>>>>>>:>)]TJ /F3 11.955 Tf 9.29 0 Td[(K3forx)]TJ /F3 11.955 Tf 9.29 0 Td[(K3forx>S3:(5)Next,fromK3=K(c2;c0)basedon ( 5 ) ,K1=K(c1;c0)basedon( 5 ), @K(c;^c0)=@c<0 basedonProposition 5.2 ,andc2K1:(5)Thus,whend=0,wehave K2(0)( 5\00036 )=K(c2;c0)( 5\00039 )=K3( 5\00041 )>K1:(5)For(ii),fromK2=K(c2;c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d)basedon( 5 )and @K(c;^c0)=@^c0<0 basedonProposition 5.3 ,wehave K02 (d)=@K(c2;^c0) @^c0@^c0 @d=@K(c2;c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d) @^c0(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)<0:(5)For(iii),wehavelimd!+1K2(d)( 5\00036 )=limd!+1K(c2;c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d) (5)=lim^c0!+1K(c2;^c0)( 5\00032 )=0( 5\00035 )
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Inaddition,weproveaby-productresultwhichwillbeusedlater.FromProposition 5.3 ,wehavethat @K(c;^c0)=@^c0<0 .NotingthatK3=K(c2;c0)basedontheaboveargument,K2(d)=K(c2;c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d)basedon( 5 ),andd>0andc1>c2basedonAssumption 5.2 ,wehaveK3>K2whend>0. 5.4.3SomeProperties Inthelastpartofthissection,welist propertiesofsi,Si,andKifori=1;2thatwillbeusedtodecideparametersandverifytheoptimalvaluefunctioninthenextsections.First,wedenenewfunctionsandprovidemorepropertiesinthefollowingproposition. Lemma5.4. Fori=1;2,let fi(x):=[ w0 (x)+ci]= 2+w(x)+cix;(5)wherew(x)isdescribedin( 5 ),1and2aredescribedin( 5 ),andci>0.Wehave limx!fi(x)=+1;(5) f00i (x)0forallx;(5) f00i (x)= w000 (x)= 2+ w00 (x)>0forx
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suchthattherst-orderderivativeoffi(x)is f0i (x)= w00 (x)= 2+ w0 (x)+ci8>>>><>>>>:<0x0x>zf;i;(5) Proof. TheprooffollowsfromLemma 5.3 withcireplacingc. Next,fromProposition 5.1 withci;0replacing^c0,andcireplacingc,wehave w(s1))]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 158.25 -194.08 Td[( 2s1=w(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 262.69 -194.08 Td[( 2S1+c0+c1(S1)]TJ /F3 11.955 Tf 11.96 0 Td[(s1);(5) w(s2))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 108.1 -260.32 Td[( 2s2=w(S2))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 212.54 -260.32 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(s2);(5) si0:(5) Lemma5.5. For(si;Si;Ki),i=1;2,decidedinCorollary 5.4.1 ,Q2(x)denedin( 5 ),and Q1(x):=[ w0 (x)+c1]=)]TJ ET BT /F3 11.955 Tf 271.67 -620.39 Td[( 2e)]TJ ET BT /F5 7.97 Tf 295.25 -615.46 Td[( 2x;(5) 122

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wehave Qi(x)=[ w0 (x)+ci]=)]TJ ET BT /F3 11.955 Tf 196.99 -99.62 Td[( 2e)]TJ ET BT /F5 7.97 Tf 220.56 -94.68 Td[( 2x8>>>>>>>>>><>>>>>>>>>>:>)]TJ /F3 11.955 Tf 9.29 0 Td[(Kiforx)]TJ /F3 11.955 Tf 9.29 0 Td[(Kiforx>Si;(5) Q0i (x)=f w00 (x)+ 2[ w0 (x)+ci]g=)]TJ ET BT /F3 11.955 Tf 194.22 -215.68 Td[( 2e)]TJ ET BT /F5 7.97 Tf 217.79 -210.74 Td[( 2x= f0i (x)=e)]TJ ET BT /F5 7.97 Tf 296.84 -210.74 Td[( 2x8>>>><>>>>:<0forx0forx>zf;i;(5)andespecially, Q0i (si)<0:(5) Proof. TheproofisgiveninAppendix D.5 Proposition5.5. For(si;Si;Ki),i=1;2,decidedinCorollary 5.4.1 , bydening gi(x)=w(x)+cix)]TJ /F3 11.955 Tf 11.95 0 Td[(Kie)]TJ ET BT /F5 7.97 Tf 263.93 -421.11 Td[( 2xfori=1;2;(5) andGi(x)=w(x))]TJ /F3 11.955 Tf 11.96 0 Td[(Kie)]TJ ET BT /F5 7.97 Tf 262.63 -475.4 Td[( 2xfori=1;2;(5)wehavethefollowingproperties:(i) g1(s1)=g1(S1)+c0;(5) g2(s2)=g2(S2)+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d;(5)(ii) 123

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g0i (x)= w0 (x)+ci+Ki 2e)]TJ ET BT /F5 7.97 Tf 184.47 -82.73 Td[( 2x8>>>>>>>>>><>>>>>>>>>>:>0forx0forx>Si:fori=1;2;(5) maxxSigi(x)=gi(si)andminxsigi(x)=gi(Si)fori=1;2;(5)(iii) fi(si)=gi(si)fori=1;2;(5) fi(Si)=gi(Si)fori=1;2;(5)wherefi(x)isdenedin( 5 )fori=1;2,(iv) f1(s1)=f1(S1)+c0;(5) f2(s2)=f2(S2)+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d:(5)(v) f01 (x)<0forxs1;(5) and f02 (x)<0forxs2:(5) Proof. TheproofisgiveninAppendix D.5 124

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5.5piecewiseLinearConcavecontrolCost Inthissection,wearereadytoproveTheroems 5.1 and 5.2 .Wediscussintwocase.Case1:dd,wheredisdescribedin( 5 )and( 5 );Case2:d>>><>>>>:<0x0x>zf;2.Thus,wehave Q02 (x)=( w00 (x)= 2+ w0 (x)+c2)=e)]TJ ET BT /F5 7.97 Tf 272.13 -484.7 Td[( 2x8>>>><>>>>:<0x0x>zf;2:(5)Second,basedon( 5 ),wehave Q2(s3)=Q2(S3)=)]TJ /F3 11.955 Tf 9.3 0 Td[(K3( 5\00041 )<)]TJ /F3 11.955 Tf 9.3 0 Td[(K1:(5) 125

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. . x . x0 . zf;2 . Q2(x) . )]TJ /F3 11.955 Tf 9.29 0 Td[(K3 . Q2(s3) . Q2(S3) . )]TJ /F3 11.955 Tf 9.29 0 Td[(K1 . s4 . S2 Figure5-7. IllustrationofQ2(x)andS2 Hence,becauseQ2(x)isunimodularbasedon( 5 )andthereexistpointss3suchthatQ2(s3)<)]TJ /F3 11.955 Tf 9.3 0 Td[(K1basedon( 5 ), asinFigure 5-7 , weknowthatthereexisttwopointss4andS2suchthatQ2(s4)=Q2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2=)]TJ /F3 11.955 Tf 9.3 0 Td[(K1andmoreover Q2(x)=[ w0 (x)+c2]=)]TJ ET BT /F3 11.955 Tf 196.32 -324.22 Td[( 2e)]TJ ET BT /F5 7.97 Tf 219.89 -319.29 Td[( 2x8>>>>>>>>>><>>>>>>>>>>:>)]TJ /F3 11.955 Tf 9.3 0 Td[(K1forx)]TJ /F3 11.955 Tf 9.3 0 Td[(K1forx>S2:(5)Next,weprovesomepropertiesofS2.Tobeginwith,dene g2(x)=w(x)+c2x)]TJ /F3 11.955 Tf 11.95 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 294.42 -441.13 Td[( 2x:(5)From( 5 )and( 5 ),wehave g02 (x)= w0 (x)+c2+K1 2e)]TJ ET BT /F5 7.97 Tf 233.51 -559.17 Td[( 2x8>>>>>>>>>><>>>>>>>>>>:>0forx0forx>S2:(5)WeprovidesomeotherpropertiesofS2inthefollowingproposition. 126

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Proposition5.6. UnderCase1,wehave s2s4
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. . x . V(x) . z2 . x2;1 . z1 . s1 . S1 . S2 Figure5-8. Illustrationofx2;1 Letting x2;1:=)]TJ /F3 11.955 Tf 9.3 0 Td[(d+g1(S1))]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2=(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2);(5)wehave 4U2;1(x)8>>>><>>>>:<0forx0forx>x2;1:(5) Thepositionofx2;1canbeunderstoodfromFigure 5-8 . Furthermore,weprovidesomepropertiesofx2;1inthefollowingproposition. Proposition5.7. UnderCase1,wehave S1)]TJ /F3 11.955 Tf 11.96 0 Td[(d
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andx2;1>s2whenK1=K2;(5)wheresiandKi,i=1;2,aredecidedinCorollary 5.4.1 ,S2isdenedin( 5 ),andx2;1isdenedin( 5 ). Proof. TheproofisgiveninAppendix D.7 . Existenceandpropertiesofs2 Inthispart,we deneanotherthresholds2,whichwillbeusedtoverifythevaluefunctionandpolicyareoptimal. Proposition5.8. UnderCase1,thereexistsauniques2suchthat s2)]TJ /F3 11.955 Tf 9.3 0 Td[(K1 2e)]TJ ET BT /F5 7.97 Tf 247.78 -577.45 Td[( 2s2whenK1>K2;(5)wheres2isdenedin( 5 )and( 5 ),and 129

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s2=s2whenK1=K2;(5)wheres2isdecidedinCorollary 5.4.1 . Proof. TheproofisgiveninAppendix D.9 . Existenceofz1,z2,A,andB Inthispart,westateatheoremcharacteringtheexistenceofz1,z2,A,andB. Theorem5.6. WhenK1>K2,thereexistz12(x2;1;s1),z20,andB>0suchthatthefollowinggroupofequationshold w(z1))]TJ /F3 11.955 Tf 11.96 0 Td[(Ae 1z1)]TJ /F3 11.955 Tf 11.96 0 Td[(Be)]TJ ET BT /F5 7.97 Tf 151.01 -247.62 Td[( 2z1=w(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 270.75 -247.62 Td[( 2S1+c0+c1(S1)]TJ /F3 11.955 Tf 11.95 0 Td[(z1) w0 (z1))]TJ ET BT /F3 11.955 Tf 60.08 -275.87 Td[( 1Ae 1z1+ 2Be)]TJ ET BT /F5 7.97 Tf 151.01 -271.53 Td[( 2z1=)]TJ /F3 11.955 Tf 9.29 0 Td[(c1w(z2))]TJ /F3 11.955 Tf 11.96 0 Td[(Ae 1z2)]TJ /F3 11.955 Tf 11.96 0 Td[(Be)]TJ ET BT /F5 7.97 Tf 151.01 -295.44 Td[( 2z2=w)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 272.6 -295.44 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(z2 w0 (z2))]TJ ET BT /F3 11.955 Tf 60.08 -323.68 Td[( 1Ae 1z2+ 2Be)]TJ ET BT /F5 7.97 Tf 151.01 -319.35 Td[( 2z2=)]TJ /F3 11.955 Tf 9.29 0 Td[(c2;(5)wherew(x)isdescribedin( 5 ),1and2aredescribedin( 5 ),s1andS1aredecidedinCorollary 5.4.1 ,S2isdenedin( 5 ),x2;1isdenedin( 5 ),ands2isdenedin( 5 )and( 5 ).WhenK1=K2,thereexistz1=s1>x2;1andz2=s2K1,thenthetwowaitingregionsbecomeoneandthepolicybecomestraditional(s2;S2)policy. Attheendofthissection,westateanimportantlemmathathelps toverifytheoptimalityoftheadditionalwaitingregion. 130

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Lemma5.6. UnderCase1,wehavethatforz2V(x)forx2C,(iv)AV(x)+h(x)0forx=2C,(v)AV(x)+h(x)=0forx2C,and(vi)limx!V(x)=h(x)<1. Wecanclaimthisbyexplaining theequivalencebetweenConditions(i)to(vi)andConditions1to6inTheorems 5.3 and 5.4 .First,Condition(i)guaranteesCondition1inTheorems 5.3 and 5.4 . Second,sincethecorrespondingguessedcontinuationregionC=fx:z2s1g asdiscussedinRemark 5.3 ,Conditions(ii)and(iii)guaranteeConditions2and6,andConditions(iv)and(v)guaranteeConditions3and5 131

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inTheorems 5.3 and 5.4 .Finally,Condition(vi)guaranteesCondition4inTheorems 5.3 and 5.4 . NowweproveConditions(i)to(vi)aresatised. ForCondition(i),from( 5 ),wenoticethatV(x)iscontinuousandsmoothinxifandonlyifV(x)iscontinuousandsmoothatx=s1,z1,andz2.First,basedon( 5 )and( 5 ),wehavethatV(x)iscontinueatx=s1.Second,basedon( 5 )and( 5 ),wehavethatV(x)issmoothatx=s1.Finally,from( 5 )andthefourequationsin( 5 ),wehavethatV(x)iscontinuousandsmoothatx=z1andz2. Inthefollowing,weverifyCondition(ii)byproving L V(x)=V(x)=V(S1)+(S1)]TJ /F3 11.955 Tf 11.96 0 Td[(x)forz1xs1inProposition 5.11 and L V(x)=V(x)=V)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2+)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(xforxz2inProposition 5.13 .WeverifyCondition(iii)byproving L V(x)>V(x)forx>s1inProposition 5.10 and L V(x)>V(x)forz2V(x)forx>s1;wheres1isdecidedinCorollary 5.4.1 . Proof. TheproofisgiveninAppendix D.11 . Proposition5.11. UnderCase1,forthefunctionsV(x)denedin( 5 )and(v)describedin( 5 ),andtheoperator L denedin( 5 ),wehave L V(x)=V(x)=V(S1)+(S1)]TJ /F3 11.955 Tf 11.96 0 Td[(x)forz1xs1;wheres1andS1aredecidedinCorollary 5.4.1 ,andz1isdecidedinTheorem 5.6 . Proof. TheproofisgiveninAppendix D.12 . Proposition5.12. UnderCase1,forthefunctionV(x)denedin( 5 )andtheoperator L denedin( 5 ),wehave L V(x)>V(x)forz2
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Proposition5.13. UnderCase1,forthefunctionsV(x)denedin( 5 )and(v)describedin( 5 ),andtheoperator L denedin( 5 ),wehave L V(x)=V(x)=V)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2+)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(xforxz2;whereS2isdescribedin( 5 )andz2isdecidedinTheorem 5.6 . Proof. TheproofisgiveninAppendix D.14 . Proposition5.14. UnderCase1,forthefunctionV(x)denedin( 5 )andtheoperator A denedin( 5 ),wehaveAV(x)+h(x)0forxs1wherez2isdecidedinTheorem 5.6 ,andlimx!1V(x)=h(x)<1. Proof. TheproofisgiveninAppendix D.15 . 5.5.2OptimalWaitingRegionParametersThissectionprovidestheproofforTheorem 5.6 .Sincetheproofislong,weseparateitintoaseriesofpropositionsandlemmas. Inourapproach, werstsimplifythefourequations in( 5 ) thatcontainA,B,z1,andz2totwoequationsthatonlycontainz1andz2. Thenweshowtheexistenceofz1andz2basedontheobtainedtwoequations. 5.5.2.1ReducefourequationstotwoequationsWerstwritethefourequationsin( 5 )tothefollowingfourequivalentequations( 5 )to( 5 ) byreorganizingterms : 133

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Ae 1z1+Be)]TJ ET BT /F5 7.97 Tf 106.13 -42.88 Td[( 2z1=w(z1))]TJ /F9 11.955 Tf 11.96 9.69 Td[(w(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 262.14 -42.88 Td[( 2S1+c0+c1(S1)]TJ /F3 11.955 Tf 11.96 0 Td[(z1) (5)( 5\00062 )=w(z1)+c1z1)]TJ /F3 11.955 Tf 11.95 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(c0; 1Ae 1z1)]TJ ET BT /F3 11.955 Tf 73.06 -101.61 Td[( 2Be)]TJ ET BT /F5 7.97 Tf 106.13 -96.68 Td[( 2z1= w0 (z1)+c1; (5)Ae 1z2+Be)]TJ ET BT /F5 7.97 Tf 106.13 -123.57 Td[( 2z2=w(z2))]TJ /F9 11.955 Tf 11.96 13.27 Td[(hw)]TJ /F4 11.955 Tf 7.48 -6.67 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 264.66 -123.57 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2)]TJ /F4 11.955 Tf 7.48 -6.67 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(z2i (5)( 5\00077 )=w(z2)+c2z2)]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d; 1Ae 1z2)]TJ ET BT /F3 11.955 Tf 73.06 -208.8 Td[( 2Be)]TJ ET BT /F5 7.97 Tf 106.13 -203.86 Td[( 2z2= w0 (z2)+c2: (5)Next,wesimplifythefourequationstogettwoequationsthatonlycontainz1andz2.WebeginwithcharacterizingAandBbasedonz1andz2.From( 5 ),wehave B= 1Ae 1z1e 2z1= 2)]TJ /F4 11.955 Tf 11.95 0 Td[([ w0 (z1)+c1]e 2z1= 2:(5)Substituting( 5 )into( 5 ),wehave A=[ 2=( 1+ 2)]fw(z1)+c1z1)]TJ /F3 11.955 Tf 11.96 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(c0+[ w0 (z1)+c1]= 2ge)]TJ ET BT /F5 7.97 Tf 387.37 -419.93 Td[( 1z1( 5\00045 )=[ 2=( 1+ 2)][f1(z1))]TJ /F3 11.955 Tf 11.96 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(c0]e)]TJ ET BT /F5 7.97 Tf 275.46 -443.84 Td[( 1z1:(5)Substituting( 5 )into( 5 ),wehaveanotherexpressionofBasfollows B=[ 1 2=( 1+ 2)]fw(z1)+c1z1)]TJ /F3 11.955 Tf 11.96 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(c0+[ w0 (z1)+c1]= 2ge 2z1= 2)]TJ /F4 11.955 Tf 11.29 0 Td[([ w0 (z1)+c1]e 2z1= 2:=[ 1=( 1+ 2)]fw(z1)+c1z1)]TJ /F3 11.955 Tf 11.96 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[([ w0 (z1)+c1]= 1ge 2z1:(5)Nowwestate anotherwaytocharacterAandBbasedonz1andz2. From( 5 ),wehave 134

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B= 1Ae 1z2e 2z2= 2)]TJ /F4 11.955 Tf 11.95 0 Td[([ w0 (z2)+c2]e 2z2= 2:(5)Substituting( 5 )into( 5 ),wehave A=[ 2=( 1+ 2)]w(z2)+c2z2)]TJ /F4 11.955 Tf 12.38 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(c0)]TJ /F4 11.955 Tf 11.95 0 Td[((c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d+[ w0 (z2)+c2]= 2e)]TJ ET BT /F5 7.97 Tf 442.86 -115.5 Td[( 1z2( 5\00045 )=[ 2=( 1+ 2)]f2(z2))]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(c0)]TJ /F4 11.955 Tf 11.95 0 Td[((c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)de)]TJ ET BT /F5 7.97 Tf 332.42 -139.41 Td[( 1z2:(5)Substituting( 5 )into( 5 ),wehaveanotherexpressionofBasfollows B=[ 1 2=( 1+ 2)]w(z2)+c2z2)]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+[ w0 (z2)+c2]= 2e 2z2= 2:)]TJ /F4 11.955 Tf 11.29 0 Td[([ w0 (z2)+c2]e 2z2= 2:=[ 1=( 1+ 2)]w(z2)+c2z2)]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d)]TJ /F4 11.955 Tf 11.95 0 Td[([ w0 (z2)+c2]= 1e 2z2:(5)Thus,ifweknowthevaluesofz1andz2,wecangetthevaluesofAandBbasedon( 5 )and( 5 )or( 5 )and( 5 ).Intheremainingpartofthissubsection,weusetheaboverelationshipstoreducetheoriginalfourequationstotwoequationsthatonlycontainz1andz2.Fortherstequation,from( 5 )and( 5 ),wehave [f1(z1))]TJ /F3 11.955 Tf 11.95 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(c0]e)]TJ ET BT /F5 7.97 Tf 153.22 -477.24 Td[( 1z1=f2(z2))]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)de)]TJ ET BT /F5 7.97 Tf 382.54 -477.24 Td[( 1z2(5)Forthesecondequation,from( 5 )and( 5 ),wehave fw(z1)+c1z1)]TJ /F3 11.955 Tf 11.95 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(c0)]TJ /F4 11.955 Tf 11.95 0 Td[([ w0 (z1)+c1]= 1ge 2z1=w(z2)+c2z2)]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d)]TJ /F4 11.955 Tf 11.95 0 Td[([ w0 (z2)+c2]= 1e 2z2(5) 135

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Finally,weonlyneedtoprovethatthereexistx2;1
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Next,wediscussthesignoftheleftsideminustherightsideof( 5 )alongtheline(z;(z)).Letthedifferencebetweentheleftsideandtherightsideof( 5 )alongtheline(z;(z))g(z):=fw(z)+c1z1)]TJ /F3 11.955 Tf 11.95 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(c0)]TJ /F4 11.955 Tf 11.95 0 Td[([ w0 (z)+c1]= 1ge 2z (5))]TJ /F9 11.955 Tf 11.96 9.69 Td[(w((z))+c2z2)]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.67 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(c0)]TJ /F4 11.955 Tf 11.95 0 Td[((c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d)]TJ /F4 11.955 Tf 11.96 0 Td[([ w0 ((z))+c2]= 1e 2(z):Inthefollowing,weprovethatg(z)hasonlyonezeropointinz2(x2;1;s1].Wediscusstheprobleminthreecases:Case1.0:K1=K2; (5)Case1.1:K1>K2ands2x2;1; (5)Case1.2:K1>K2ands2>x2;1: (5)ForCases1.1and1.2,weprovethat g0 (z)>0forz2(x2;1;s1]inLemma 5.8 ,g(x2;1)<0inLemma 5.9 ,andthatthereexistsazr2(x2;1;s1]suchthatg(zr)>0inLemma 5.10 .Accordingly,thereshouldexistauniquez1suchthat x2;1K2:(5)Bydening z2=(z1);(5)wehave( 5 )and( 5 )hold.Inaddition,wehavez2
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ForCase1.0whereK1=K2,theproofissimilarwithabove.Wehaveg(s1)=0inLemma 5.10 .Since g0 (z)>0forz2(x2;1;s1]basedonLemma 5.8 ,thezeropointofg(x)isuniqueandso z1=s1whenK1=K2:(5)Bydening z2=(z1);(5)wehave( 5 )and( 5 )hold.Inaddition,wehavez2=s2inLemma 5.11 .ThisnishestheproofforTheorem 5.6 . Lemma5.8. Forthefunctionsg(x)denedin( 5 )and(x)denedin( D ),wehave 0 (z)>0and g0 (z)>0forz2(x2;1;s1],wherex2;1isdenedin( 5 )ands1isdecidedinCorollary 5.4.1 . Proof. TheproofisgiveninAppendix D.17 . Lemma5.9. Forx2;1denedin( 5 )andthefunctiong(x)denedin( 5 ),wehaveg(x2;1)<0. Proof. TheproofisgiveninAppendix D.18 . Lemma5.10. Forthefunctionsg(x)denedin( 5 )and(x)denedin( D ),s1decidedinCorollary 5.4.1 ,andx2;1denedin( 5 ),wehaveg(s1)=0forCase1.0,g(s1)>0forCase1.1,andg(zr)>0forCase1.2,wherezr2(x2;1;s1]isdenedsuchthat(zr)=x2;1. Proof. TheproofisgiveninAppendix D.19 . Lemma5.11. Forx2;1denedin( 5 ),s2denedin( 5 )and( 5 ),andz1andz2denedin( 5 )and( 5 ),wehavez2
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Proof. TheproofisgiveninAppendix D.20 . 5.5.3Case2:ds2gunderCase1asdiscussedinRemark 5.3 .AsdiscussedinSubsection 5.5.1 ,inordertoshowthatthevaluefunctionandcontrolpolicyareoptimal,itsufcestoverifythatthefollowingconditionsaresatised:(i)V(x)iscontinuousandsmooth,(ii)LV(x)=V(x)forx=2CandLV(x)=V(S2)+(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(x),(iii)LV(x)>V(x)forx2C,(iv)AV(x)+h(x)0forx=2C,(v)AV(x)+h(x)=0forx2C,and(vi)limx!V(x)=h(x)<1.TheequivalencebetweenConditions(i)to(vi)andConditions1to6inTheorems 5.3 and 5.4 hasbeenexplainedinSection 5.5.1 . 5.5.3.1PreliminaryResultsInthissubsection,westatesomepropertiesthatcanbeusedtoverifyConditions(i)to(vi).Notethatbasedon( 5 ),wehave L V(x)=inff L1 V(x); L2 V(x)g,whereL1V(x)andL2V(x)aredenedin( 5 )and( 5 ).Thus,inordertocompare L V(x)andV(x)toverifyConditions(ii)and(iii),weneedtocomputeandcompare L1 V(x)and L2 V(x)rst. Intuitively,LiV(x)istheminimumsummationvalueofVi(y)andthecontrol/controlcostsofincreasingthediffusion/inventorylevelfromxtoy.Thecontrol/controlcostsincludeaconstantset-upcostci;0,withci;0describedin( 5 ),andalinearcontrolcostwithslopecifori=1;2. ExistenceofS1 InthispartwedeneS1andshowsomeresultsthatwillbeusedtocompute L1 V(x).From( 5 ),wehave 139

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w0 (x)+c1+K1 2e)]TJ ET BT /F5 7.97 Tf 212.59 -70.77 Td[( 2x8>>>>>>>>>><>>>>>>>>>>:>0forx0forx>S1:Hence,withinCase2whereK1>>>>>>>>><>>>>>>>>>>:>0forx0forx>S1:(5)Inordertoprove( 5 ),itsufcestoshowthatthereexisttwopointss5andS1suchthat[ w0 (x)+c1]=)]TJ ET BT /F3 11.955 Tf 140.34 -549.5 Td[( 2e)]TJ ET BT /F5 7.97 Tf 163.92 -545.16 Td[( 2x8>>>>>>>>>><>>>>>>>>>>:>)]TJ /F3 11.955 Tf 9.3 0 Td[(K2forx)]TJ /F3 11.955 Tf 9.3 0 Td[(K2forx>S1.NotethatQ1(x)=[ w0 (x)+c1]=)]TJ ET BT /F3 11.955 Tf 73.25 -617.44 Td[( 2e)]TJ ET BT /F5 7.97 Tf 96.82 -613.1 Td[( 2xasdenedin( 5 ),itsufcestoshowthatthereexisttwopointss5andS1suchthat 140

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Q1(x)8>>>>>>>>>><>>>>>>>>>>:>)]TJ /F3 11.955 Tf 9.3 0 Td[(K2forx)]TJ /F3 11.955 Tf 9.3 0 Td[(K2forx>S1:(5)Tostartwith,notingthatbasedon( 5 )and( 5 ),wehavetherstderivativeofQ1(x) Q01 (x)=( w00 (x)= 2+ w0 (x)+c1)=e)]TJ ET BT /F5 7.97 Tf 273.12 -231.43 Td[( 2x8>>>><>>>>:<0x0x>zf;1;(5)wherezf;1isdescrbedinLemma 5.4 .Moreover,basedon( 5 )and( 5 ),wehave Q1(^x0)<)]TJ /F3 11.955 Tf 9.3 0 Td[(K2:(5)Thus,basedon( 5 )and( 5 ),wehavethatthereexisttwopointss5andS1suchthat( 5 )holds. SinceCase2.1isrelativelyeasytoverify,inthefollowing,weprovidesomepreliminaryresultsunderCase2.2. Denitionofx2;1 Inthispart,wedeneathresholdx2;1thatwillbeusedtocompare L1 (x)and L2 (x)underCase2.2.Tobeginwith,dene g1(x):=w(x)+c1x)]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 296.04 -598.32 Td[( 2x;(5) 141

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U1(x):=w)]TJ /F4 11.955 Tf 7.48 -6.67 Td[(S1)]TJ /F3 11.955 Tf 11.95 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 240.12 -42.88 Td[( 2S1+c0+c1)]TJ /F4 11.955 Tf 7.47 -6.67 Td[(S1)]TJ /F3 11.955 Tf 11.96 0 Td[(x (5)( 5\000128 )=g1)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S1+c0)]TJ /F3 11.955 Tf 11.95 0 Td[(c1x;andU2(x):=w(S2))]TJ /F3 11.955 Tf 11.95 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 198.27 -141.5 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d+c2(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(x) (5)( 5\00062 )=g2(S2)+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d)]TJ /F3 11.955 Tf 11.96 0 Td[(c2x:InordertocompareU1(x)andU2(x),let4U2;1(x):=U2(x))]TJ /F4 11.955 Tf 13.95 3.02 Td[(U1(x) (5)=g2(S2))]TJ /F4 11.955 Tf 12.37 0 Td[(g1)]TJ /F4 11.955 Tf 7.48 -6.67 Td[(S1+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)x:Letting x2;1:=)]TJ /F3 11.955 Tf 9.3 0 Td[(d+g1)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S1)]TJ /F3 11.955 Tf 11.96 0 Td[(g2(S2)=(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2);(5)wehave 4U2;1(x)8>>>><>>>>:<0forx0forx>x2;1:(5)PropertiesofS1andx2;1 Inthispart,weprovidesomepropertiesofS1andx2;1inthefollowingpropositionsthatwillbeusedtoverifyConditions(ii)and(iii)underCase2.2. Proposition5.15. UnderCase2.2,wehave s2
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g1(x)s2anddd.Therefore,whentheinventoryxisonorbelows2,controlS2)]TJ /F3 11.955 Tf 12.14 0 Td[(xwithsomediscount(i.e.,withcostc0+c1d+c2(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(x)]TJ /F3 11.955 Tf 11.95 0 Td[(d))becomespossible. 5.5.3.2VerifyConditionsInthissubsection,weverifyConditions(i)to(vi).ForCondition(i),from( 5 ),wenotethatV(x)issmoothinxifandonlyifV(x)issmoothatx=s2,(i.e.,V(x)iscontinuousatx=s2anditsleftandrightderivativesareequalatx=s2),whichrequiresw(s2))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 127 -565 Td[( 2s2=w(S2))]TJ /F3 11.955 Tf 11.95 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 231.44 -565 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(s2)and w0 (s2)+K2 2e)]TJ ET BT /F5 7.97 Tf 108.29 -601.46 Td[( 2s2=)]TJ /F3 11.955 Tf 9.3 0 Td[(c2:Andthesetwoequationsholdbasedon( 5 )and( 5 )inProposition 5.1 . 143

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Inthefollowing,weverifyCondition(ii)inProposition 5.17 ,Condition(iii)inProposition 5.18 ,andConditions(iv)(vi)inProposition 5.19 . Proposition5.17. UnderCase2,forthefunctionV(x)denedin( 5 )andtheoperator L denedin( 5 ),wehave L V(x)=V(x)=V(S2)+(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(x)forxs2,wheres2andS2aredecidedinCorollary 5.4.1 . Proof. TheproofisgiveninAppendix D.24 . Proposition5.18. UnderCase2,forthefunctionV(x)denedin( 5 )andtheoperator L denedin( 5 ),wehave L V(x)>V(x)forx>s2,wheres2isdecidedinCorollary 5.4.1 . Proof. TheproofisgiveninAppendix D.23 . Proposition5.19. UnderCase2,forthefunctionV(x)denedin( 5 )andtheoperatorAdenedin( 5 ),wehaveAV(x)+h(x)0forxs2,wheres2isdecidedinCorollary 5.4.1 .Inaddition,wehavelimx!V(x)=h(x)<1. Proof. TheproofisgiveninAppendix D.25 . 5.6ConcludingRemarkInthischapter,wediscussthestochasticinventorycontrolproblemwithBrowniandemandandpiecewiselinearconcavecontrolcost.Wederiveclosedformoptimalpolicytominimizethediscountedtotalcost.Weshowthat,dependingonthebasicparameters,theoptimalpolicyiseitheran(s;S)policyoranewtypepolicywhichisdifferent from traditionalpolicies.Thenewpolicylookslike acombinationof two(s;S)policieswithanadditionalwaitingarea.Forfutureresearch,it willbe interestingtoexploretheproblemunderlong-runaveragecostsettingorstudytheproblemwhentheinventorycanalsobedecreasedbythecontroller. 144

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CHAPTER6CONCLUSIONSANDFUTURERESEARCHInthisdissertation,westudiedoptimalPHEVbatterycontrolproblems fordifferent electricitymarkets.Intherstpartofthisdissertation,weconsideredseveraldiscretenite-horizonmodelsincludingmarketimpact,aggregator'sriskattitude,andparticipatinginboththereal-timeandday-aheadmarkets.Throughconcavityanalysis,weprovedtheexistenceoftheoptimalpolicyandanalyzedtheoptimalpolicystructure.Furthermore,basedonthederivedtheoreticalresultsandnumericalstudiesperformed,wediscussedhowtocontrolPHEVsoptimally,foresawhowPHEVswouldaffectthemarket,andprovidedsuggestionsforownersofPHEVsonhowtousePHEVsbetter.Inparticular,weexploredthefollowingtwoinsights.First,throughnumericalstudies,weshowedthat,withtheconsiderationofmarketimpact,anaggregatoruseslesselectricitystorageandarisk-averseaggregatorusesevenless.Ontheotherhand,toomanyindividualactivitiesmightincreasereal-timepricevolatilityandfurthermakereal-timepriceuctuateintenselyinthecurrentmarketmechanism.Soallowingoneortwoaggregators,withtheconsiderationofmarketimpact,tocontroltheelectricitystorageisbenecialtothesystem.Second,whentheelectricitystoragecapacityissufcientlylarge,participatingonlyinthereal-timemarketis neither optimalforaggregators norfor thesystemoperator.Largeelectricitystorageownersshouldparticipateinboththeday-aheadandreal-timemarkets.Inthesecondpartofthisdissertation,inordertooptimallycontrolthePHEVbatterystorage for electricityreserve markets ,we studied generalimpulsecontrolproblemswithapplicationsininventorymanagement.We started withthestochasticinventorycontrolproblemwhenthecontrolquantityhastobeintegraltimesof a basequantity.We discussed twocases withtherstin 145

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which theinventorycanonlybeincreasedand thesecondinwhich theinventorycanbe either increased or decreased.Ineachcase,we provided optimal policies inclosed forms andthe expressions oftheoptimalvaluefunction.Next,we discussed theBrownianinventorycontrolproblemwithpiecewiselinearconcavecontrol costs .Wederiveclosedformoptimal policies tominimizethediscountedtotalcost.We showed that,dependingonthebasicparameters,theoptimalpolicyiseitheran(s;S)policyoranewtype ofpolicies which are differentfromtraditional ones .Thenewpolicylookslike acombinationof two(s;S)policieswithanadditionalwaitingarea.Infutureresearch,wewillexplorehowtocontrolPHEVsoptimallywhentherearecompetitorsandhowcompetitioninuencesthesystem.Inaddition,wewilldevelopefcientapproximatedynamicprogrammingalgorithms(cf.[ 62 ])tosolvelarge-scaleproblems.Moreover,itisinterestingtoexplorethestochasticcontrolproblemwithpiecewiselinearconcavecontrolcost where theinventorycanalsobe adjusteddownwards . 146

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APPENDIXAPROOFS Inthisappendix,fornotationbrevity,weusex,y,andztoreplacexn,yn,andzn. A.1ProofofProposition 2.1 (a)Letthechargingamountz=y)]TJ /F3 11.955 Tf 11.21 0 Td[(x.Fromequation( 2 ),theprotfunctionofthecurrentperiodisLn(z;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)=8><>:)]TJ /F3 11.955 Tf 9.3 0 Td[(dz(Epnj~pn)]TJ /F10 5.978 Tf 5.75 0 Td[(m[pn]+bndz)forz0)]TJ /F5 7.97 Tf 13.53 4.7 Td[(z u(Epnj~pn)]TJ /F10 5.978 Tf 5.75 0 Td[(m[pn]+bnz u)forz0TherstorderderivativeofLn(z;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)withrespecttozis @Ln(z;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m) @z=8><>:)]TJ /F4 11.955 Tf 9.3 0 Td[(2bn2dz)]TJ /F3 11.955 Tf 11.96 0 Td[(dEpnj~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m[pn]forz<0)]TJ /F7 7.97 Tf 10.5 4.7 Td[(2bn 2uz)]TJ /F7 7.97 Tf 16.2 4.7 Td[(1 uEpnj~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m[pn]forz>0:(A)ThesecondorderderivativeofLn(z;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)withrespecttozis@2Ln(z;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m) @z2=8><>:)]TJ /F4 11.955 Tf 9.3 0 Td[(2bn2dforz<0)]TJ /F4 11.955 Tf 9.3 0 Td[(2bn=2uforz>0:Wealsonotice that @Ln(z;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m) @zjz=0)]TJ /F4 11.955 Tf 10.41 1.79 Td[(=)]TJ /F3 11.955 Tf 9.29 0 Td[(dEpnj~pn)]TJ /F10 5.978 Tf 5.75 0 Td[(m[pn])]TJ /F4 11.955 Tf 26.61 8.09 Td[(1 uEpnj~pn)]TJ /F10 5.978 Tf 5.75 0 Td[(m[pn]=@Ln(z;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m) @zjz=0+:Thus,wecanobservethatinfz:z<0gandfz:z>0g,thesecondorderderivativesofLn(z;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)withrespecttozarestrictlynegative.Togetherwiththefactthatatz=0theleftderivativeofLn(z;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)isnotsmallerthantherightderivativeofLn(z;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m),wehavethatLn(z;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)isstrictlyconcaveinz.Nowweconsiderthemyopicoptimalsolutionforthecurrentperiod.Sincethereisonlyoneperiod,chargingisneveranoptimaldecision.Sowithoutanyconstraint,setting@Ln(z;~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m) @z=)]TJ /F4 11.955 Tf 9.3 0 Td[(2bn2dz)]TJ /F3 11.955 Tf 12.06 0 Td[(dEpnj~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m[pn]=0providestheunconstrainedglobalmaximizer 147

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ofLn(z;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m),z4=)]TJ /F5 7.97 Tf 10.5 8.03 Td[(Epnj~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m[pn] 2bnd.Thus,thecorrespondingtargetstoragelevelwithoutanyconstraint,ymyop;4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)=x+z4=x)]TJ /F5 7.97 Tf 13.15 8.03 Td[(Epnj~pn)]TJ /F10 5.978 Tf 5.75 0 Td[(m[pn] 2bnd.NotethatLn(z;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)isstrictlyconcaveinz,whichindicatesthatLn(y)]TJ /F3 11.955 Tf 12.15 0 Td[(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)isstrictlyconcaveinyforanygivenx.Hence,wehavethemyopicoptimaltargetstoragelevelwithinthefeasiblesetymyopn(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)=8><>:(x)]TJ /F3 11.955 Tf 11.96 0 Td[(Cd)+ifymyop;4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)<(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Cd)+ymyop;4n(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)if(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Cd)+ymyop;4n(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m):Thatisymyopn(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)=maxnx)]TJ /F3 11.955 Tf 11.96 0 Td[(Cd;x)]TJ /F5 7.97 Tf 13.15 8.04 Td[(Epnj~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m[pn] 2bnd;0o.(b)Theproofisbyinduction.Theclaimholdstriviallyforn=N+1becauseVN+1(x; ~pN)]TJ /F5 7.97 Tf 6.58 0 Td[(m+1 )isconcavelyincreasinginxinoursetting.Nowassumethattheclaimholdsforperiodn+1,i.e.,Vn+1(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1)isconcaveinx.From(a),Ln(z;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)isstrictlyconcaveinz,implyingthatLn(y)]TJ /F3 11.955 Tf 12.91 0 Td[(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)isjointlyconcavein(x;y)andstrictlyconcaveinyforanygivenx.Thus,Jn(x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)=Ln(y)]TJ /F3 11.955 Tf 13.45 0 Td[(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)+Epn)]TJ /F10 5.978 Tf 5.76 0 Td[(m+1j~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m[Vn+1(y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1)]isjointlyconcavein(x;y)andstrictlyconcaveinyforanygivenx.Furthermore,Vn(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)=maxy2A(x)fJn(x;y;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)gisconcaveinxsincef(x;y):0xC;y2A(x)=[(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Cd)+;minfC;x+Cug]gisaconvexset,whichcompletestheinductionstep.Sinceforanygivenx,Jn(x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)isstrictlyconcaveinyandthefeasiblesetisaboundedinterval,thereexistsauniqueoptimalsolutionyn(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m).(c)WeprovethatVn(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)non-decreasesinxbyinduction.NotethatVN+1(x)non-decreasesinxbasedonourinitialsetting.Letusassumetheclaimholdsforperiodn+1.Inthefollowing,weprovethatforanypair(x1;x2)suchthat0x1
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Certainlyy12A(x1).Nowwediscusstwocasestoproveourclaimbasedontherangeofx2.Notethaty1+x2)]TJ /F3 11.955 Tf 11.95 0 Td[(x10becausey10andx2>x1.Ify1+x2)]TJ /F3 11.955 Tf 11.96 0 Td[(x1C ,lety2:=y1+x2)]TJ /F3 11.955 Tf 12.12 0 Td[(x1.Forthiscase,itiseasytoobservethaty2isfeasibleandy2y1.Thenbyinductionassumption,wehave Vn+1(y1;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1)Vn+1(y2;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1):(A)Thus,Vn(x1;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)=Ln(y1)]TJ /F3 11.955 Tf 11.96 0 Td[(x1;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)+Epn)]TJ /F10 5.978 Tf 5.76 0 Td[(m+1j~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m[Vn+1(y1;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m+1)]Ln(y2)]TJ /F3 11.955 Tf 11.95 0 Td[(x2;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)+Epn)]TJ /F10 5.978 Tf 5.75 0 Td[(m+1j~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m[Vn+1(y2;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1)]Vn(x2;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m);wheretherstequalityholdsbecauseoftheoptimalityofy1,i.e.,equation( A )andtherstinequalityholdsbecauseoftheinductionassumption,i.e.,equation( A ),andy1)]TJ /F3 11.955 Tf 11.95 0 Td[(x1=y2)]TJ /F3 11.955 Tf 11.95 0 Td[(x2.Ify1+x2)]TJ /F3 11.955 Tf 11.96 0 Td[(x1>C ,lety2=C.Forthiscase,becausey1+x2)]TJ /F3 11.955 Tf 12.37 0 Td[(x1>C=y2,wehavey2)]TJ /F3 11.955 Tf 11.94 0 Td[(x20asshownin( A ).Fromthefeasibilityofy1,wehavey1C=y2,andbyinductionassumption,wehave Vn+1(y1;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1)Vn+1(y2;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1):(A)Thus,Vn(x1;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)=Ln(y1)]TJ /F3 11.955 Tf 11.96 0 Td[(x1;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)+Epn)]TJ /F10 5.978 Tf 5.76 0 Td[(m+1j~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m[Vn+1(y1;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m+1)]Ln(y2)]TJ /F3 11.955 Tf 11.95 0 Td[(x2;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)+Epn)]TJ /F10 5.978 Tf 5.75 0 Td[(m+1j~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m[Vn+1(y2;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1)]Vn(x2;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m); 149

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wheretherstequalityholdsbecauseoftheoptimalityofy1,i.e.,equation( A ),andtherstinequalityholdsbecauseofequations( A )and( A ).Thiscompletestheproof. A.2ProofofTheorem 2.1 Notethatforanygiven xnand~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m ,theoptimaltargetstoragelevelyn(xn ;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m )isunique.Inaddition,wehavexn+1=yn(xn ;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m ).Therefore,oncex1 and~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(mare given, x2 =y1(x1 ;~p1)]TJ /F5 7.97 Tf 6.58 0 Td[(m ), x3=y2(x2 ;~p2)]TJ /F5 7.97 Tf 6.58 0 Td[(m )=y2(y1(x1 ;~p1)]TJ /F5 7.97 Tf 6.59 0 Td[(m ); ~p2)]TJ /F5 7.97 Tf 6.59 0 Td[(m ) ,:::,andxn= yn)]TJ /F7 7.97 Tf 6.58 0 Td[(1(yn)]TJ /F7 7.97 Tf 6.59 0 Td[(2(:::y1(x1;~p1)]TJ /F5 7.97 Tf 6.58 0 Td[(m);:::;~pn)]TJ /F7 7.97 Tf 6.58 0 Td[(2)]TJ /F5 7.97 Tf 6.58 0 Td[(m); ~pn)]TJ /F7 7.97 Tf 6.59 0 Td[(1)]TJ /F5 7.97 Tf 6.59 0 Td[(m)areuniquelydecidedforn=2;:::;N+1. A.3ProofofProposition 2.2 (a)NoticethatVn+1(y;~pn+1)]TJ /F5 7.97 Tf 6.59 0 Td[(m)andJn(x;y;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)arebothconcaveiny,sotheirleftandrightderivativesatyalwaysexist.Fornotationbrevity,wewriteEpn)]TJ /F10 5.978 Tf 5.76 0 Td[(m+1j~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m[Vn+1(y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1)]as~Vn+1(y).Let~V0n+1;l(y)and~V0n+1;r(y)denotetheleftandrightderivativesof~Vn+1(y)atyrespectively.Inthefollowing,werstprovethattheglobalmaximizerofJn(x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)withoutanyconstraint, denotedas y4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m),non-decreasesinx.Foranyx1andx2suchthat0x1
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andtherightderivativeofJn(x1;y;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)withrespecttoyattheunconstrainedglobalmaximizery4n(x1;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)shouldbenon-positive,i.e., @Jn(x1;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m) @yjy=y4n(x1;~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m)+=@Ln(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x1;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m) @yjy=y4n(x1;~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m)++~V0n+1;l(y4n(x1;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m))0: (A)FromProposition 2.1 (a),Ln(z;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)isstrictlyconcaveinz,sotheleftderivative@Ln(z;~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m) @zjz)]TJ /F1 11.955 Tf 10.41 1.8 Td[(decreasesinz.Togetherwiththefactthaty)]TJ /F3 11.955 Tf 11.95 0 Td[(x2@Ln(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x1;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m) @yjy=y4n(x1;~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m))]TJ /F3 11.955 Tf 7.08 3.94 Td[(:(A)SotheleftderivativeofJn(x2;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)withrespecttoyaty4n(x1;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)satisesthat@Jn(x2;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m) @yjy=y4n(x1;~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m))]TJ /F4 11.955 Tf 10.41 3.94 Td[(=@Ln(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x2;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m) @yjy=y4n(x1;~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m))]TJ /F4 11.955 Tf 9.74 3.94 Td[(+~V0n+1;l(y4n(x1;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m))>@Ln(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x1;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m) @yjy=y4n(x1;~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m))]TJ /F4 11.955 Tf 9.74 3.94 Td[(+~V0n+1;l(y4n(x1;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m))0;wheretheequalityholdsbecauseofthedenitionofJn(x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m);therstinequalityholdsbecauseof( A ),andthelastinequalityholdsbecauseof( A ).TogetherwiththeconcavityofJn(x2;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)iny,wehavey4n(x2;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)y4n(x1;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)andsoy4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)non-decreasesinx.NotethatthemaximizerofJn(x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)withinthefeasiblesetA(x), denotedas yn(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m),istheclosestpointtotheglobalmaximizerofJn(x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)withoutanyconstraint,y4n(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m),limitedtobebetween theupperandlowerlimits ,i.e., yn(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)=8>>>><>>>>:(x)]TJ /F3 11.955 Tf 11.96 0 Td[(Cd)+ify4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)<(x)]TJ /F3 11.955 Tf 11.96 0 Td[(Cd)+y4n(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)if(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Cd)+y4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)minfx+Cu;Cgminfx+Cu;Cgify4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)>minfx+Cu;Cg:(A) 151

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Sincethelowerbound,(x)]TJ /F3 11.955 Tf 13.6 0 Td[(Cd)+,andtheupperbound,minfx+Cu;Cg,arenon-decreasinginx,soisyn(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m). Toproveyn(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)ymyopn(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m),it issufcient toshowtheunconstrainedglobalmaximizer of Jn(x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)isnotsmallerthantheunconstrainedglobalmaximizerofLn(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m),i.e.,y4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)ymyop;4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m). FromtheproofofProposition 2.1 (a),weknowthat@Ln @yjymyop;4n(x;~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m)=0.FromProposition 2.1 (c),weknowthat~V0n+1; r (ymyop;4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m))0.Sowehave @Jn(x;y;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m) @yjy=ymyop;4n(x;~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m)+=@Ln(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m) @yjy=ymyop;4n(x;~pn)]TJ /F10 5.978 Tf 5.75 0 Td[(m)+~V0n+1; r (ymyop;4n(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)) 0: (A) BasedontheconcavityofJn(x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)inyprovedinProposition 2.1 (b),togetherwith( A )and( A ),wehavey4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)ymyop;4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m). (b)Similar to theabovesection,wewriteE pn)]TJ /F10 5.978 Tf 5.75 0 Td[(m+1 j~pn)]TJ /F10 5.978 Tf 5.75 0 Td[(m[Vn+1(y;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m+1)]as~Vn+1(y),andlet~V0n+1;l(y)and~V0n+1;r(y)denotetheleftandright derivatives of~Vn+1(y)atyrespectively.WewillrstprovethattheglobalmaximizerofJn(x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)withoutanyconstraint,y4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m),satises that y4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m))]TJ /F3 11.955 Tf 12.71 0 Td[(xnon-increasesinx,whichisequivalenttoy4n(x+4x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m))]TJ /F4 11.955 Tf 12.21 0 Td[((x+4x)y4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m))]TJ /F3 11.955 Tf 12.22 0 Td[(xforanyfeasiblexandanypositive4xsuchthatx+4xisstillfeasible.Foranygivenfeasiblex,fromtheconcavityofJn(x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)iny,theglobalmaximizerofJn(x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)withoutanyconstraint,y4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)satisesthattheleftderivativeofJn(x;y;~pn)]TJ /F5 7.97 Tf 6.59 .01 Td[(m)withrespecttoyaty4n(x;~pn)]TJ /F5 7.97 Tf 6.59 .01 Td[(m)isnon-negative,i.e.,@Jn(x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m) @yjy=y4n(x;~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m))]TJ /F4 11.955 Tf 10.4 3.95 Td[(=@Ln(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m) @yjy=y4n(x;~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m))]TJ /F4 11.955 Tf 9.74 3.95 Td[(+~V0n+1;l(y4n(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m))0; 152

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andtherightderivativeofJn(x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)withrespecttoyaty4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)isnon-positive,i.e., @Jn(x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m) @yjy=y4n(x;~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m)+=@Ln(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m) @yjy=y4n(x;~pn)]TJ /F10 5.978 Tf 5.75 0 Td[(m)++~V0n+1;r(y4n(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m))0:(A)Inthefollowing,wewillrstprovethatforanypositive4xsuchthatx+4xC,therightderivativeofJn(x+4x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)withrespecttoyaty4n(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)+4xisnon-positive,andsoy4n(x+4x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)y4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)+4x.TherightderivativeofJn(x+4x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)withrespecttoyaty4n(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)+4xis@Jn(x+4x;y;~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m) @yjy=y4n(x;~pn)]TJ /F10 5.978 Tf 5.75 0 Td[(m)+4x+=@Ln(y)]TJ /F5 7.97 Tf 6.59 0 Td[(xx;~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m) @yjy=y4n(x;~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m)+4x++~V0n+1;r(y4n(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)+4x)=@Ln(y)]TJ /F5 7.97 Tf 6.59 0 Td[(x;~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m) @yjy=y4n(x;~pn)]TJ /F10 5.978 Tf 5.75 0 Td[(m)++~V0n+1;r(y4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)+4x)@Ln(y)]TJ /F5 7.97 Tf 6.58 0 Td[(x;~pn)]TJ /F10 5.978 Tf 5.75 0 Td[(m) @yjy=y4n(x;~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m)++~V0n+1;r(y4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m))0;wherethesecondequalityholdsbecause(y4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)+4x+))]TJ /F4 11.955 Tf 9.73 0 Td[((x+4x)=y4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)+)]TJ /F3 11.955 Tf 9.3 0 Td[(x,therstinequalityholdsbecause~Vn+1(y)isconcaveinyandsotherstorderrightderivativeof~Vn+1(y)non-increasesiny,andthelastinequalityholdsbecause of equation( A ).SinceJn(x+4x;y;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)isconcaveinyandtherightderivativeofJn(x+4x;y;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)withrespecttoyaty4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)+4xisnon-positive,wehavey4n(x+4x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)y4n(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)+4x,andsoy4n(x+4x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m))]TJ /F4 11.955 Tf 12.05 0 Td[((x+4x)y4n(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m))]TJ /F3 11.955 Tf 12.05 0 Td[(x,whichmeansy4n(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m))]TJ /F3 11.955 Tf 11.95 0 Td[(xnon-increasesinx.From( A)-222()]TJ /F4 11.955 Tf 21.26 0 Td[(9 ),wehaveyn(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m))]TJ /F3 11.955 Tf 11.95 0 Td[(x=8>>>><>>>>:(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Cd)+)]TJ /F3 11.955 Tf 11.95 0 Td[(xify4n(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)<(x)]TJ /F3 11.955 Tf 11.96 0 Td[(Cd)+y4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m))]TJ /F3 11.955 Tf 11.95 0 Td[(xif(x)]TJ /F3 11.955 Tf 11.96 0 Td[(Cd)+y4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)minfx+Cu;Cgminfx+Cu;Cg)]TJ /F3 11.955 Tf 20.59 0 Td[(xify4n(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)>minfx+Cu;Cg: 153

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Since(x)]TJ /F3 11.955 Tf 12.25 0 Td[(Cd)+)]TJ /F3 11.955 Tf 12.25 0 Td[(x=maxf)]TJ /F3 11.955 Tf 15.28 0 Td[(x;)]TJ /F3 11.955 Tf 9.3 0 Td[(Cdg,andminfx+Cu;Cg)]TJ /F3 11.955 Tf 21.17 0 Td[(x=minfCu;C)]TJ /F3 11.955 Tf 12.25 0 Td[(xg,arebothnon-increasinginx,soisyn(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m))]TJ /F3 11.955 Tf 11.95 0 Td[(x.(c)Ifthereal-timepricespnareindependentbetweenperiods,thenweuseJn(x;y;n),wheren=E[pn],toreplaceJn(x;y;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m).WewillrstprovethattheunconstrainedglobalmaximizerofJn(x;y;n),y4n(x;n),non-increasesinn.Foranypair(1n;2n)suchthat1n<2n,letyn(x;in)betheconstrainedmaximizerofJn(x;y;in),i.e.,Vn(x;in)=Jn(x;yn(x;in);in)=maxy2A(x)Jn(x;y;in)andlety4n(x;in)betheunconstrainedglobalmaximizerofJn(x;y;in),i.e.,Jn(x;y4n(x;in);in)=maxy2RJn(x;y;in)fori=1;2.IftheunconstrainedglobalmaximizerofJn(x;y;1n)issmallerthanorequaltox,i.e.,y4n(x;1n)x,thenfromtheconcavityofJn(x;y;1n)iny,wehavethattheleftderivativeofJn(x;y;1n)withrespecttoyaty4n(x;1n)isnon-negativeandtherightderivativeofJn(x;y;1n)withrespecttoyaty4n(x;1n)isnon-positive,i.e., @Jn(x;y;1n) @yjy=y4n(x;1n))]TJ /F4 11.955 Tf 10.41 3.94 Td[(=)]TJ /F4 11.955 Tf 9.29 0 Td[(2bn2d(y4n(x;1n))]TJ /F3 11.955 Tf 11.39 0 Td[(x))]TJ /F3 11.955 Tf 11.39 0 Td[(d1n+~V0n+1;l(y4n(x;1n))0(A)and @Jn(x;y;1n) @yjy=y4n(x;1n)+=)]TJ /F4 11.955 Tf 9.29 0 Td[(2bn2d(y4n(x;1n))]TJ /F3 11.955 Tf 10.61 0 Td[(x))]TJ /F3 11.955 Tf 10.61 0 Td[(d1n+~V0n+1;r(y4n(x;1n))0:(A)NotethattherightderivativeofJn(x;y;2n)withrespecttoyaty4n(x;1n)is @Jn(x;y;2n) @yjy=y4n(x;1n)+=)]TJ /F4 11.955 Tf 9.3 0 Td[(2bn2d(y4n(x;1n))]TJ /F3 11.955 Tf 11.96 0 Td[(x))]TJ /F3 11.955 Tf 11.96 0 Td[(d2n+~V0n+1;r(y4n(x;1n)):(A)Since1n<2n,from( A )and( A )wehave that @Jn(x;y;1n) @yjy=y4n(x;1n)+>@Jn(x;y;2n) @yjy=y4n(x;1n)+:(A) 154

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Inequalities( A )and( A )givetherightderivativeofJn(x;y;2n)withrespecttoyaty4n(x;1n), @Jn(x;y;2n) @yjy=y4n(x;1n)+<0:(A)IftheleftderivativeofJn(x;y;2n)withrespecttoyaty4n(x;1n)isnon-negative,i.e., @Jn(x;y;2n) @yjy=y4n(x;1n))]TJ /F4 11.955 Tf 10.41 3.94 Td[(=)]TJ /F4 11.955 Tf 9.29 0 Td[(2bn2d(y4n(x;1n))]TJ /F3 11.955 Tf 10.85 0 Td[(x))]TJ /F3 11.955 Tf 10.85 0 Td[(d2n+~V0n+1;l(y4n(x;1n))0;(A)thenfrom( A )and( A )togetherwiththeconcavityofJn(x;y;2n)iny,wehavey4n(x;2n)=y4n(x;1n).OtherwisetheleftderivativeofJn(x;y;2n)withrespecttoyaty4n(x;1n)isnegative,i.e., @Jn(x;y;2n) @yjy=y4n(x;1n))]TJ /F4 11.955 Tf 10.41 3.95 Td[(=)]TJ /F4 11.955 Tf 9.29 0 Td[(2bn2d(y4n(x;1n))]TJ /F3 11.955 Tf 10.88 0 Td[(x))]TJ /F3 11.955 Tf 10.88 0 Td[(d2n+~V0n+1;l(y4n(x;1n))<0:(A)Then,from( A )andtheconcavityofJn(x;y;2n)iny,wehavey4n(x;2n)x, wecanuseasimilarapproachto provey4n(x;2n)y4n(x;1n).Soy4n(x;n)non-increasesinn.NotethattheconstrainedmaximizerofJn(x;y;n)withinthefeasiblesetA,yn(x;n),istheclosestpointtotheunconstrainedglobalmaximizerofJn(x;y;n),y4n(x;n),limitedtobebetweentheupperandlowerlimits,i.e.,yn(x;n)=8>>>><>>>>:(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Cd)+ify4n(x;n)<(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Cd)+y4n(x;n)if(x)]TJ /F3 11.955 Tf 11.96 0 Td[(Cd)+y4n(x; n )minfx+Cu;Cgminfx+Cu;Cgify4n(x;n)>minfx+Cu;Cg;wehavethatyn(x;n)non-increasesinn. 155

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A.4ProofofProposition 2.3 (a)Letthechargingamountz=y)]TJ /F3 11.955 Tf 12.39 0 Td[(x,whichisthedifferencebetweenthetargetstoragelevelandtheinitialstoragelevelinoneperiod.From( 2)-221()]TJ /F4 11.955 Tf 21.25 0 Td[(4 ),wehave L0n(z;pn)=8><>:)]TJ /F3 11.955 Tf 9.3 0 Td[(dz(pn+bndz)forz0)]TJ /F5 7.97 Tf 13.53 4.71 Td[(z u(pn+bnz u)forz0:(A)TherstandsecondorderderivativesofL0n(z;pn)withrespecttozare@L0n(z;pn) @z=8><>:)]TJ /F4 11.955 Tf 9.3 0 Td[(2bn2dz)]TJ /F3 11.955 Tf 11.96 0 Td[(dpnforz<0)]TJ /F7 7.97 Tf 10.5 4.71 Td[(2bn 2uz)]TJ /F7 7.97 Tf 16.21 4.71 Td[(1 upnforz>0and@2L0n(z;pn) @z2=8><>:)]TJ /F4 11.955 Tf 9.3 0 Td[(2bn2dforz<0)]TJ /F7 7.97 Tf 10.49 4.71 Td[(2bn 2uforz>0:Nowwehavetherstorderandsecondorderderivativesofthemyopicutilityfunction,Ln(z;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m),withrespecttozare@Ln(z;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m) @z=Epnj~pn)]TJ /F10 5.978 Tf 5.75 0 Td[(me)]TJ /F5 7.97 Tf 6.59 0 Td[(L0n(z;pn)@L0n(z;pn) @z=8><>:Epnj~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(mhe)]TJ /F5 7.97 Tf 6.58 0 Td[(L0n(z;pn)()]TJ /F4 11.955 Tf 9.3 0 Td[(2bn2dz)]TJ /F3 11.955 Tf 11.96 0 Td[(dpn)iforz<0Epnj~pn)]TJ /F10 5.978 Tf 5.75 0 Td[(mhe)]TJ /F5 7.97 Tf 6.59 0 Td[(L0n(z;pn)()]TJ /F7 7.97 Tf 10.5 4.71 Td[(2bn 2uz)]TJ /F7 7.97 Tf 16.21 4.71 Td[(1 upn)iforz>0 (A)and@2Ln(z;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m) @z2=Epnj~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m")]TJ /F3 11.955 Tf 9.3 0 Td[(2e)]TJ /F5 7.97 Tf 6.59 0 Td[(L0n(z;pn)@L0n(z;pn) @z2+e)]TJ /F5 7.97 Tf 6.59 0 Td[(L0n(z;pn)@2L0n(z;pn) @z2# (A)8><>:<0forz<0<0forz>0: 156

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From( A )and( A ),wehave that @Ln(z;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m) @zjz=0)]TJ /F4 11.955 Tf 10.4 1.79 Td[(=)]TJ /F3 11.955 Tf 9.3 0 Td[(dEpnj~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m[pn])]TJ /F3 11.955 Tf 21.91 0 Td[(1 uEpnj~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m[pn]=@Ln(z;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m) @zjz=0+:Hence,weobservethatinbothfz:z<0gandfz:z>0g,thesecondorderderivativesofLn(z;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)withrespecttozarestrictlynegative.Togetherwiththefactthatatz=0theleftderivativeofLn(z;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)isnotsmallerthantherightderivativeofLn(z;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m),wehavethatLn(z;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)isstrictlyconcaveinz.(b)Theproofisbyinduction.Theclaimholdstriviallyforn=N+1,asVN+1(x; ~pN+1)]TJ /F5 7.97 Tf 6.59 0 Td[(m )isconcaveandincreasesinxinoursetting.Nowassumethattheclaimholdsforperiodn+1,i.e.,Vn+1(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1)isconcaveinx.From(a),Ln(z;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)isstrictlyconcaveinz,implyingthatLn(y)]TJ /F3 11.955 Tf 12.26 0 Td[(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)isjointlyconcavein(x;y) (notehere that @L=@z=@L=@yand@L=@z=)]TJ /F3 11.955 Tf 9.3 0 Td[(@L=@x,whichimpliesthattheHessianmatrixofLwithrespectto(x;y)isnegativesemi-denite) ,andstrictlyconcaveinyforanygivenx.Thus,Jn(x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)=Ln(y)]TJ /F3 11.955 Tf 12.41 0 Td[(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)+Epn)]TJ /F10 5.978 Tf 5.75 0 Td[(m+1j~pn)]TJ /F10 5.978 Tf 5.75 0 Td[(m[Vn+1(y;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m+1)]isjointlyconcavein(x;y)andstrictlyconcaveinyforanygivenx.Furthermore,Vn(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)=maxy2A(x)fJn(x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)gisconcaveinxsince(x;y):0xC;y2A(x)=(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Cd)+;minfC;x+Cugisaconvexset,whichcompletestheinductionstep.Sinceforanygivenx,Jn(x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)isstrictlyconcaveinyandthefeasiblesetisaboundedinterval,thereexistsauniqueoptimalsolutionyn(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m).(c)TheproofissimilartothatofTheorem 2.1 (c). A.5ProofofTheorem 2.2 Notethatforanygiven xnand~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m ,theoptimaltargetstoragelevelyn(xn ;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m )isunique.Inaddition,wehavexn+1=yn(xn ;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m ).Therefore,oncex1 and~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(mare given, x2 =y1(x1 ;~p1)]TJ /F5 7.97 Tf 6.58 0 Td[(m ), x3=y2(x2 ;~p2)]TJ /F5 7.97 Tf 6.58 0 Td[(m )=y2(y1(x1 ;~p1)]TJ /F5 7.97 Tf 6.59 0 Td[(m ); ~p2)]TJ /F5 7.97 Tf 6.59 0 Td[(m ) ,:::,andxn= yn)]TJ /F7 7.97 Tf 6.58 0 Td[(1(yn)]TJ /F7 7.97 Tf 6.59 0 Td[(2(:::y1(x1;~p1)]TJ /F5 7.97 Tf 6.58 0 Td[(m);:::;~pn)]TJ /F7 7.97 Tf 6.59 0 Td[(2)]TJ /F5 7.97 Tf 6.59 0 Td[(m); ~pn)]TJ /F7 7.97 Tf 6.59 0 Td[(1)]TJ /F5 7.97 Tf 6.59 0 Td[(m)areuniquelydecidedforn=2;:::;N+1. 157

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A.6ProofofProposition 2.4 (a) Itissufcienttoprovethat ymyopn(xn;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m;) )]TJ /F3 11.955 Tf 12.24 0 Td[(xnnon-decreasesin.Toshowthis,withthedenitionofz=y)]TJ /F3 11.955 Tf 12.91 0 Td[(x,werstprovethat(z4)0()0,wherez4()representstheunconstrainedmaximizerofLn(z;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m),andthenextendthistotheconstrainedcase.Firstofall,basedonthefactthat Ln(z;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m) isstrictlyconcaveinz from ( A )anddischargingisoptimal myopically,togetherwith( A ) wehavethatz4()satises @Ln(z;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m) @zjz=z4()=Epnj~pn)]TJ /F10 5.978 Tf 5.75 0 Td[(mhe)]TJ /F5 7.97 Tf 6.59 0 Td[(L0n(z4();pn)()]TJ /F4 11.955 Tf 9.3 0 Td[(2bn2dz4())]TJ /F3 11.955 Tf 11.95 0 Td[(dpn)i=0:(A) Bytakingthederivativeof( A )intermsof ,wehavethat (z4)0()=)]TJ /F9 11.955 Tf 11.29 16.86 Td[(@2Ln(z;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m) @z@=@2Ln(z;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m) @z2jz=z4():(A) Considering @2Ln( z ;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m) @z2<0 (A) basedon ( A ),thesignof(z4)0() in( A ) isthesamewiththatof@2Ln(z;~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m) @z@j z= z4(). Inthefollowing,weonlyneedtoprove@2Ln(z;~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m) @z@j z= z4()0. Beforeprovingthisclaim,werstgettherst-orderderivativeofLn(z;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)withrespecttoz,@Ln(z;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m) @z=Epnj~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(me)]TJ /F5 7.97 Tf 6.58 0 Td[(L0n(z;pn)@L0n(z;pn) @z=Epnj~pn)]TJ /F10 5.978 Tf 5.75 0 Td[(mhe)]TJ /F5 7.97 Tf 6.58 0 Td[(L0n(z;pn)()]TJ /F4 11.955 Tf 9.3 0 Td[(2bn2dz)]TJ /F3 11.955 Tf 11.96 0 Td[(dpn)i;basedon( A ) forthecasez<0,becausewehave z4()<0.Nowwehave 158

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@2Ln(z;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m) @z@=Epnj~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(mhe)]TJ /F5 7.97 Tf 6.59 0 Td[(L0n(z;pn)()]TJ /F4 11.955 Tf 9.3 0 Td[(2bn2dz)]TJ /F3 11.955 Tf 11.95 0 Td[(dpn)i)]TJ /F3 11.955 Tf 11.96 0 Td[(Epnj~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(mhe)]TJ /F5 7.97 Tf 6.58 0 Td[(L0n(z;pn)L0n(z;pn)()]TJ /F4 11.955 Tf 9.29 0 Td[(2bn2dz)]TJ /F3 11.955 Tf 11.95 0 Td[(dpn)i=Epnj~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(mhe)]TJ /F5 7.97 Tf 6.59 0 Td[(L0n(z;pn)()]TJ /F4 11.955 Tf 9.3 0 Td[(2bn2dz)]TJ /F3 11.955 Tf 11.95 0 Td[(dpn)i+Epnj~pn)]TJ /F10 5.978 Tf 5.75 0 Td[(mhe)]TJ /F5 7.97 Tf 6.59 0 Td[(L0n(z;pn)dz(pn+bndz)()]TJ /F4 11.955 Tf 9.3 0 Td[(2bn2dz)]TJ /F3 11.955 Tf 11.95 0 Td[(dpn)i=Epnj~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(mhe)]TJ /F5 7.97 Tf 6.59 0 Td[(L0n(z;pn)()]TJ /F4 11.955 Tf 9.3 0 Td[(2bn2dz)]TJ /F3 11.955 Tf 11.95 0 Td[(dpn)i)]TJ /F3 11.955 Tf 11.96 0 Td[(Epnj~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(mhe)]TJ /F5 7.97 Tf 6.58 0 Td[(L0n(z;pn)2dz(pn+2bndz)]TJ /F3 11.955 Tf 11.96 0 Td[(bndz)(2bndz+pn)i=Epnj~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(mhe)]TJ /F5 7.97 Tf 6.59 0 Td[(L0n(z;pn)()]TJ /F4 11.955 Tf 9.3 0 Td[(2bn2dz)]TJ /F3 11.955 Tf 11.95 0 Td[(dpn)i (A))]TJ /F3 11.955 Tf 11.96 0 Td[(Epnj~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(mhe)]TJ /F5 7.97 Tf 6.58 0 Td[(L0n(z;pn)2dz(2bndz+pn)2i+3dz2bnEpnj~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(mhe)]TJ /F5 7.97 Tf 6.58 0 Td[(L0n(z;pn)(2bndz+pn)i;wherethesecondequalityholdsbecauseof( 2)-222()]TJ /F4 11.955 Tf 21.25 0 Td[(4 )andthethirdequalityholdsbecausebndz=2bndz)]TJ /F3 11.955 Tf 11.95 0 Td[(bndz. Notehere that therstandthirdtermsof( A )arezerosatz=z4 () ,followingtheexpressiondescribedin( A )andthefactthatdisaconstant.Then, @2Ln(z;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m) @z@jz=z4()=Epnj~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(mhe)]TJ /F5 7.97 Tf 6.59 0 Td[(L0n(z4();pn)2d()]TJ /F3 11.955 Tf 9.3 0 Td[(z4())(2bndz4()+pn)2i0;becausez4()0. Therefore,theoriginalconclusionholdsfortheunconstrainedcase.Inaddition,notethat inthiscase thefeasibleregionofzis[)]TJ /F3 11.955 Tf 9.3 0 Td[(Cd;0],whichdoesnotdependon.Therefore,theconclusionholdsfortheconstrainedcaseaswell. (b)Fornotationbrevity,wewriteEpn)]TJ /F10 5.978 Tf 5.76 0 Td[(m+1j~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m[Vn+1(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m+1)]as~Vn+1(x).Let(~Vn+1;l)0(y)and(~Vn+1;r)0(y)denotetheleftandrightderivativesatyrespectively.NoticethatVn+1(y;~pn+1)]TJ /F5 7.97 Tf 6.59 0 Td[(m)andJn(x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)arebothconcaveiny,sotheleftandrightderivativesatyalwaysexist. 159

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Inthefollowing,werstprovethattheglobalmaximizerofJn(x;y;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)withoutanyconstraint, denotedas y4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m),non-decreasesinx. Foranyx1andx2suchthat0x1@Ln(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x1;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m) @yjy=y4n(x1;~pn)]TJ /F10 5.978 Tf 5.75 0 Td[(m))]TJ /F3 11.955 Tf 7.09 3.95 Td[(: (A)Thus,theleftderivativeofJn(x2;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)withrespecttoyaty4n(x1;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)satisesthat@Jn(x2;y;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m) @yjy=y4n(x1;~pn)]TJ /F10 5.978 Tf 5.75 0 Td[(m))]TJ /F4 11.955 Tf 10.41 3.94 Td[(=@Ln(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x2;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m) @yjy=y4n(x1;~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m))]TJ /F4 11.955 Tf 9.74 3.94 Td[(+(~Vn+1;l)0(y4n(x1;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m))>@Ln(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x1;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m) @yjy=y4n(x1;~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m))]TJ /F4 11.955 Tf 9.74 3.94 Td[(+(~Vn+1;l)0(y4n(x1;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m))0; wheretheequalityholdsbecauseofthedenitionofJn(x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m);therstinequalityholdsbecauseof( A ),andthelastinequalityholdsbecauseof( A ). TogetherwiththeconcavityofJn(x2;y;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)iny,wehavey4n(x2;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)y4n(x1;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)andsoy4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)non-decreasesinx. Nowweprovetheconclusion holds fortheconstrainedcase. NotethatthemaximizerofJn(x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)withinthefeasiblesetA(x),yn(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m),istheclosestpointtotheglobalmaximizerofJn(x;y;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)withoutanyconstraint,y4n(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m),limited 160

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tobebetweentheupper andlowerlimits , yn(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)=8>>>><>>>>:(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Cd)+ify4n(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)<(x)]TJ /F3 11.955 Tf 11.96 0 Td[(Cd)+y4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)if(x)]TJ /F3 11.955 Tf 11.96 0 Td[(Cd)+y4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)minfx+Cu;Cgminfx+Cu;Cgify4n(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)>minfx+Cu;Cg:(A)Sincethelowerbound,(x)]TJ /F3 11.955 Tf 13.6 0 Td[(Cd)+,andtheupperbound,minfx+Cu;Cg,arenon-decreasinginx,soisyn(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m).(c)Similar to theabovesection,wewriteE pn)]TJ /F10 5.978 Tf 5.75 0 Td[(m+1 j~pn)]TJ /F10 5.978 Tf 5.75 0 Td[(m[Vn+1(y;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m+1)]as~Vn+1(y),andlet(~Vn+1;l)0(y)and(~Vn+1;r)0(y)denotetheleftandrightderivativesof~Vn+1(y)atyrespectively.WewillrstprovethattheunconstrainedglobalmaximizerofJn(x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m),y4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m),satisesthaty4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m))]TJ /F3 11.955 Tf 12.71 0 Td[(xnon-increasesinx,whichisequivalenttoy4n(x+4x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m))]TJ /F4 11.955 Tf 12.21 0 Td[((x+4x)y4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m))]TJ /F3 11.955 Tf 12.22 0 Td[(xforanyfeasiblexandanypositive4xsuchthatx+4xisstillfeasible.FromtheconcavityofJn(x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m),theunconstrainedglobalmaximizerofJn(x;y;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m),y4n(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)satisesthattheleftderivativeofJn(x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)withrespecttoyaty4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)isnon-negative,i.e.,@Jn(x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m) @yjy=y4n(x;~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m))]TJ /F4 11.955 Tf 10.4 3.94 Td[(=@Ln(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m) @yjy=y4n(x;~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m))]TJ /F4 11.955 Tf 7.08 3.94 Td[(+(eVn+1;l)0(y4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m))0andtherightderivativeofJn(x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)withrespecttoyaty4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)isnon-positive,i.e., @Jn(x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m) @yjy=y4n(x;~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m)+=@Ln(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m) @yjy=y4n(x;~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m)++(eVn+1;r)0(y4n(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m))0:(A)Inthefollowing,wewillprovethatforpositive4xsuchthatx+4xC,therightderivativeofJn(x+4x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)withrespecttoyaty4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)+4xisnon-positiveandsoy4n(x+4x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)y4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)+4x.Thisisbecause 161

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@Jn(x+4x;y;~pn)]TJ /F10 5.978 Tf 5.75 0 Td[(m) @yjy=y4n(x;~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m)+4x+=@Ln(y)]TJ /F5 7.97 Tf 6.59 0 Td[(xx;~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m) @yjy=y4n(x;~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m)+4x++(eVn+1;r)0(y4n(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)+4x)=@Ln(y)]TJ /F5 7.97 Tf 6.59 0 Td[(x;~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m) @yjy=y4n(x;~pn)]TJ /F10 5.978 Tf 5.75 0 Td[(m)++(eVn+1;r)0(y4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)+4x)@Ln(y)]TJ /F5 7.97 Tf 6.58 0 Td[(x;~pn)]TJ /F10 5.978 Tf 5.75 0 Td[(m) @yjy=y4n(x;~pn)]TJ /F10 5.978 Tf 5.76 0 Td[(m)++(eVn+1;r)0(y4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m))0;wheretherstequalityholdsbecauseofthedenitionofJn(x+4x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m),thesecondequalityholdsbecause (y4n(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)+4x)]TJ /F4 11.955 Tf 9.3 0 Td[())]TJ /F4 11.955 Tf 11.95 0 Td[((x+4x)=(y4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m))]TJ /F4 11.955 Tf 9.3 0 Td[())]TJ /F3 11.955 Tf 11.95 0 Td[(x ,therstinequalityholdsbecause~Vn+1(y)isconcaveiny,andthelastinequityholdsbecauseof( A ).SinceJn(x+4x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)isconcaveinyandtherightderivativeofJn(x+4x;y;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)withrespecttoyaty4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)+4xisnon-positive,wehavey4n(x+4x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)y4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)+4x,becausey4n(x+4x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m)istheunconstrainedglobalmaximizerofJn(x+4x;y;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m).Thus,wehavey4n(x+4x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m))]TJ /F4 11.955 Tf 12.82 0 Td[((x+4x)y4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m))]TJ /F3 11.955 Tf 11.95 0 Td[(x,whichmeansy4n(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m))]TJ /F3 11.955 Tf 11.96 0 Td[(xnon-increasesinx. Fortheconstrainedcase, basedon( A ) ,wehavethat yn(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m))]TJ /F3 11.955 Tf 11.96 0 Td[(x=8>>>><>>>>:(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Cd)+)]TJ /F3 11.955 Tf 11.95 0 Td[(xify4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)<(x)]TJ /F3 11.955 Tf 11.96 0 Td[(Cd)+y4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m))]TJ /F3 11.955 Tf 11.96 0 Td[(xif(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Cd)+y4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)minfx+Cu;CgminfCu;C)]TJ /F3 11.955 Tf 11.96 0 Td[(xgify4n(x;~pn)]TJ /F5 7.97 Tf 6.59 0 Td[(m)>minfx+Cu;Cg: Since (x)]TJ /F3 11.955 Tf 12.25 0 Td[(Cd)+)]TJ /F3 11.955 Tf 12.25 0 Td[(x=maxf)]TJ /F3 11.955 Tf 15.28 0 Td[(x;)]TJ /F3 11.955 Tf 9.3 0 Td[(Cdg,andminfx+Cu;Cg)]TJ /F3 11.955 Tf 21.17 0 Td[(x=minfCu;C)]TJ /F3 11.955 Tf 12.25 0 Td[(xg,arenon-increasinginx,soisyn(x;~pn)]TJ /F5 7.97 Tf 6.58 0 Td[(m))]TJ /F3 11.955 Tf 11.96 0 Td[(x. A.7ProofsofProposition 2.5 andTheorem 2.3 (a) Theproofsforthetwoclaimedcasesaresimilar.Weonlyprovethe1kK)]TJ /F3 11.955 Tf -458.19 -23.91 Td[(`+mcase. ToprovethatLn;k(z;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m; ~fn ;qn;k)isstrictlyconcavein(z;qn;k),itsufcestoshowthattheHessianmatricesofLn;k(z;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m; ~fn ;qn;k)withrespectto(z;qn;k)arenegativedeniteinf(z;qn;k):z<0gandf(z;qn;k):z>0g,theleftderivativeof 162

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Ln;k(z;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m; ~fn ;qn;k)isnotsmallerthantherightderivativeofLn;k(z;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m; ~fn ;qn;k)withrespecttoqn;katz=0,andtheleftderivativeofLn;k(z;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m; ~fn ;qn;k)isnotsmallerthantherightderivativeofLn;k(z;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m; ~fn ;qn;k)withrespecttozatz=0. FromthedenitionofLn;k(z;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m; ~fn ;qn;k),theHessianmatricesofLn;k(z;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m; ~fn ;qn;k)at(z;qn;k)are264)]TJ /F7 7.97 Tf 10.5 6.27 Td[(2bn;k 2u)]TJ /F7 7.97 Tf 10.49 6.27 Td[(2bn;k u)]TJ /F7 7.97 Tf 10.5 6.27 Td[(2bn;k u)]TJ /F4 11.955 Tf 9.3 0 Td[(2(an;k+bn;k)375whenz>0and264)]TJ /F4 11.955 Tf 9.3 0 Td[(2bn;k2d)]TJ /F4 11.955 Tf 9.3 0 Td[(2bn;kd)]TJ /F4 11.955 Tf 9.3 0 Td[(2bn;kd)]TJ /F4 11.955 Tf 9.29 0 Td[(2(an;k+bn;k)375whenz<0:Botharenegativedenite.Therstorderderivativewithrespecttozis @Ln;k @z=8><>:)]TJ /F7 7.97 Tf 10.49 6.27 Td[(2bn;k 2uz)]TJ /F7 7.97 Tf 13.15 6.27 Td[(2bn;k uqn;k)]TJ /F5 7.97 Tf 13.15 10.1 Td[(Epn;kj(~pn;k)]TJ /F10 5.978 Tf 5.76 0 Td[(m; ~fn )[pn;k] uforz>0)]TJ /F4 11.955 Tf 9.3 0 Td[(2bn;k2dz)]TJ /F4 11.955 Tf 11.96 0 Td[(2bn;kdqn;k)]TJ /F3 11.955 Tf 11.95 0 Td[(dEpn;kj(~pn;k)]TJ /F10 5.978 Tf 5.76 0 Td[(m; ~fn )[pn;k]forz<0(A)andtherstorderderivativewithrespecttoqn;kis@Ln;k @qn;k=8><>:)]TJ /F4 11.955 Tf 9.3 0 Td[(2(an;k+bn;k)qn;k)]TJ /F7 7.97 Tf 13.15 6.27 Td[(2bn;k uz+fn;k)]TJ /F3 11.955 Tf 11.96 0 Td[(Epn;kj(~pn;k)]TJ /F10 5.978 Tf 5.76 0 Td[(m; ~fn )[pn;k]forz>0)]TJ /F4 11.955 Tf 9.3 0 Td[(2(an;k+bn;k)qn;k)]TJ /F4 11.955 Tf 11.95 0 Td[(2bn;kdz+fn;k)]TJ /F3 11.955 Tf 11.95 0 Td[(Epn;kj(~pn;k)]TJ /F10 5.978 Tf 5.75 0 Td[(m; ~fn )[pn;k]forz<0:Wehave,byourassumption,@Ln;k @zjz=0)]TJ /F2 11.955 Tf 9.74 1.79 Td[()]TJ /F5 7.97 Tf 13.15 6.28 Td[(@Ln;k @zjz=0+=(1 u)]TJ /F3 11.955 Tf 11.96 0 Td[(d)(2bn;kqn;k+Epn;kj(~pn;k)]TJ /F10 5.978 Tf 5.75 0 Td[(m; ~fn )[pn;k])(1 u)]TJ /F3 11.955 Tf 11.95 0 Td[(d)(2bn;kqn;k+p n;k)0;and@Ln;k @qn;kjz=0)]TJ /F4 11.955 Tf 10.41 1.79 Td[(=)]TJ /F4 11.955 Tf 9.3 0 Td[(2(an;k+bn;k)qn;k+fn;k)]TJ /F3 11.955 Tf 11.96 0 Td[(Epn;kj(~pn;k)]TJ /F10 5.978 Tf 5.76 0 Td[(m; ~fn )[pn;k]=@Ln;k @qn;kjz=0+: Therefore,theconclusionholds. 163

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Inthefollowing,weprove(b) and (c) ofProposition 2.5 andTheorem 2.3 together .We rstprovethreelemmas. LemmaA.1. WhenK)]TJ /F3 11.955 Tf 11.95 0 Td[(`+m+1kK,ifVn;k+1( y ;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1;~fn+1;~qn;k+1:K;~qn+1;1:K)isstrictlyconcavein( y ;~qn;k+1:K;~qn+1;1:K)andLn;k(z;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m; ~fn+1 ;qn;k)isstrictlyconcavein(z;qn;k),thenJn;k(x;y;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m;~fn+1;~qn;k:K;~qn+1;1:K)=Ln;k(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m; ~fn+1 ;qn;k)+Epn;k+1)]TJ /F10 5.978 Tf 5.75 0 Td[(mj(~pn;k)]TJ /F10 5.978 Tf 5.76 0 Td[(m; ~fn+1 )[Vn;k+1(y;~pn;k+1)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn+1;~qn;k+1:K;~qn+1;1:K)] (A)isstrictlyconcavein(x;y;~qn;k:K;~qn+1;1:K).Whenk=K)]TJ /F3 11.955 Tf 11.59 0 Td[(`+m,ifVn;k+1( y ;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m+1;~fn+1;~qn;k+1:K;~qn+1;1:K)isstrictlyconcavein( y ;~qn;k+1:K;~qn+1;1:K)andLn;k(z;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m; ~fn ;qn;k)isstrictlyconcavein(z;qn;k),thenJn;k(x;y;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m;~fn;~qn;k:K;~qn+1;1:K)=Ln;k(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m; ~fn ;qn;k)+E(pn;k)]TJ /F10 5.978 Tf 5.76 0 Td[(m+1; ~fn+1 )j(~pn;k)]TJ /F10 5.978 Tf 5.75 0 Td[(m;~fn)[Vn;k+1(y;~pn;k+1)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn+1;~qn;k+1:K;~qn+1;1:K)] (A)isstrictlyconcavein(x;y;~qn;k:K;~qn+1;1:K).When1kK)]TJ /F3 11.955 Tf 12.16 0 Td[(`+m)]TJ /F4 11.955 Tf 12.16 0 Td[(1,ifVn;k+1( y ;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1;~fn;~qn;k+1:K)isstrictlyconcavein( y ;~qn;k+1:K)andLn;k(z;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m; ~fn ;qn;k)isstrictlyconcavein(z;qn;k),thenJn;k(x;y;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn;~qn;k:K)=Ln;k(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m; ~fn ;qn;k)+Epn;k+1)]TJ /F10 5.978 Tf 5.75 0 Td[(mj(~pn;k)]TJ /F10 5.978 Tf 5.76 0 Td[(m;~fn )[Vn;k+1(y;~pn;k+1)]TJ /F5 7.97 Tf 6.58 0 Td[(m;~fn;~qn;k+1:K)] (A)isstrictlyconcavein(x;y;~qn;k:K). Proof. Theproofsforthethreeclaimedcasesaresimilar.Weonlyprovethe K)]TJ /F3 11.955 Tf 11.1 0 Td[(`+m+1kKcase. 164

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Foranygiven2(0;1),w1=(x1;y1;~q1n;k:K;~q1n+1;1:K),andw2=(x2;y2;~q2n;k:K;~q2n+1;1:K)withw16=w2.Wewillprovethat Jn;k(~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn+1;w1+(1)]TJ /F3 11.955 Tf 11.95 0 Td[()w2)>Jn;k(~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m;~fn+1;w1)+(1)]TJ /F3 11.955 Tf 11.96 0 Td[()Jn;k(~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn+1;w2);(A)andsoJn;k(x;y;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn+1;~qn;k:K;~qn+1;1:K)isstrictlyconcavein(x;y;~qn;k:K;~qn+1;1:K). Letui=(yi;~qin;k+1:K;~qin+1;1:K),zi=yi)]TJ /F3 11.955 Tf 12.59 0 Td[(xi,andvi=(zi;qin;k)fori=1;2.Inthefollowing,weprovetheclaimbasedontwoscenariosdependingonifu1=u2. Ifu16=u2,then basedontheassumptionthat Vn;k+1( y ;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1;~fn+1;~qn;k+1:K;~qn+1;1:K) isstrictlyconcave in( y ;~qn;k+1:K;~qn+1;1:K),wehave Vn;k+1(~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1;~fn+1;u1+(1)]TJ /F3 11.955 Tf 9.3 0 Td[()u2)>Vn;k+1(~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1;~fn+1;u1)+(1)]TJ /F3 11.955 Tf 9.3 0 Td[()Vn;k+1(~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m+1;~fn+1;u2);(A)andfromthestrictconcavityofLn;k(z;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m; ~fn+1 ;qn;k)in(z;qn;k),wehave that Ln;k(~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m; ~fn+1 ;v1+(1)]TJ /F3 11.955 Tf 11.96 0 Td[()v2)Ln;k(~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m; ~fn+1 ;v1)+(1)]TJ /F3 11.955 Tf 11.96 0 Td[()Ln;k(~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m; ~fn+1 ;v2):(A) From ( A ),( A ),and( A ),wehave ( A )holds. Otherwiseifu1=u2,thenv16=v2becausew16=w2.FromthestrictconcavityofLn;k(z;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m; ~fn+1 ;qn;k)in(z;qn;k),wehave that Ln;k(~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m; ~fn+1 ;v1+(1)]TJ /F3 11.955 Tf 11.96 0 Td[()v2)>Ln;k(~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m; ~fn+1 ;v1)+(1)]TJ /F3 11.955 Tf 11.95 0 Td[()Ln;k(~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m; ~fn+1 ;v2);(A)andbasedonu1=u2,wehave that Vn;k+1(~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1;~fn+1;u1+(1)]TJ /F3 11.955 Tf 9.3 0 Td[()u2)=Vn;k+1(~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1;~fn+1;u1)+(1)]TJ /F3 11.955 Tf 9.3 0 Td[()Vn;k+1(~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m+1;~fn+1;u2):(A) From ( A ),( A ),and( A ),wehave ( A )holds. Thus,theconclusionholds. 165

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LemmaA.2. When1kK)]TJ /F3 11.955 Tf 11.22 0 Td[(`+m)]TJ /F4 11.955 Tf 11.22 0 Td[(1,ifJn;k(x;y;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m;~fn;~qn;k:K)isstrictlyconcavein(x;y;~qn;k:K)for(x;y;~qn;k:K)2AQ,wherebothQRK)]TJ /F5 7.97 Tf 6.59 0 Td[(k+1andA:=f(x;y):0xC;y2A(x)gareboundedclosedconvexsets,then Vn;k(x;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn;~qn;k:K)=maxy2A(x)fJn;k(x;y;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn;~qn;k:K)g(A)isstrictlyconcavein(x;~qn;k:K)for(x;~qn;k:K)2[0;C]Q.WhenK)]TJ /F3 11.955 Tf 9.44 0 Td[(`+m+1kK,ifJn;k(x;y;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn+1;~qn;k:K;~qn+1;1:K)isstrictlyconcavein(x;y;~qn;k:K;~qn+1;1:K)for(x;y;~qn;k:K;~qn+1;1:K)2AQ0,wherebothQ0R2K)]TJ /F5 7.97 Tf 6.59 0 Td[(k+1andA:=f(x;y):0xC;y2A(x)gareboundedclosedconvexsets,then Vn;k(x;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m; ~fn+1 ;~qn;k:K;~qn+1;1:K)=maxy2A(x)fJn;k(x;y;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m; ~fn+1 ;~qn;k:K;~qn+1;1:K)g(A)isstrictlyconcavein(x;~qn;k:K;~qn+1;1:K)for(x;~qn;k:K;~qn+1;1:K)2[0;C]Q0. Proof. Theproofsforthetwocasesaresimilar.Weonlyprove the1kK)]TJ /F3 11.955 Tf 10.75 0 Td[(`+m)]TJ /F4 11.955 Tf 10.75 0 Td[(1case.First,forany1=(x1;~q1n;k:K) and 2=(x2;~q2n;k:K)2[0;C]Qwith16=2 and2(0;1) ,wewillprovethatVn;k(~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn;1)+(1)]TJ /F3 11.955 Tf 11.95 0 Td[()Vn;k(~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn;2)
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andbothAandQareconvexsets.Itiseasyto observe thaty(1)+(1)]TJ /F3 11.955 Tf 12.11 0 Td[()y(2)isafeasiblesolutionfor( A )whentheinitialstateis1+(1)]TJ /F3 11.955 Tf 11.95 0 Td[()2,i.e., (x1+(1)]TJ /F3 11.955 Tf 11.95 0 Td[()x2;y(1)+(1)]TJ /F3 11.955 Tf 11.95 0 Td[()y(2);~q1n;k:K+(1)]TJ /F3 11.955 Tf 11.96 0 Td[()~q2n;k:K)2AQ:(A) Thus, wehave Jn;k(~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m;~fn;y(1)+(1)]TJ /F3 11.955 Tf 11.96 0 Td[()y(2);1+(1)]TJ /F3 11.955 Tf 11.96 0 Td[()2)Jn;k(~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m;~fn;y(1+(1)]TJ /F3 11.955 Tf 11.96 0 Td[()2);1+(1)]TJ /F3 11.955 Tf 11.95 0 Td[()2):(A)Finally,we canprovetheconcavityasfollows :Vn;k(~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn;1)+(1)]TJ /F3 11.955 Tf 11.95 0 Td[()Vn;k(~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn;2)=Jn;k(~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m;~fn;y(1);1)+(1)]TJ /F3 11.955 Tf 11.95 0 Td[()Jn;k(~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn;y(2);2)
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Forany1=(~q1n;k:K;x1) and 2=(~q2n;k:K;x2)2Q[0;C]with16=2 and2(0;1) ,wewillprovethatVn;k(~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn;1)+(1)]TJ /F3 11.955 Tf 11.95 0 Td[()Vn;k(~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn;2)
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Finally,wecanprovetheconcavityasfollows:Vn;k(~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn;1)+(1)]TJ /F3 11.955 Tf 11.95 0 Td[()Vn;k(~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn;2)=Jn;k(1;y(1);~qn+1;1:K(1))+(1)]TJ /F3 11.955 Tf 11.95 0 Td[()Jn;k(2;y(2);~qn+1;1:K(2))
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VN;K(x;~pN;K)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fN;qN;K)isstrictlyconcavein(x;qN;K)for(x;qN;K)2[0;C]Q(N;K).Nowassumethattheclaimholdsfor periods (n;k+1)with 1 nNandK)]TJ /F3 11.955 Tf 11.96 0 Td[(`+m+1kK)]TJ /F4 11.955 Tf 11.96 0 Td[(1,i.e.,Vn;k+1(x;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1;~fn+1;~qn;k+1:K;~qn+1;1:K)isstrictlyconcavein(x;~qn;k+1:K;~qn+1;1:K)for(x;~qn;k+1:K;~qn+1;1:K)2[0;C] Q(n;k+1)Q(n+1;1) .From(a),Ln;k(z;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m; ~fn+1 ;qn;k)isstrictlyconcavein(z;qn;k).Thus, basedon( A )in Lemma1,wehavethatJn;k(x;y;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m;~fn+1;~qn;k:K;~qn+1;1:K)=Ln;k(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m; ~fn+1 ;qn;k)+Epn; k)]TJ /F10 5.978 Tf 5.76 0 Td[(m+1 j(~pn;k)]TJ /F10 5.978 Tf 5.76 0 Td[(m; ~fn+1 )[Vn;k+1(y;~pn;k+1)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn+1;~qn;k+1:K;~qn+1;1:K)] (A)isstrictlyconcavein(x;y;~qn;k:K;~qn+1;1:K)for(x;y;~qn;k:K;~qn+1;1:K)2A Q(n;k+1)Q(n+1;1) ,andsothereisauniqueoptimaltargetstoragelevelyn;k(x;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m;~fn+1;~qn;k:K;~qn+1;1:K).SincebothQ0(n;k)andAareconvexsets,from ( A ) Lemma2,wehavethat Vn;k(x;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn+1;~qn;k:K;~qn+1;1:K)=maxy2A(x)fJn(x;y;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m;~fn+1;~qn;k:K;~qn+1;1:K)g(A)isstrictlyconcavein(x;~qn;k:K;~qn+1;1:K)for(x;~qn;k:K;~qn+1;1:K)2[0;C] Q(n;k+1)Q(n+1;1) .Whenk=K)]TJ /F3 11.955 Tf 9.29 0 Td[(`+m,byinductionassumption,Vn;k+1(x;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m+1;~fn+1;~qn;k+1:K;~qn+1;1:K)isstrictlyconcavein(x;~qn;k+1:K;~qn+1;1:K)for(x;~qn;k+1:K;~qn+1;1:K)2[0;C]Q0(n;k+1). Similarly,from (a),Ln;k(z;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m; ~fn ;qn;k)isstrictlyconcavein(z;qn;k).From ( A ) inLemma1,wehavethatJn;k(x;y;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m;~fn;~qn;k:K;~qn+1;1:K):=Ln;k(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m; ~fn ;qn;k)+E(pn;k)]TJ /F10 5.978 Tf 5.75 0 Td[(m+1; ~fn+1 )j(~pn;k)]TJ /F10 5.978 Tf 5.76 0 Td[(m;~fn)[Vn;k+1(y;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1;~fn+1;~qn;k+1:K;~qn+1;1:K)] 170

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isstrictlyconcavein(x;y;~qn;k:K;~qn+1;1:K)for(x;y;~qn;k:K;~qn+1;1:K)2A Q(n;k+1)Q(n+1;1) . TogetherwiththefactthatAandQ(n+1;1)arebounded convex sets,wehavethat there exist auniqueoptimaltargetstoragelevelyn;k(x;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn;~qn;k:K)andauniquegroupofoptimalcharging/discharging amounts intheday-aheadmarketforthenextday~qn+1;1:K(x;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m;~fn;~qn;k:K). Similarly,sinceA, Q(n;k),andQ(n+1;1)areallboundedconvexsets,from Lemma 3,wehavethatVn;k(x;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn;~qn;k:K)=max(y;qn+1;1:K)2A(x)Q(n+1;1)fJn(x;y;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn;~qn;k:K;~qn+1;1:K)gisstrictlyconcavein(x;~qn;k:K)for(x;~qn;k:K)2[0;C]Q(n;k). Nowweassume thattheclaimholdsforperiods(n;k+1),1kK)]TJ /F3 11.955 Tf 10.42 0 Td[(`+m)]TJ /F4 11.955 Tf 10.43 0 Td[(1,i.e.,Vn;k+1(x;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1;~fn;~qn;k+1:K)isstrictlyconcavein(x;~qn;k+1:K)for(x;~qn;k+1:K)2[0;C]Q(n;k+1).From(a),Ln;k(z;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m; ~fn ;qn;k)isstrictlyconcavein(z;qn;k).Thus,from ( A )in Lemma1,wehavethatJn(x;y;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn;~qn;k:K)=Ln;k(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m; ~fn ;qn;k)+Epn;k+1)]TJ /F10 5.978 Tf 5.76 0 Td[(mj(~pn;k)]TJ /F10 5.978 Tf 5.75 0 Td[(m; ~fn )[Vn;k+1(y;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m+1;~fn;~qn;k+1:K)]isstrictlyconcavein(x;y;~qn;k:K)for(x;y;~qn;k:K)2AQ(n;k),andsothereisauniqueoptimaltargetstoragelevelyn;k(x;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn;~qn;k:K). SinceQ(n;k)andAarebondedconvexsets,from( A )inLemma2 ,wehavethatVn;k(x;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m;~fn;~qn;k:K)=maxy2A(x)fJn(x;y;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m;~fn;~qn;k:K)gisstrictlyconcavein(x;~qn;k:K)for(x;~qn;k:K)2[0;C]Q(n;k). Thus,theproofiscompleted. (d)Theproofisbyinduction. 171

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Byourassumption,thesalvagefunctionVN;K+1(x; ~pN;K+1)]TJ /F5 7.97 Tf 6.59 0 Td[(m )non-decreasesinx.Letusassumetheclaimholdsforperiod(n;k+1)for1nNandK)]TJ /F3 11.955 Tf 10.54 0 Td[(`+m+1kK.Inthefollowing,weprovethat foranypair(x1;x2) suchthat0x1x1.Ify1+x2)]TJ /F3 11.955 Tf 11.96 0 Td[(x1C ,lety2:=y1+x2)]TJ /F3 11.955 Tf 12.12 0 Td[(x1.Forthiscase,itiseasytoobservethaty2isfeasible ( e.g. ,0y2Candy2)]TJ /F3 11.955 Tf 12.07 0 Td[(x2=y1)]TJ /F3 11.955 Tf 12.07 0 Td[(x1,whichimplies that y2isfeasible whentheinitialstoragelevelisx2 aslongasy1isfeasible whentheinitialstoragelevelisx1 ) andy2y1.Thenbyinductionassumption,wehave that Vn;k+1(y1;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1;~fn+1;~qn;k+1:K;~qn+1;1:K)Vn;k+1(y2;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m+1;~fn+1;~qn;k+1:K;~qn+1;1:K):(A) 172

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Thus,Vn;k(x1;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn+1;~qn;k:K;~qn+1;1:K)=Ln;k(y1)]TJ /F3 11.955 Tf 11.95 0 Td[(x1;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m; ~fn+1 ;qn;k)+Epn;k+1)]TJ /F10 5.978 Tf 5.76 0 Td[(mj(~pn;k)]TJ /F10 5.978 Tf 5.75 0 Td[(m; ~fn+1 )[Vn;k+1(y1;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1;~fn+1;~qn;k+1:K;~qn+1;1:K)]Ln;k(y2)]TJ /F3 11.955 Tf 11.96 0 Td[(x2;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m; ~fn+1 ;qn;k)+Epn;k+1)]TJ /F10 5.978 Tf 5.76 0 Td[(mj(~pn;k)]TJ /F10 5.978 Tf 5.75 0 Td[(m; ~fn+1 )[Vn;k+1(y2;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1;~fn+1;~qn;k+1:K;~qn+1;1:K)]Vn;k(x2;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn+1;~qn;k:K;~qn+1;1:K);wheretherstequalityholdsbecauseoftheoptimalityofy1,i.e.,( A ),therstinequalityholdsbecauseoftheinductionassumption,i.e.,( A )andy1)]TJ /F3 11.955 Tf 12.05 0 Td[(x1=y2)]TJ /F3 11.955 Tf 12.05 0 Td[(x2, andthelastinequalityfollowsfromthefactthaty2isafeasiblesolution. Ify1+x2)]TJ /F3 11.955 Tf 11.96 0 Td[(x1>C ,lety2=C.Forthiscase,becausey1+x2)]TJ /F3 11.955 Tf 10.49 0 Td[(x1>C=y2,wehavey2)]TJ /F3 11.955 Tf 10.75 0 Td[(x20asshownin( A ).Fromthefeasibilityofy1,wehavey1C=y2. Then, byinductionassumption,wehave that Vn;k+1(y1;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1;~fn+1;~qn;k+1:K;~qn+1;1:K)Vn;k+1(y2;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m+1;~fn+1;~qn;k+1:K;~qn+1;1:K):(A) 173

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Thus,Vn;k(x1;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn+1;~qn;k:K;~qn+1;1:K)=Ln;k(y1)]TJ /F3 11.955 Tf 11.95 0 Td[(x1;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m; ~fn+1 ;qn;k)+Epn;k+1)]TJ /F10 5.978 Tf 5.76 0 Td[(mj(~pn;k)]TJ /F10 5.978 Tf 5.75 0 Td[(m; ~fn+1 )[Vn;k+1(y1;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1;~fn+1;~qn;k+1:K;~qn+1;1:K)]Ln;k(y2)]TJ /F3 11.955 Tf 11.96 0 Td[(x2;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m; ~fn+1 ;qn;k)+Epn;k+1)]TJ /F10 5.978 Tf 5.76 0 Td[(mj(~pn;k)]TJ /F10 5.978 Tf 5.75 0 Td[(m; ~fn+1 )[Vn;k+1(y2;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1;~fn+1;~qn;k+1:K;~qn+1;1:K)]Vn;k(x2;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m;~fn+1;~qn;k:K;~qn+1;1:K);wheretherstequalityholdsbecauseoftheoptimalityofy1,i.e.,( A ),therstinequalityholdsbecauseof( A )and( A ).SimilarlywecanprovethatVn;k( x1 ;~pn;k)]TJ /F5 7.97 Tf 6.59 0 Td[(m+1;~fn;~qn;k:K)Vn;k( x2 ;~pn;k)]TJ /F5 7.97 Tf 6.58 0 Td[(m+1;~fn;~qn;k:K)for1kK)]TJ /F3 11.955 Tf 11.96 0 Td[(`+m.Finally, note here that theproofofTheorem 2.3 isembeddedinthe proofsfor (b)and(c).Thus,wecompletetheproof. 174

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APPENDIXBPROOFS B.1ProofofLemma 3.1 Proof. AsprovedbyMichaelHarrisonintheproofofProposition2.13in[ 32 ], Ito's formularemainsvalidwhenVhasanabsolutelycontinuousderivative. Inourproblemsetting,Vsatisesthisconditionbecause Vhascontinuoussecondderivativeatallx2Rbutcountablepoints.Thus, wecanapplyIto's formulatot2[n;n+1)forn=0;1;:::;N(T))]TJ /F4 11.955 Tf 11.95 0 Td[(1,(0=0),we obtainthefollowing results e)]TJ /F5 7.97 Tf 6.59 0 Td[(rn+1V)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Z)]TJ /F3 11.955 Tf 5.48 -9.68 Td[()]TJ /F5 7.97 Tf -1.34 -7.89 Td[(n+1=e)]TJ /F5 7.97 Tf 6.59 0 Td[(rnV(z(n))+Zn+1ne)]TJ /F5 7.97 Tf 6.58 0 Td[(rtAV(z(t))dt+Zn+1ne)]TJ /F5 7.97 Tf 6.59 0 Td[(rtV0(z(t))dB(t);(B) forn=0;1;:::;N(T))]TJ /F4 11.955 Tf 11.96 0 Td[(1.Similarly,a pplying Ito's formulatot2[N(T);T),wehavethat e)]TJ /F5 7.97 Tf 6.58 0 Td[(rTV(Z(T))=e)]TJ /F5 7.97 Tf 6.59 0 Td[(rN(T)V(z(N(T)))+ZTN(T)e)]TJ /F5 7.97 Tf 6.59 0 Td[(rtAV(z(t))dt+ZTN(T)e)]TJ /F5 7.97 Tf 6.59 0 Td[(rtV0(z(t))dB(t):(B)Summingup( B )overn=0;1;:::;N(T))]TJ /F4 11.955 Tf 11.95 0 Td[(1,and( B ),wehave( 3 ). B.2ProofofTheorem 3.2 ProofofTheorem 3.2 Proof. WeprovethatV(x)ExR10e)]TJ /F5 7.97 Tf 6.59 0 Td[(rth(Z(t))dt+P1n=1e)]TJ /F5 7.97 Tf 6.59 0 Td[(rn)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Z)]TJ /F3 11.955 Tf 5.48 -9.69 Td[()]TJ /F5 7.97 Tf -1.34 -8.01 Td[(j;vjforanyw=((1;v1);(2;v2);:::)2W.Let W1:=(w2W:Ex"Z10e)]TJ /F5 7.97 Tf 6.59 0 Td[(rth(Z(t))dt+1Xn=1e)]TJ /F5 7.97 Tf 6.59 0 Td[(rn)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Z)]TJ /F3 11.955 Tf 5.48 -9.68 Td[()]TJ /F5 7.97 Tf -1.33 -8.01 Td[(j;vj#<1):(B)ItsufcestoprovethatV(x)ExR10e)]TJ /F5 7.97 Tf 6.58 0 Td[(rth(Z(t))dt+P1n=1e)]TJ /F5 7.97 Tf 6.58 0 Td[(rn)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Z)]TJ /F3 11.955 Tf 5.48 -9.68 Td[()]TJ /F5 7.97 Tf -1.34 -8.02 Td[(j;vjforanyw2W1. 175

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Foranyxed w2W1,werstproveaninequalitythatwillbeusedlater.FromthedenitionofLV(x)=infv2fV(x+v)+(x;v)g, bysettingx=Z)]TJ /F3 11.955 Tf 5.48 -9.68 Td[()]TJ /F5 7.97 Tf -1.34 -8.02 Td[(j ,wehave LV)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Z)]TJ /F3 11.955 Tf 5.48 -9.69 Td[()]TJ /F5 7.97 Tf -1.34 -8.01 Td[(jV(Z(j))+)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Z)]TJ /F3 11.955 Tf 5.48 -9.69 Td[()]TJ /F5 7.97 Tf -1.34 -8.01 Td[(j;vj;(B)whereZ(j)=Z)]TJ /F3 11.955 Tf 5.48 -9.68 Td[()]TJ /F5 7.97 Tf -1.34 -8.02 Td[(j+vj.SubtractingV)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Z)]TJ /F3 11.955 Tf 5.47 -9.68 Td[()]TJ /F5 7.97 Tf -1.33 -8.01 Td[(jinbothsidesof( B ),wehaveLV)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Z)]TJ /F3 11.955 Tf 5.48 -9.69 Td[()]TJ /F5 7.97 Tf -1.33 -8.01 Td[(j)]TJ /F3 11.955 Tf 11.95 0 Td[(V)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Z)]TJ /F3 11.955 Tf 5.48 -9.69 Td[()]TJ /F5 7.97 Tf -1.34 -8.01 Td[(jV(Z(j)))]TJ /F3 11.955 Tf 11.95 0 Td[(V)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Z)]TJ /F3 11.955 Tf 5.48 -9.69 Td[()]TJ /F5 7.97 Tf -1.33 -8.01 Td[(j+)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Z)]TJ /F3 11.955 Tf 5.48 -9.69 Td[()]TJ /F5 7.97 Tf -1.33 -8.01 Td[(j;vj:Moving)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Z)]TJ /F3 11.955 Tf 5.48 -9.68 Td[()]TJ /F5 7.97 Tf -1.34 -8.01 Td[(j;vjtotheotherside,wehaveV(Z(j)))]TJ /F3 11.955 Tf 11.95 0 Td[(V)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Z)]TJ /F3 11.955 Tf 5.48 -9.68 Td[()]TJ /F5 7.97 Tf -1.33 -8.01 Td[(jLV)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Z)]TJ /F3 11.955 Tf 5.48 -9.68 Td[()]TJ /F5 7.97 Tf -1.33 -8.01 Td[(j)]TJ /F3 11.955 Tf 11.96 0 Td[(V)]TJ /F3 11.955 Tf 5.47 -9.68 Td[(Z)]TJ /F3 11.955 Tf 5.48 -9.68 Td[()]TJ /F5 7.97 Tf -1.34 -8.01 Td[(j)]TJ /F3 11.955 Tf 11.95 0 Td[()]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Z)]TJ /F3 11.955 Tf 5.47 -9.68 Td[()]TJ /F5 7.97 Tf -1.33 -8.01 Td[(j;vj)]TJ /F3 11.955 Tf 21.91 0 Td[()]TJ /F3 11.955 Tf 5.47 -9.69 Td[(Z)]TJ /F3 11.955 Tf 5.48 -9.69 Td[()]TJ /F5 7.97 Tf -1.34 -8.01 Td[(j;vj; (B)wherethelastinequalityholdsbecauseV(x)LV(x)basedoncondition2.Nowwestarttoprovethemainresults. ForanyT>0,f romLemma 3.1 andnotingthatZ(0)]TJ /F4 11.955 Tf 9.3 0 Td[()=x,wehave e)]TJ /F5 7.97 Tf 6.59 0 Td[(rTV(Z(T))=V(x)+RT0e)]TJ /F5 7.97 Tf 6.58 0 Td[(rtAV(Z(t))dt+RT0e)]TJ /F5 7.97 Tf 6.59 0 Td[(rtV0(Z(t))dB(t)+PN(T)n=1e)]TJ /F5 7.97 Tf 6.58 0 Td[(rn[V(Z(n)))]TJ /F3 11.955 Tf 11.95 0 Td[(V(Z(n)]TJ /F4 11.955 Tf 9.3 0 Td[())]V(x))]TJ /F9 11.955 Tf 11.95 9.64 Td[(RT0e)]TJ /F5 7.97 Tf 6.58 0 Td[(rth(Z(t))dt+RT0e)]TJ /F5 7.97 Tf 6.58 0 Td[(rtV0(Z(t))dB(t)+PN(T)n=1e)]TJ /F5 7.97 Tf 6.58 0 Td[(rn[V(Z(n)))]TJ /F3 11.955 Tf 11.95 0 Td[(V(Z(n)]TJ /F4 11.955 Tf 9.3 0 Td[())]V(x))]TJ /F9 11.955 Tf 11.95 9.63 Td[(RT0e)]TJ /F5 7.97 Tf 6.58 0 Td[(rth(Z(t))dt+RT0e)]TJ /F5 7.97 Tf 6.58 0 Td[(rtV0(Z(t))dB(t))]TJ /F9 11.955 Tf 11.29 8.97 Td[(PN(T)n=1e)]TJ /F5 7.97 Tf 6.59 0 Td[(rn)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Z)]TJ /F3 11.955 Tf 5.48 -9.68 Td[()]TJ /F5 7.97 Tf -1.33 -8.01 Td[(j;vj;(B)wheretherstinequalityholdsbecauseAV(x) )]TJ /F3 11.955 Tf 21.92 0 Td[(h(x) basedon C ondition3andthelastinequalityholdsbecauseof( B ) andEhRT0e)]TJ /F5 7.97 Tf 6.58 0 Td[(rtV0(Z(t))dB(t)i=0 .Takingexpectationonbothsidesof( B ),wehave 176

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Ee)]TJ /F5 7.97 Tf 6.59 0 Td[(rTV(Z(T))V(x))]TJ /F3 11.955 Tf 11.95 0 Td[(Ex24ZT0e)]TJ /F5 7.97 Tf 6.59 0 Td[(rth(Z(t))dt+N(T)Xn=1e)]TJ /F5 7.97 Tf 6.58 0 Td[(rn)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Z)]TJ /F3 11.955 Tf 5.48 -9.68 Td[()]TJ /F5 7.97 Tf -1.34 -8.01 Td[(j;vj35; forallT>0. MovingExhRT0e)]TJ /F5 7.97 Tf 6.58 0 Td[(rth(Z(t))dt+PN(T)n=1e)]TJ /F5 7.97 Tf 6.59 0 Td[(rn)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Z)]TJ /F3 11.955 Tf 5.48 -9.69 Td[()]TJ /F5 7.97 Tf -1.34 -8.01 Td[(j;vjitotheothersideandlettingT!+1,wehaveV(x)liminfT!+1Ex24ZT0e)]TJ /F5 7.97 Tf 6.59 0 Td[(rth(Z(t))dt+N(T)Xn=1e)]TJ /F5 7.97 Tf 6.58 0 Td[(rn)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Z)]TJ /F3 11.955 Tf 5.48 -9.68 Td[()]TJ /F5 7.97 Tf -1.34 -8.01 Td[(j;vj35+liminfT!+1Exe)]TJ /F5 7.97 Tf 6.59 0 Td[(rTV(Z(T)):Finally,becauseh(x) 0 and(x;v) 0 forallx2Randv0, bothRT0e)]TJ /F5 7.97 Tf 6.58 0 Td[(rth(Z(t))dtandPN(T)n=1e)]TJ /F5 7.97 Tf 6.59 0 Td[(rn)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Z)]TJ /F3 11.955 Tf 5.48 -9.69 Td[()]TJ /F5 7.97 Tf -1.33 -8.01 Td[(j;vjarenon-negativeandincreaseinT, andso liminfT!+1andExcanbeexchanged. Therefore,w ehaveV(x)Ex"Z10e)]TJ /F5 7.97 Tf 6.59 0 Td[(rth(Z(t))dt+1Xn=1e)]TJ /F5 7.97 Tf 6.59 0 Td[(rn)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Z)]TJ /F3 11.955 Tf 5.48 -9.68 Td[()]TJ /F5 7.97 Tf -1.34 -8.01 Td[(j;vj#+liminfT!+1Exe)]TJ /F5 7.97 Tf 6.59 0 Td[(rTV(Z(T)) (B)=Ex"Z10e)]TJ /F5 7.97 Tf 6.58 0 Td[(rth(Z(t))dt+1Xn=1e)]TJ /F5 7.97 Tf 6.58 0 Td[(rn)]TJ /F3 11.955 Tf 5.47 -9.68 Td[(Z)]TJ /F3 11.955 Tf 5.48 -9.68 Td[()]TJ /F5 7.97 Tf -1.34 -8.01 Td[(j;vj#;wheretheequalityholds becauseliminfT!+1Exe)]TJ /F5 7.97 Tf 6.59 0 Td[(rTV(Z(T))=0,whichcanbejustiedbasedonthefollowingarguments: FromCondition4,wehavethereexistconstantsm0andm1suchthatjV(x)jm0+m1h(x)forallx,whichimpliesExe)]TJ /F5 7.97 Tf 6.58 0 Td[(rTjV(Z(T))jm0e)]TJ /F5 7.97 Tf 6.59 0 Td[(rT+m1Exe)]TJ /F5 7.97 Tf 6.58 0 Td[(rTh(Z(T)).Meanwhile,ExR10e)]TJ /F5 7.97 Tf 6.59 0 Td[(rTh(Z(T))dT<1basedon( B ),whichimpliesthatliminfT!+1Exe)]TJ /F5 7.97 Tf 6.58 0 Td[(rTh(Z(T))=0.Thus,wehave 177

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0liminfT!+1Exe)]TJ /F5 7.97 Tf 6.59 0 Td[(rTV(Z(T))liminfT!+1Exe)]TJ /F5 7.97 Tf 6.58 0 Td[(rTjV(Z(T))jm0liminfT!+1e)]TJ /F5 7.97 Tf 6.58 0 Td[(rT+m1liminfT!+1Exe)]TJ /F5 7.97 Tf 6.59 0 Td[(rTh(Z(T))=0: B.3ProofofTheorem 3.3 Proof. Weonlyneedtoprovetheinequilitiesin( B )equalbasedonCondition5and6. Inordertoprovetheequalityachieved,weneedtoprovetheinequalityin( B )equals,whichrequiresthetwoinequalitiesin( B )equal. Therstinequalityin( B )canbeprovedtoequalbecauseAV(x)+h(x)=0forx2CbasedonCondition5andZ(t)2Cforallt6=^iwithi=1;2;:::basedonCondition6. Thesecondinequalityin( B )canbeprovedtoequalifthetwoinequalitiesin( B )equal,whichrequirestheinequalityin( B )equalsandLV)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Z)]TJ /F4 11.955 Tf 6.08 -9.69 Td[(^)]TJ /F5 7.97 Tf -1.34 -8.01 Td[(j=V)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Z)]TJ /F4 11.955 Tf 6.09 -9.68 Td[(^)]TJ /F5 7.97 Tf -1.34 -8.01 Td[(j.Notethatthefactthattheinequalityin( B )equalsholdsdirectlybasedonthedenitionof^jand^qjinCondition6.ThefactthatLV)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Z)]TJ /F4 11.955 Tf 6.09 -9.68 Td[(^)]TJ /F5 7.97 Tf -1.34 -8.02 Td[(j=V)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Z)]TJ /F4 11.955 Tf 6.09 -9.68 Td[(^)]TJ /F5 7.97 Tf -1.33 -8.02 Td[(jholdsbecauseofthedenitionofCinCondition5andthedenitionof^jinCondition6. TogetherwiththeresultsinTheorem 3.2 ,wehavetheconclusion. B.4SomeLemmasandProofs LemmaB.1. Let f0()=)]TJ /F3 11.955 Tf 9.3 0 Td[(r++22=2;(B)and2=h+p 2+2r2i=2.Thenf0()]TJ /F3 11.955 Tf 9.3 0 Td[(2)=0and2>0. 178

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Proof. Theconclusionimmediatelyfollowsthefactthatf0()isaquadraticfunctionofand1=h)]TJ /F3 11.955 Tf 9.3 0 Td[(+p 2+2r2i=2and)]TJ /F3 11.955 Tf 9.3 0 Td[(2=)]TJ /F9 11.955 Tf 11.29 13.27 Td[(h+p 2+2r2i=2aretworootsoff0()=0 . LemmaB.2. ForthefunctionH(x)=ExR10e)]TJ /F5 7.97 Tf 6.59 0 Td[(rth(X(t))dt denedin( 3 ) ,wehaveAH(x)+h(x)=0 ,whereAisdenedin( 3 ). Proof. Thisconclusionfollowsfrompage45in[ 31 ]. LemmaB.3. ForthefunctionKe)]TJ /F5 7.97 Tf 6.59 0 Td[(2x,where2=h+p 2+2r2i=2,wehaveA)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Ke)]TJ /F5 7.97 Tf 6.59 0 Td[(2x=0 ,whereAisdenedin( 3 ). Proof. From( 3 ),wehave B.5ProofofLemma 3.2 Proof. Frist,f romthepropositioninpage45of[ 31 ],wehaveH(x)=e1xZ1xh(y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1ydy+e)]TJ /F5 7.97 Tf 6.59 0 Td[(2xZxh(y)e2ydy=p 2+2r2: Now ,wehavetherstorderderivativeof H (x) H0(x) =1e1xR1xh(y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1ydy)]TJ /F3 11.955 Tf 11.95 0 Td[(e1xh(x)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1x )]TJ /F3 11.955 Tf 9.3 0 Td[(2e)]TJ /F5 7.97 Tf 6.58 0 Td[(2xRxh(y)e2ydy+e)]TJ /F5 7.97 Tf 6.59 0 Td[(2xh(x)e2xo=p 2+2r2 =h1e1xR1xh(y)e)]TJ /F5 7.97 Tf 6.58 0 Td[(1ydy)]TJ /F3 11.955 Tf 11.95 0 Td[(2e)]TJ /F5 7.97 Tf 6.59 0 Td[(2xRxh(y)e2ydyi=p 2+2r2 =he1xR1xh0(y)e)]TJ /F5 7.97 Tf 6.58 0 Td[(1ydy+h(x)+e)]TJ /F5 7.97 Tf 6.59 0 Td[(2xRxh0(y)e2ydy)]TJ /F3 11.955 Tf 11.96 0 Td[(h(x)i=p 2+2r2 =he1xR1xh0(y)e)]TJ /F5 7.97 Tf 6.58 0 Td[(1ydy+e)]TJ /F5 7.97 Tf 6.58 0 Td[(2xRxh0(y)e2ydyi=p 2+2r2;(B) wherethethirdequalityfollowsfromintegrationbyparts,limy!1h(y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1y=0andlimy!h(y)e2y=0basedonAssumption 3.2 (e). BasedonAssumption 3.2 (b),wediscussintwocases:x
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Case1:x
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H0(x)=8>>>>>>><>>>>>>>:he1xR1xh00(y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1ydy=1)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F5 7.97 Tf 6.58 0 Td[(2xRxh00(y)e2ydy=2+)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(h0(a+))]TJ /F3 11.955 Tf 11.96 0 Td[(h0(a)]TJ /F4 11.955 Tf 9.3 0 Td[()e1(x)]TJ /F5 7.97 Tf 6.58 0 Td[(a)=1+(1=1+1=2)h0(x)=p 2+2r2forx>>>>>><>>>>>>>:he1xR1xh00(y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1ydy+e)]TJ /F5 7.97 Tf 6.59 0 Td[(2xRxh00(y)e2ydy+)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(h0(a+))]TJ /F3 11.955 Tf 11.96 0 Td[(h0(a)]TJ /F4 11.955 Tf 9.3 0 Td[()e1(x)]TJ /F5 7.97 Tf 6.58 0 Td[(a)=p 2+2r2forx0 bycontradiction .IfH00(x)>0doesnothold,thereexistsay0suchthatH00(y0)=0. From( B ),wehave h0(a+)=h0(a)]TJ /F4 11.955 Tf 9.3 0 Td[()(B)and Zy0h00(y)e2ydy=0andZ1y0h00(y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1ydy=0:(B) From( B )and h0(x)0forxaandh0(x)0forx
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h(x)=h(a)forallx,whichcontradictswithlimx!+1h(x)=+1inAssumption 3.2 . Thatisoriginalconclusionholds. B.6ProofofProposition 3.1 BeforeweproveProposition 3.1 ,werstprovealemma. LemmaB.4. Forf(x)denedin( 3 ),wehave f00(x)0;(B)andthereexistsa ~z suchthattherst-orderderivativeoff(x) f0(x)=H00(x)=2+H0(x)+c8>>>><>>>>:<0x< ~z =0x= ~z >0x> ~z :(B) Proof. WerstderiveH000(x). From( B ),wehavethethirdorderderivativeof H (x) H000(x)=8>>>>>>><>>>>>>>:h1e1xR1xh00(y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1ydy)]TJ /F3 11.955 Tf 11.95 0 Td[(2e)]TJ /F5 7.97 Tf 6.58 0 Td[(2xRxh00(y)e2ydy+1)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(h0(a+))]TJ /F3 11.955 Tf 11.95 0 Td[(h0(a)]TJ /F4 11.955 Tf 9.3 0 Td[()e1(x)]TJ /F5 7.97 Tf 6.59 0 Td[(a)=p 2+2r2forxa:(B) Basedonthedenitionof1and2inTheorem 3.1 ,wehave (1=1+1=2)=p 2+2r2=1=r:(B) F rom( 3 ),wehave 182

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f0(x)=H00(x)=2+H0(x)+c =8>>>><>>>>:c+e1xR1xh00(y)e)]TJ /F5 7.97 Tf 6.58 0 Td[(1ydy+)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(h0(a+))]TJ /F3 11.955 Tf 11.96 0 Td[(h0(a)]TJ /F4 11.955 Tf 9.29 0 Td[()e1(x)]TJ /F5 7.97 Tf 6.58 0 Td[(a)+h0(x)(1=1+1=2)=p 2+2r2forx<>:c+e1xR1xh00(y)e)]TJ /F5 7.97 Tf 6.58 0 Td[(1ydy+)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(h0(a+))]TJ /F3 11.955 Tf 11.95 0 Td[(h0(a)]TJ /F4 11.955 Tf 9.3 0 Td[()e1(x)]TJ /F5 7.97 Tf 6.59 0 Td[(a)+h0(x)=rforx<>:1e1xR1xh00(y)e)]TJ /F5 7.97 Tf 6.58 0 Td[(1ydy+)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(h0(a+))]TJ /F3 11.955 Tf 11.95 0 Td[(h0(a)]TJ /F4 11.955 Tf 9.3 0 Td[()e1(x)]TJ /F5 7.97 Tf 6.59 0 Td[(a)=rforxa; (B) wherethelastequalityholdsbecauseof( B ),( B ),and( B ). Fromh00(x)0basedonAssumption 3.2 (a)andh0(a+)h0(a)]TJ /F4 11.955 Tf 9.3 0 Td[()basedonAssumption 3.2 (c),wehave f00(x)0:(B) Now,we provethat( B )holds byproving(i)f0(x)>0forxa,(ii)limx!f0(x)<0,and(iii)f00(x)>0forx0;(B) 183

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wheretherstinequalityholdsbecauseofh0(x)0forxabasedonAssumption 3.2 (c)andh00(x)0basedonAssumption 3.2 (a)and(b). Second,weprove limx!f0(x)<0.Notethatfrom( B ),wehavethatforx0forx0forx
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Sinceh0(x)0forxaandh0(x)0forxa.Thus,h0(x)=h0(a+)=0andsoh(x)=h(a)forallx>a,whichcontradictswithlimx!+1h(x)=+1inAssumption 3.2 (d).Finally,from( B ),( B ),and( B ),wehavethatthereexistsa ~z >>><>>>>:<0x< ~z =0x= ~z >0x> ~z :(B) NowwestarttoproveProposition 3.1 . Proof. From( B ),( B ),and( B ),wehaveforanyx< ~z ,thereexistsauniquey(x)> ~z suchthatf(x)=f(y(x)) asshowninFigure B-1 .Letu(x)=y(x))]TJ /F3 11.955 Tf 12.43 0 Td[(x,wehaveu( ~z )=0andu( ~z )]TJ /F3 11.955 Tf 11.96 0 Td[(Q)>Q.SothereexistsauniqueR< ~z suchthatu(R)=Qandsof(R)=f(R+Q).AlsonotethatR+Q=y(R)> ~z .Dene F(x):=f(x))]TJ /F3 11.955 Tf 11.95 0 Td[(f(x+Q):(B)Inthefollowing,weprove( 3 )holdsbyproving 185

PAGE 186

. . s . f(s) . ~z . x . y(x) . f(x) . f(y(x)) FigureB-1. Existenceofy(x) F (x):=f(x))]TJ /F3 11.955 Tf 11.95 0 Td[(f(x+Q)8>>>><>>>>:>0forxR:(B)First,wehaveproved F (R)=0andalsohave F 0(x)=f0(x))]TJ /F3 11.955 Tf 11.95 0 Td[(f0(x+Q)=)]TJ /F3 11.955 Tf 9.29 0 Td[(f00()Q0(B)basedon( B ),where2[x;x+Q].Inthefollowing,weprovethat F 0(x)<0forx< ~z and F (x)<0forx ~z .Second,forx ~z )]TJ /F3 11.955 Tf 12.32 0 Td[(Q
0basedon( B ).Thus,notingthatR< ~z ,wehave F (x)> F (R)=0forx F ( x ) forR ~z ,wehave F (x) F ( ~z )<0basedon( B ). B.7ProofofLemma 3.3 BeforeweproveLemma 3.3 ,werststateandproveanotherlemmaasfollows. 186

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LemmaB.5. For (x)denedas (x)=(H(x))]TJ /F3 11.955 Tf 11.96 0 Td[(H(x+Q))]TJ /F3 11.955 Tf 11.96 0 Td[(cQ)=e)]TJ /F5 7.97 Tf 6.59 0 Td[(2x)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F5 7.97 Tf 6.58 0 Td[(2Q;(B)wehave 0(x)=2[f(x))]TJ /F3 11.955 Tf 11.96 0 Td[(f(x+Q)]=e)]TJ /F5 7.97 Tf 6.59 0 Td[(2x)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2Q8>>>><>>>>:<0forx>R=0forx=R>0forx0,f(x)andRaredenedinLemma 3.1 . Proof. Fromthedenitionofq(x) asin( B ) ,wehave 0(x)=)]TJ /F3 11.955 Tf 10.46 -9.69 Td[(H0(x))]TJ /F3 11.955 Tf 11.95 0 Td[(H0(x+Q))]TJ /F4 11.955 Tf 11.96 0 Td[((H(x))]TJ /F3 11.955 Tf 11.95 0 Td[(H(x+Q))]TJ /F3 11.955 Tf 11.95 0 Td[(cQ)()]TJ /F3 11.955 Tf 9.3 0 Td[(2)=e)]TJ /F5 7.97 Tf 6.59 0 Td[(2x)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2Q=2[f(x))]TJ /F3 11.955 Tf 11.96 0 Td[(f(x+Q)]=e)]TJ /F5 7.97 Tf 6.59 0 Td[(2x)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2Q=2q1(x)=e)]TJ /F5 7.97 Tf 6.58 0 Td[(2x)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2Q;wherethesecondequalityholdsbecauseof( 3 )andthelastequalityholdsbecausesof( B ).Fromtheaboveequationand( B ),wehave 0(x)8>>>><>>>>:>0forxR: NowweproveLemma 3.3 asfollows. Proof. Beforeweprovethelemma,weprovesomepreliminaryresults. First,from( B ),wehave 187

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(R)> (x)forallx6=R:(B)Second,from( 3 ),wehaveK=(H(R+Q))]TJ /F3 11.955 Tf 11.95 0 Td[(H(R)+cQ)=)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(e)]TJ /F5 7.97 Tf 6.58 0 Td[(2(R+Q))]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2R= (R); wherethesecondequalityholdsbecauesof( B ). Nowweprovetheclaim. From( 3 ),wehavethatforl0andl2Z+,g(x;l+1))]TJ /F3 11.955 Tf 11.96 0 Td[(g(x;l)=H(x+(l+1)Q))]TJ /F3 11.955 Tf 11.96 0 Td[(H(x+lQ)+cQ)]TJ ET BT /F3 11.955 Tf 244.38 -256.58 Td[( (R))]TJ /F3 11.955 Tf 5.48 -9.68 Td[(e)]TJ /F5 7.97 Tf 6.58 0 Td[(2(x+(l+1)Q))]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2(x+lQ)=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2(x+lQ))]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2(x+(l+1)Q) (R))]TJ /F4 11.955 Tf 11.96 0 Td[([H(x+lQ))]TJ /F3 11.955 Tf 11.95 0 Td[(H(x+(l+1)Q))]TJ /F3 11.955 Tf 11.96 0 Td[(cQ]=)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2(x+lQ))]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2(x+(l+1)Q)=)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2(x+lQ))]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2(x+(l+1)Q)( (R))]TJ ET BT /F3 11.955 Tf 233.36 -328.31 Td[( (x+lQ));wherethelastequalityholdsbecauseof( B ). Nowweonlyneedtoshowx+lQ6=Rbecausee)]TJ /F5 7.97 Tf 6.58 0 Td[(2(x+lQ))]TJ /F3 11.955 Tf 12.25 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2(x+(l+1)Q)>0and (R)> (x)forallx6=Rbasedon( B ). Ifx>R,wehave x+lQ>R .ForxRandl( x;R ),wehavex+lQ>R basedonthedenitionof(x;R)in( 3 ) .Insum,wehaveg(x;l+1)>g(x;l)forl( x;R ): 188

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APPENDIXCPROOFS C.1ProofofLemma 4.2 Proof. Frist,f romthepropositioninpage45of[ 31 ],wehavew(x)=e1xZ1xh(y)e)]TJ /F5 7.97 Tf 6.58 0 Td[(1ydy+e)]TJ /F5 7.97 Tf 6.58 0 Td[(2xZxh(y)e2ydy=p 2+2r2: Now ,wehavetherstorderderivativeof H (x) w0 (x) =1e1xR1xh(y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1ydy)]TJ /F3 11.955 Tf 11.96 0 Td[(e1xh(x)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1x )]TJ /F3 11.955 Tf 9.3 0 Td[(2e)]TJ /F5 7.97 Tf 6.59 0 Td[(2xRxh(y)e2ydy+e)]TJ /F5 7.97 Tf 6.58 0 Td[(2xh(x)e2xo=p 2+2r2 =h1e1xR1xh(y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1ydy)]TJ /F3 11.955 Tf 11.95 0 Td[(2e)]TJ /F5 7.97 Tf 6.58 0 Td[(2xRxh(y)e2ydyi=p 2+2r2 =he1xR1x h0 (y)e)]TJ /F5 7.97 Tf 6.58 0 Td[(1ydy+h(x)+e)]TJ /F5 7.97 Tf 6.59 0 Td[(2xRx h0 (y)e2ydy)]TJ /F3 11.955 Tf 11.96 0 Td[(h(x)i=p 2+2r2 =he1xR1x h0 (y)e)]TJ /F5 7.97 Tf 6.58 0 Td[(1ydy+e)]TJ /F5 7.97 Tf 6.58 0 Td[(2xRx h0 (y)e2ydyi=p 2+2r2;(C) wherethethirdequalityholdsbecauseofintegrationbyparts,limy!1h(y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1y=0andlimy!h(y)e2y=0basedonAssumption 4.2 (e). BasedonAssumption 4.2 (b),wediscussintwocases:x
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w0 (x)=e1xRax h0 (y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1ydy+e1xR1a h0 (y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1ydy+e)]TJ /F5 7.97 Tf 6.59 0 Td[(2xRx h0 (y)e2ydyi=p 2+2r2=e1x h0 (y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1y=()]TJ /F3 11.955 Tf 9.3 0 Td[(1)jax)]TJ /F3 11.955 Tf 11.95 0 Td[(e1xRax h00 (y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1ydy=()]TJ /F3 11.955 Tf 9.3 0 Td[(1)+e1x h0 (y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1y=()]TJ /F3 11.955 Tf 9.29 0 Td[(1)j1a)]TJ /F3 11.955 Tf 11.96 0 Td[(e1xR1a h00 (y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1ydy=()]TJ /F3 11.955 Tf 9.3 0 Td[(1)+e)]TJ /F5 7.97 Tf 6.59 0 Td[(2x h0 (y)e2y=2jx)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2xRx h00 (y)e2ydy=2i=p 2+2r2= h0 (x)=1)]TJ /F3 11.955 Tf 11.96 0 Td[(e1x h0 (a)]TJ /F4 11.955 Tf 9.3 0 Td[()e)]TJ /F5 7.97 Tf 6.58 0 Td[(1a=1+e1xRax h00 (y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1ydy=1+e1x h0 (a+)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1a=1+e1xR1a h00 (y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1ydy=1+ h0 (x)=2)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F5 7.97 Tf 6.58 0 Td[(2xRx h00 (y)e2ydy=2i=p 2+2r2=he1xR1x h00 (y)e)]TJ /F5 7.97 Tf 6.58 0 Td[(1ydy=1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2xRx h00 (y)e2ydy=2+( h0 (a+))]TJ ET BT /F3 11.955 Tf 142.9 -259.21 Td[(h0 (a)]TJ /F4 11.955 Tf 9.29 0 Td[())e1(x)]TJ /F5 7.97 Tf 6.58 0 Td[(a)=1+(1=1+1=2) h0 (x)=p 2+2r2: Similarly,wehavethatunderCase2wherexa, w0 (x)=e1xR1x h0 (y)e)]TJ /F5 7.97 Tf 6.58 0 Td[(1ydy+e)]TJ /F5 7.97 Tf 6.59 0 Td[(2xRa h0 (y)e2ydy+e)]TJ /F5 7.97 Tf 6.59 0 Td[(2xRxa h0 (y)e2ydyi=p 2+2r2=e1x h0 (y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1y=()]TJ /F3 11.955 Tf 9.3 0 Td[(1)j1x)]TJ /F3 11.955 Tf 11.95 0 Td[(e1xR1x h00 (y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1ydy=()]TJ /F3 11.955 Tf 9.3 0 Td[(1)+e)]TJ /F5 7.97 Tf 6.58 0 Td[(2x h0 (y)e2y=2ja)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2xRa h00 (y)e2ydy=2+e)]TJ /F5 7.97 Tf 6.59 0 Td[(2x h0 (y)e2y=2jxa)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2xRxa h00 (y)e2ydy=2=p 2+2r2= h0 (x)=1+e1xR1x h00 (y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1ydy=1+e)]TJ /F5 7.97 Tf 6.58 0 Td[(2x h0 (a)]TJ /F4 11.955 Tf 9.3 0 Td[()e2a=2)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2xRa h00 (y)e2ydy=2+ h0 (x)=2)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F5 7.97 Tf 6.58 0 Td[(2x h0 (a+)e2a=2)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2xRxa h00 (y)e2ydy=2=p 2+2r2=he1xR1x h00 (y)e)]TJ /F5 7.97 Tf 6.58 0 Td[(1ydy=1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2xRx h00 (y)e2ydy=2)]TJ /F4 11.955 Tf 11.29 0 Td[(( h0 (a+))]TJ ET BT /F3 11.955 Tf 133.42 -558.1 Td[(h0 (a)]TJ /F4 11.955 Tf 9.3 0 Td[())e)]TJ /F5 7.97 Tf 6.59 0 Td[(2(x)]TJ /F5 7.97 Tf 6.59 0 Td[(a)=2+(1=1+1=2) h0 (x)=p 2+2r2: Insum,wehave 190

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w0 (x)=8>>>>>>><>>>>>>>:he1xR1x h00 (y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1ydy=1)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2xRx h00 (y)e2ydy=2+( h0 (a+))]TJ ET BT /F3 11.955 Tf 125.93 -66.15 Td[(h0 (a)]TJ /F4 11.955 Tf 9.29 0 Td[())e1(x)]TJ /F5 7.97 Tf 6.59 0 Td[(a)=1+(1=1+1=2) h0 (x)=p 2+2r2forx>>>>>><>>>>>>>:he1xR1x h00 (y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1ydy+e)]TJ /F5 7.97 Tf 6.59 0 Td[(2xRx h00 (y)e2ydy+( h0 (a+))]TJ ET BT /F3 11.955 Tf 165.49 -233.55 Td[(h0 (a)]TJ /F4 11.955 Tf 9.3 0 Td[())e1(x)]TJ /F5 7.97 Tf 6.58 0 Td[(a)=p 2+2r2forx0 bycontradiction .If w00 (x)>0doesnothold,thereexistsay0suchthat w00 (y0)=0. From( C ),wehave h0 (a+)= h0 (a)]TJ /F4 11.955 Tf 9.3 0 Td[()(C)and Zy0 h00 (y)e2ydy=0andZ1y0 h00 (y)e)]TJ /F5 7.97 Tf 6.58 0 Td[(1ydy=0:(C) From( C )and h0 (x)0forxaand h0 (x)0forx
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forallx,whichcontradictswithlimx!+1h(x)=+1inAssumption 4.2 . Thatisoriginalconclusionholds. C.2ProofofLemma 4.3 Proof. (i) Werstderive w000 (x). From( C ),wehavethethirdorderderivativeof w (x) w000 (x)=8>>>>>>><>>>>>>>:h1e1xR1x h00 (y)e)]TJ /F5 7.97 Tf 6.58 0 Td[(1ydy)]TJ /F3 11.955 Tf 11.96 0 Td[(2e)]TJ /F5 7.97 Tf 6.59 0 Td[(2xRx h00 (y)e2ydy+1( h0 (a+))]TJ ET BT /F3 11.955 Tf 166.64 -180.34 Td[(h0 (a)]TJ /F4 11.955 Tf 9.3 0 Td[())e1(x)]TJ /F5 7.97 Tf 6.59 0 Td[(a)=p 2+2r2forxa:(C) Basedonthedenitionof1and2inTheorem 4.1 ,wehave (1=1+1=2)=p 2+2r2=1=r:(C) F rom( 4 ),wehave f0 (x)= w00 (x)=2+ w0 (x) =8>>>><>>>>:e1xR1x h00 (y)e)]TJ /F5 7.97 Tf 6.58 0 Td[(1ydy+( h0 (a+))]TJ ET BT /F3 11.955 Tf 135.91 -437.22 Td[(h0 (a)]TJ /F4 11.955 Tf 9.29 0 Td[())e1(x)]TJ /F5 7.97 Tf 6.59 0 Td[(a)+ h0 (x)(1=1+1=2)=p 2+2r2forx<>:e1xR1x h00 (y)e)]TJ /F5 7.97 Tf 6.58 0 Td[(1ydy+( h0 (a+))]TJ ET BT /F3 11.955 Tf 249.09 -486.24 Td[(h0 (a)]TJ /F4 11.955 Tf 9.3 0 Td[())e1(x)]TJ /F5 7.97 Tf 6.59 0 Td[(a)+ h0 (x)=rforx
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f00 (x)= w000 (x)=2+ w00 (x)=8><>:1e1xR1x h00 (y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1ydy+( h0 (a+))]TJ ET BT /F3 11.955 Tf 264.04 -76.91 Td[(h0 (a)]TJ /F4 11.955 Tf 9.3 0 Td[())e1(x)]TJ /F5 7.97 Tf 6.59 0 Td[(a)=rforxa; (C) wherethelastequalityholdsbecauseof( C ),( C ),and( C ). From h00 (x)0basedonAssumption 4.2 (a)and h0 (a+) h0 (a)]TJ /F4 11.955 Tf 9.3 0 Td[()basedonAssumption 4.2 (c),wehave f00 (x)0:(C) Now,we provethat( 4 )holds byproving(i) f0 (x)>0forxa,(ii)limx! f0 (x)<0,and(iii) f00 (x)>0forx
0forxa:(C)Weprovethisbycontradiction.Ifthisclaimdoesnothold,thereexistsay0asuchthat f0 (y0)=0.FromAssumption 4.2 (a)and(b),wehavethat h00 (x)0.From( C ),wehavethat f0 (y0)=0implies h0 (y0)=0andR1y0 h00 (y)e)]TJ /F5 7.97 Tf 6.59 -.01 Td[(1ydy=0.Furthermore,fromthefactsthatR1y0 h00 (y)e)]TJ /F5 7.97 Tf 6.58 0 Td[(1ydy=0,h(x)2C2(R)exceptatx=a,and h00 (x)0forallx,wehave h00 (x)=0forallxy0.Thus, h0 (x)= h0 (y0)=0andsoh(x)=h(y0)forallxy0,whichcontradictswithlimx!+1h(x)=+1inAssumption 4.2 (d). Second,weprove limx! f0 (x)<0.Notethatfrom( C ),wehavethatforx
PAGE 194

f0 (x)=e1xZ1x h00 (y)e)]TJ /F5 7.97 Tf 6.58 0 Td[(1ydy+( h0 (a+))]TJ ET BT /F3 11.955 Tf 234.08 -47.82 Td[(h0 (a)]TJ /F4 11.955 Tf 9.3 0 Td[())e1(x)]TJ /F5 7.97 Tf 6.59 0 Td[(a)=r=e1x h0 (y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1yj1x)]TJ /F4 11.955 Tf 11.95 0 Td[(()]TJ /F3 11.955 Tf 9.3 0 Td[(1)e1xZ1x h0 (y)e)]TJ /F5 7.97 Tf 6.58 0 Td[(1ydy+( h0 (a+))]TJ ET BT /F3 11.955 Tf 361.26 -80.5 Td[(h0 (a)]TJ /F4 11.955 Tf 9.3 0 Td[())e1(x)]TJ /F5 7.97 Tf 6.59 0 Td[(a)=r=1e1xZ1x h0 (y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1ydy+( h0 (a+))]TJ ET BT /F3 11.955 Tf 243.34 -113.17 Td[(h0 (a)]TJ /F4 11.955 Tf 9.3 0 Td[())e1(x)]TJ /F5 7.97 Tf 6.58 0 Td[(a)=r: From( C ),wehave limx! f0 (x)=limx!e1xR1x h00 (y)e)]TJ /F5 7.97 Tf 6.58 0 Td[(1ydy+( h0 (a+))]TJ ET BT /F3 11.955 Tf 327.56 -202.63 Td[(h0 (a)]TJ /F4 11.955 Tf 9.3 0 Td[())e1(x)]TJ /F5 7.97 Tf 6.59 0 Td[(a)+ h0 (x)=r=limx!1e1xR1x h0 (y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1ydy=r=limx! h0 (x)=r<)]TJ /F3 11.955 Tf 9.3 0 Td[(cu; (C) wherethelastequalityholdsbecauseofHospital'sRuleandtheinequalityholdsbecauseofAssumption 4.2 (d). Third,weprove f00 (x)>0forx0forxa.Thus, h0 (x)= h0 (a+)=0andso 194

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h(x)=h(a)forallx>a,whichcontradictswithlimx!+1h(x)=+1inAssumption 4.2 (d).Finally,from( C ),( C ),and( C ),wehavethatthereexistsa zf>>><>>>>:<0x0x>zf:(C)(ii)Similarwiththeproofforf(x),from( 4 ),wehave g0 (x)= w0 (x))]TJ ET BT /F3 11.955 Tf 86.87 -319.29 Td[(w00 (x)=1=8><>: h0 (x)=r)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2xRx h00 (y)e2ydy=rforx
<>:2e)]TJ /F5 7.97 Tf 6.59 0 Td[(2xRx h00 (y)e2ydy=rforxa:Theremainingprooffollowssimilarargumentwiththatoff(x).(iii)From( 4 ),( 4 ),and( C ),wehavef(x)=(1+1=2)e1xZ1xh(y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1ydy=p 2+2r2: 195

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Takingtherstorderderivativeoff(x),wehave f0 (x)=1e1xZ1xh(y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1ydy)]TJ /F3 11.955 Tf 11.95 0 Td[(h(x)(1+1=2)=p 2+2r2:Thus,from( 4 ),wehaveF(x)=f(x))]TJ ET BT /F3 11.955 Tf 135.63 -137.97 Td[(f0 (x)=1=h(x)(1=1+1=2)=p 2+2r2=h(x)=r:SimilarlywecanproveG(x)=h(x)=r=F(x). C.3ProofofLemma 4.4 Proof. Intherstpartofthisproof,weanalyze thesignofAd(x).FromthedenitionofAd(x)in( 4 ),weknowthatthesignofAd(x)isthesameasthesignoff(x)]TJ /F3 11.955 Tf 11.96 0 Td[(Q))]TJ /F3 11.955 Tf -441.23 -23.9 Td[(f(x)+cdQ.Therefore,weonlyneedtoprovethereexistsaaD0suchthatf(x)]TJ /F3 11.955 Tf 11.96 0 Td[(Q))]TJ /F3 11.955 Tf -447.38 -42.84 Td[(f(x)+cdQ8>>>><>>>>:>0ifxD0:Inthefollowing,weprove(i) [f(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(x)+cdQ]0 <0,(ii)limx![f(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(x)+cdQ]>0,and(iii)limx!+1[f(x)]TJ /F3 11.955 Tf 11.96 0 Td[(Q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(x)+cdQ]<0.(i)Weprove [f(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(x)+cdQ]0 <0:(C)Thisisbecause [f(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(x)+cdQ]0 = f0 (x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ ET BT /F3 11.955 Tf 299.68 -519.42 Td[(f0 (x)=)]TJ ET BT /F3 11.955 Tf 352.11 -519.42 Td[(f00 ()Q<0,where2[x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q;x].(ii)Weprovelimx![f(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(x)+cdQ]>0.Notethat f(x)]TJ /F3 11.955 Tf 11.96 0 Td[(Q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(x)+cdQ=(cd)]TJ ET BT /F3 11.955 Tf 177.12 -627 Td[(f0 (1))Q2[(cd)]TJ ET BT /F3 11.955 Tf 269.31 -627 Td[(f0 (x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))Q;(cd)]TJ ET BT /F3 11.955 Tf 368.91 -627 Td[(f0 (x))Q];(C) 196

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From( C ),wehave limx![f(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(x)+cdQ]limx!(cd)]TJ ET BT /F3 11.955 Tf 319.03 -65.55 Td[(f0 (x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))Q>(cd+cu)Q>0;(C)wherethesecondinequalityholdsbecauseof( C ).(iii)Weprovelimx!+1[f(x)]TJ /F3 11.955 Tf 11.96 0 Td[(Q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(x)+cdQ]<0.From( C ),wehavelimx!+1[f(x)]TJ /F3 11.955 Tf 11.96 0 Td[(Q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(x)+cdQ]limx!+1(cd)]TJ ET BT /F3 11.955 Tf 323.59 -225.98 Td[(f0 (x))Qlimx!+1(cd)]TJ ET BT /F3 11.955 Tf 323.59 -255.9 Td[(h0 (x)=r)Q (C)<0;wherethesecondinequalityholdsbecause f0 (x) h0 (x)=rforxabasedon( C ),andthelastinequalityholdsbecauseof(d)ofAssumption 4.2 .Thus,from( 4 ),( 4 )( C ),wehavethatthereexistsaD0suchthat Ad(x)8>>>><>>>>:>0ifxD0:(C) Inthesecondpartofthisproof, weanalyzethesignof A0d (x).From( 4 )and( 4 ),wehave A0d (x)=[12=(1+2)][F(x))]TJ /F3 11.955 Tf 11.95 0 Td[(F(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(cdQ]=e1x)]TJ /F3 11.955 Tf 11.96 0 Td[(e1(x)]TJ /F5 7.97 Tf 6.59 0 Td[(Q):(C) 197

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Sothesignof A0d (x)isthesameasthesignofF(x))]TJ /F3 11.955 Tf 10.62 0 Td[(F(x)]TJ /F3 11.955 Tf 10.63 0 Td[(Q))]TJ /F3 11.955 Tf 10.63 0 Td[(cdQ.Therefore,weonlyneedtoprovethereexistsaaD1suchthatF(x))]TJ /F3 11.955 Tf 11.96 0 Td[(F(x)]TJ /F3 11.955 Tf 11.96 0 Td[(Q))]TJ /F3 11.955 Tf 11.95 0 Td[(cdQ8>>>><>>>>:<0ifx0ifx>D1.Inthefollowing,weprove(i) [F(x))]TJ /F3 11.955 Tf 11.95 0 Td[(F(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(cdQ]0 >0,(ii)limx![F(x))]TJ /F3 11.955 Tf 11.96 0 Td[(F(x)]TJ /F3 11.955 Tf 11.96 0 Td[(Q))]TJ /F3 11.955 Tf 11.95 0 Td[(cdQ]<0,and(iii)limx!+1[F(x))]TJ /F3 11.955 Tf 11.95 0 Td[(F(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(cdQ]>0.(i)Weprove [F(x))]TJ /F3 11.955 Tf 11.95 0 Td[(F(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(cdQ]0 >0:(C)Thisisbecause [F(x))]TJ /F3 11.955 Tf 11.95 0 Td[(F(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(cdQ]0 = F0 (x))]TJ ET BT /F3 11.955 Tf 299.91 -248.95 Td[(F0 (x)]TJ /F3 11.955 Tf 11.22 0 Td[(Q)= F00 ()Q>0,where2[x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q;x].(ii)Weprovelimx![F(x))]TJ /F3 11.955 Tf 11.95 0 Td[(F(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(cdQ]<0.Notethat F(x))]TJ /F3 11.955 Tf 11.96 0 Td[(F(x)]TJ /F3 11.955 Tf 11.96 0 Td[(Q))]TJ /F3 11.955 Tf 11.95 0 Td[(cdQ=( F0 (1))]TJ /F3 11.955 Tf 11.96 0 Td[(cd)Q2[( F0 (x)]TJ /F3 11.955 Tf 11.96 0 Td[(Q))]TJ /F3 11.955 Tf 11.95 0 Td[(cd)Q;( F0 (x))]TJ /F3 11.955 Tf 11.96 0 Td[(cd)Q]:(C)From( C ),wehavelimx![F(x))]TJ /F3 11.955 Tf 11.95 0 Td[(F(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(cdQ]limx!( F0 (x))]TJ /F3 11.955 Tf 11.96 0 Td[(cd)Q (C)=limx!( h0 (x)=r)]TJ /F3 11.955 Tf 11.95 0 Td[(cd)Q<)]TJ /F4 11.955 Tf 11.29 0 Td[((cd+cu)Q<0;wheretheequalityholdsbecauseF(x)=h(x)basedonLemma 4.3 ,andthesecondinequalityholdsbecauseof(d)ofAssumption 4.2 .(iii)Weprovelimx!+1[F(x))]TJ /F3 11.955 Tf 11.95 0 Td[(F(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(cdQ]>0.From( C ),wehave 198

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limx!+1[F(x))]TJ /F3 11.955 Tf 11.95 0 Td[(F(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(cdQ]limx!+1( F0 (x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(cd)Q (C)=limx!+1( h0 (x)]TJ /F3 11.955 Tf 11.96 0 Td[(Q)=r)]TJ /F3 11.955 Tf 11.95 0 Td[(cd)Q>0:wheretheequalityholdsbecauseF(x)=h(x)basedonLemma 4.3 ,andthelastinequalityholdsbecauseof(d)ofAssumption 4.2 .Thus,from( C )( C ),wehavethatthereexistsaD1suchthat A0d (x)8>>>><>>>>:<0ifx0ifx>D1:Intheremainingpartofthisproof,weproveD0>>><>>>>:>0ifxD2:Inthefollowing,weprove(i) [g(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(g(x)+cdQ]0 <0, 199

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(ii)limx![g(x)]TJ /F3 11.955 Tf 11.96 0 Td[(Q))]TJ /F3 11.955 Tf 11.95 0 Td[(g(x)+cdQ]>0,and(iii)limx!+1[g(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ /F3 11.955 Tf 11.95 0 Td[(g(x)+cdQ]<0.(i)Weprove [g(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(g(x)+cdQ]0 <0:(C)Thisisbecause [g(x)]TJ /F3 11.955 Tf 11.96 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(g(x)+cdQ]0 = g0 (x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ ET BT /F3 11.955 Tf 297.32 -137.97 Td[(g0 (x)=)]TJ ET BT /F3 11.955 Tf 349.23 -137.97 Td[(g00 ()Q<0,where2[x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q;x].(ii)and(iii)Notethat g(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(g(x)+cdQ=(cd)]TJ /F3 11.955 Tf 11.96 0 Td[(g(1))Q2[(cd)]TJ ET BT /F3 11.955 Tf 267.41 -245.56 Td[(g0 (x)]TJ /F3 11.955 Tf 11.96 0 Td[(Q))Q;(cd)]TJ ET BT /F3 11.955 Tf 366 -245.56 Td[(g0 (x))Q];(C)SimilarlywiththeproofforthecaseAd(x),from( C ),wehave limx![g(x)]TJ /F3 11.955 Tf 11.96 0 Td[(Q))]TJ /F3 11.955 Tf 11.95 0 Td[(g(x)+cdQ]limx!(cd)]TJ ET BT /F3 11.955 Tf 318.53 -335.01 Td[(g0 (x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))Q>(cd+cu)Q>0;(C)andlimx!+1[g(x)]TJ /F3 11.955 Tf 11.96 0 Td[(Q))]TJ /F3 11.955 Tf 11.95 0 Td[(g(x)+cdQ]limx!+1(cd)]TJ ET BT /F3 11.955 Tf 322.58 -471.36 Td[(g0 (x))Qlimx!+1(cd)]TJ ET BT /F3 11.955 Tf 322.58 -501.28 Td[(h0 (x)=r)Q (C)<0:Thus,from( 4 ),( C )( C ),wehavethatthereexistsaD2suchthat 200

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Bd(x)8>>>><>>>>:<0ifx0ifx>D2:(C) Inthesecondpartofthisproof, weanalyzethesignof B0d (x).From( 4 )and( 4 ),wehave B0d (x)=[12=(1+2)][G(x))]TJ /F3 11.955 Tf 11.96 0 Td[(G(x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ /F3 11.955 Tf 11.95 0 Td[(cdQ]=e)]TJ /F5 7.97 Tf 6.59 0 Td[(2(x)]TJ /F5 7.97 Tf 6.59 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2x:(C)Thus,thesignof B0d (x)issamewiththesignofG(x))]TJ /F3 11.955 Tf 11.96 0 Td[(G(x)]TJ /F3 11.955 Tf 11.96 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(cdQ.NotethatG(x)=F(x)basedonLemma 4.3 (iii).From( C ),wehave B0d (x)hasthesamesignwith A0d (x).Especially,from( C )and( C ),wehave B0d (x)= A0d (x)e1x)]TJ /F3 11.955 Tf 11.95 0 Td[(e1(x)]TJ /F5 7.97 Tf 6.58 0 Td[(Q)=e)]TJ /F5 7.97 Tf 6.59 0 Td[(2(x)]TJ /F5 7.97 Tf 6.59 0 Td[(Q))]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F5 7.97 Tf 6.58 0 Td[(2x: Intheremainingpartofthisproof, weproveD2>D1.Notethatg(D2)]TJ /F3 11.955 Tf 11.96 0 Td[(Q))]TJ /F3 11.955 Tf 10.56 0 Td[(g(D2)+cdQ=0basedonthe( 4 )and( C ).From( 4 ),wehaveG(D2))]TJ /F3 11.955 Tf 12.21 0 Td[(G(D2)]TJ /F3 11.955 Tf 12.21 0 Td[(Q))]TJ /F3 11.955 Tf -449.92 -23.91 Td[(cdQ=[ g0 (D2))]TJ ET BT /F3 11.955 Tf 85.2 -406.45 Td[(g0 (D2)]TJ /F3 11.955 Tf 11.95 0 Td[(Q)]=2= g00 (3)Q=2>0,where32[D2)]TJ /F3 11.955 Tf 11.96 0 Td[(Q;D2].SowehaveD0D1. C.6ProofofLemma 4.7 Proof. TheproofissimilarwiththeproofofLemma 4.6 .From( 4 )and( 4 ),wehave A0u (x)=[12=(1+2)][F(x+Q))]TJ /F3 11.955 Tf 11.96 0 Td[(F(x)+cuQ]=e1(x+Q))]TJ /F3 11.955 Tf 11.95 0 Td[(e1x:(C)From( 4 )and( 4 ),wehave 201

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B0u (x)=[12=(1+2)][G(x+Q))]TJ /F3 11.955 Tf 11.96 0 Td[(G(x)+cuQ]=e)]TJ /F5 7.97 Tf 6.59 0 Td[(2x)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2(x+Q):(C)NotethatG(x)=F(x),wehave B0u (x)= A0u (x)e1(x+Q))]TJ /F3 11.955 Tf 11.96 0 Td[(e1x=e)]TJ /F5 7.97 Tf 6.58 0 Td[(2x)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F5 7.97 Tf 6.58 0 Td[(2(x+Q): C.7ProofofLemma 4.8 Proof. WerstgivetheexpressionofBd(D0)basedonthepropertiesofD0.NotingthatfromLemma 4.4 ,wehave f(D0)]TJ /F3 11.955 Tf 11.96 0 Td[(Q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(D0)+cdQ=0:(C)From( 4 ),wehave w(D0)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(w(D0)+cdQ= w0 (D0)=2)]TJ ET BT /F3 11.955 Tf 305.89 -311.93 Td[(w0 (D0)]TJ /F3 11.955 Tf 11.95 0 Td[(Q)=2:(C)From( 4 ),wehaveg(D0)]TJ /F3 11.955 Tf 11.95 0 Td[(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(g(D0)+cdQ=w(D0)]TJ /F3 11.955 Tf 11.96 0 Td[(Q))]TJ /F3 11.955 Tf 11.95 0 Td[(w(D0)+cdQ)]TJ ET BT /F3 11.955 Tf 311.49 -402.08 Td[(w0 (D0)]TJ /F3 11.955 Tf 11.95 0 Td[(Q)=1+ w0 (D0)=1=(1=1+1=2)[ w0 (D0))]TJ ET BT /F3 11.955 Tf 283.27 -428.98 Td[(w0 (D0)]TJ /F3 11.955 Tf 11.95 0 Td[(Q)];wherethesecondequalityholdsbecauseof( C ).Thus,from( 4 ),wehaveBd(D0)=[1=(1+2)][g(D0)]TJ /F3 11.955 Tf 11.96 .01 Td[(Q))]TJ /F3 11.955 Tf 11.95 -.01 Td[(g(D0)+cdQ]=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2D0)]TJ /F3 11.955 Tf 11.96 -.01 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2(D0)]TJ /F5 7.97 Tf 6.59 0 Td[(Q)=[ w0 (D0))]TJ ET BT /F3 11.955 Tf 147.95 -551.51 Td[(w0 (D0)]TJ /F3 11.955 Tf 11.95 0 Td[(Q)]=2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(e)]TJ /F5 7.97 Tf 6.58 0 Td[(2D0)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F5 7.97 Tf 6.58 0 Td[(2(D0)]TJ /F5 7.97 Tf 6.58 0 Td[(Q)=2WF(D0); (C)whereWF(x)isdenedas 202

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WF(x):=[ w0 (x)]TJ /F3 11.955 Tf 11.96 0 Td[(Q))]TJ ET BT /F3 11.955 Tf 224.95 -35.86 Td[(w0 (x)]=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2(x)]TJ /F5 7.97 Tf 6.59 0 Td[(Q))]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F5 7.97 Tf 6.58 0 Td[(2x:(C)Next,wegivetheexpressionsofBu(S0)basedonthepropertiesofS0.Similarly,fromLemma 4.5 ,wehave f(S0+Q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(S0)+cuQ=0:(C)From( 4 ),wehave w(S0+Q))]TJ /F3 11.955 Tf 11.96 0 Td[(w(S0)+cuQ= w0 (S0)=2)]TJ ET BT /F3 11.955 Tf 303.64 -228.12 Td[(w0 (S0+Q)=2:(C)From( 4 ),wehaveg(S0+Q))]TJ /F3 11.955 Tf 11.95 0 Td[(g(S0)+cuQ=w(S0+Q))]TJ /F3 11.955 Tf 11.96 0 Td[(w(S0)+cuQ)]TJ ET BT /F3 11.955 Tf 309.42 -318.28 Td[(w0 (S0+Q)=1+ w0 (S0)=1=(1=1+1=2)[ w0 (S0))]TJ ET BT /F3 11.955 Tf 283.36 -345.18 Td[(w0 (S0+Q)];wherethesecondequalityholdsbecauseof( C ).Hence,from( 4 ),wehaveBu(S0)=[1=(1+2)][g(S0+Q))]TJ /F3 11.955 Tf 11.95 0 Td[(g(S0)+cuQ]=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(e)]TJ /F5 7.97 Tf 6.58 0 Td[(2S0)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2(S0+Q)=[ w0 (S0))]TJ ET BT /F3 11.955 Tf 148.59 -467.71 Td[(w0 (S0+Q)]=2)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(e)]TJ /F5 7.97 Tf 6.58 0 Td[(2S0)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2(S0+Q)=2WF(S0+Q); (C)whereWF(x)isdenedin( C ).NowweareabletocompareBd(D0)andBu(S0).From( C )and( C ),wehaveBd(D0))]TJ /F3 11.955 Tf 13.06 0 Td[(Bu(S0)=2[WF(D0))]TJ /F3 11.955 Tf 11.96 0 Td[(WF(S0+Q)].Hence,inordertoproveBd(D0)Bu(S0),itsufcestoprove (WF)0 (x)0andD0>S0+Q.First,weprove (WF)0 (x)0.TakingderivativeofWF(x),wehave 203

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(WF)0 (x)=2[ f0 (x)]TJ /F3 11.955 Tf 11.96 0 Td[(Q))]TJ ET BT /F3 11.955 Tf 227.15 -35.86 Td[(f0 (x)]=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(e)]TJ /F5 7.97 Tf 6.58 0 Td[(2(x)]TJ /F5 7.97 Tf 6.59 0 Td[(Q))]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2x0;wheretheequalityholdsbecauseof( 4 )and( C )andtheinequalityholdsbecauseof f00 (x)0basedonLemma 4.3 .Second,weproveD0>S0+Q.DeneWf(x):=f(x))]TJ /F3 11.955 Tf 12.74 0 Td[(f(x)]TJ /F3 11.955 Tf 11.96 0 Td[(Q).Wehave (Wf)0 (x)= f0 (x))]TJ ET BT /F3 11.955 Tf 107.32 -137.97 Td[(f0 (x)]TJ /F3 11.955 Tf 11.95 0 Td[(Q)0because f00 (x)0basedonLemma 4.3 .Finally,from( C )and( C ),wehaveWf(D0)=f(D0))]TJ /F3 11.955 Tf 12.2 0 Td[(f(D0)]TJ /F3 11.955 Tf 11.95 0 Td[(Q)=cdQ>)]TJ /F3 11.955 Tf 9.3 0 Td[(cuQ=f(S0+Q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(S0)=Wf(S0+Q).Thus,wehaveD0>S0+Q. C.8ProofofLemma 4.9 Proof. Weprovideprooffor(i)here.(ii)canbeprovedsimilarly.Werstprovesomesimpleresults.From( 4 )and( 4 ),wehave qu(S)=0and q0u (S)=0:(C)Wealsohave qu(D)]TJ /F3 11.955 Tf 11.95 0 Td[(Q)=(D))]TJ /F3 11.955 Tf 11.96 0 Td[((D)]TJ /F3 11.955 Tf 11.96 0 Td[(Q)+cuQ=cuQ+cdQ>0;(C)wherethesecondequalityholdsbecauseofthethirdequationin( 4 ),and q0u (D)]TJ /F3 11.955 Tf 11.95 0 Td[(Q)= 0 (D))]TJ ET BT /F3 11.955 Tf 252.62 -468.33 Td[(0 (D)]TJ /F3 11.955 Tf 11.96 0 Td[(Q)=0;(C)wherethelastequalityholdsbecauseofthefourthequationin( 4 ).Inordertoprovequ(x)>0forS
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1(x):=[f(x+Q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(x)+cuQ]e)]TJ /F5 7.97 Tf 6.59 0 Td[(1x(C) 2(x):=[g(x+Q))]TJ /F3 11.955 Tf 11.95 0 Td[(g(x)+cuQ]e2x(C)Notethatqu(x)=[2Fqu(x)+1Gqu(x)]=(1+2)and q0u (x)=12[Fqu(x))]TJ /F3 11.955 Tf 11.96 0 Td[(Gqu(x)]=(1+2).Inordertoprovequ(x)forS
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samesignwith A0u (x)basedon( 4 )inLemma 4.5 ,wehave 01 (x)8>>>><>>>>:>0ifxS1.NotethatSS1suchthat1(x)8>>>>>>><>>>>>>>:<1(S)ifx1(S)ifSSFandso Fqu(x)8>>>>>>>>>><>>>>>>>>>>:<0ifx0ifSSF:(C)Furthermore,from( C ),( C ),( C ),and( C ),wehaveFqu(S)=qu(S)+ q0u (S)=2=0andFqu(D)]TJ /F3 11.955 Tf 11.95 0 Td[(Q)=qu(D)]TJ /F3 11.955 Tf 11.96 0 Td[(Q)+ q0u (D)]TJ /F3 11.955 Tf 11.96 0 Td[(Q)=2=cuQ+cdQ>0.From( C ),wehaveD)]TJ /F3 11.955 Tf 11.95 0 Td[(Q0forS
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Next,weexplorethesignofGqu(x).SimilarlywiththeproofofFqu(x),from( C ),wehaveGqu(x)=qu(x))]TJ ET BT /F3 11.955 Tf 124.23 -71.73 Td[(q0u (x)=1=g(x+Q))]TJ /F3 11.955 Tf 11.96 0 Td[(g(x)+cuQ)]TJ /F4 11.955 Tf 11.95 0 Td[((1+2=1)B)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2Qe)]TJ /F5 7.97 Tf 6.58 0 Td[(2x( 4\00059 )=g(x+Q))]TJ /F3 11.955 Tf 11.96 0 Td[(g(x)+cuQ)]TJ /F4 11.955 Tf 11.96 0 Td[([g(S+Q))]TJ /F3 11.955 Tf 11.96 0 Td[(g(S)+cuQ]e2Se)]TJ /F5 7.97 Tf 6.59 0 Td[(2x=e)]TJ /F5 7.97 Tf 6.59 0 Td[(2x[g(x+Q))]TJ /F3 11.955 Tf 11.96 0 Td[(g(x)+cuQ]e2x)]TJ /F4 11.955 Tf 11.95 0 Td[([g(S+Q))]TJ /F3 11.955 Tf 11.95 0 Td[(g(S)+cuQ]e2S( C\00047 )=e)]TJ /F5 7.97 Tf 6.59 0 Td[(2x[2(x))]TJ /F3 11.955 Tf 11.96 0 Td[(2(S)];wherethesecondequalityholdsbecauseof( 4 )and( 4 ).Thus,Gqu(x)hasthesamesignwith2(x))]TJ /F3 11.955 Tf 11.96 0 Td[(2(S).Inordertocompare2(x)with2(S),westudythederivativeof2(x).From( C ),wehave 02 (x)=[ g0 (x+Q))]TJ ET BT /F3 11.955 Tf 174.18 -322.76 Td[(g0 (x)]e2x+2[g(x+Q))]TJ /F3 11.955 Tf 11.96 0 Td[(g(x)+cuQ]e2x=2e2xf[g(x+Q))]TJ /F3 11.955 Tf 11.95 0 Td[(g(x)+cuQ]+[ g0 (x+Q))]TJ ET BT /F3 11.955 Tf 352.42 -349.66 Td[(g0 (x)]=2g( 4\00031 )=2e2x[G(x+Q))]TJ /F3 11.955 Tf 11.95 0 Td[(G(x)+cuQ];=2e2x[F(x+Q))]TJ /F3 11.955 Tf 11.96 0 Td[(F(x)+cuQ] (C)wherethelastequalityholdsbecauseG(x)=F(x)basedon(iv)ofLemma 4.3 .Thenweanalyzethesignof2(x).From( 4 )and( C ),wehavethat 02 (x)hasthesamesignwith[F(x+Q))]TJ /F3 11.955 Tf 11.96 0 Td[(F(x)+cuQ].Since[F(x+Q))]TJ /F3 11.955 Tf 11.95 0 Td[(F(x)+cuQ]hasthesamesignwith A0u (x)basedon( 4 )inLemma 4.5 ,wehave 02 (x)8>>>><>>>>:<0ifx0ifx>S1. 207

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NotethatSS1suchthat2(x)8>>>>>>><>>>>>>>:>2(S)ifx2(S)ifx>SGandso Gqu(x)8>>>>>>>>>><>>>>>>>>>>:>0ifx0ifx>SG:(C)Moreover,from( C ),( C ),( C ),and( C ),wehaveGqu(S)=qu(S))]TJ ET BT /F3 11.955 Tf 0 -328.71 Td[(q0u (S)=1=0andGqu(D)]TJ /F3 11.955 Tf 11.96 0 Td[(Q)=qu(D)]TJ /F3 11.955 Tf 11.96 0 Td[(Q))]TJ ET BT /F3 11.955 Tf 251.44 -328.71 Td[(q0u (D)]TJ /F3 11.955 Tf 11.96 0 Td[(Q)=1=cuQ+cdQ>0.SinceD)]TJ /F3 11.955 Tf 11.96 0 Td[(Q>S,from( C ),wehaveD)]TJ /F3 11.955 Tf 11.95 0 Td[(Q>SG.Hence,from( C ),wehave Gqu(x)0forS0forSG0forS0forSG0forSG0forS
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12[Fqu(x))]TJ /F3 11.955 Tf 11.95 0 Td[(Gqu(x)]=(1+2)>0forSqu(S)=0forS0forSV(x)forxS Iffl2N:u(S;x)+1lu(D)]TJ /F4 11.955 Tf 11.95 0 Td[(0;x))]TJ /F4 11.955 Tf 11.95 0 Td[(1g=;,Lu2V(x)=1>V(x)obviouslyholds.Inthefollowing,weassumefl2N:u(S;x)+1lu(D)]TJ /F4 11.955 Tf 11.96 0 Td[(0;x))]TJ /F4 11.955 Tf 11.95 0 Td[(1g6=;.Foru(S;x)+1lu(D)]TJ /F4 11.955 Tf 11.95 0 Td[(0;x))]TJ /F4 11.955 Tf 12.29 0 Td[(1,fromthedenitionofu(y;x)in( 4 ),wehaveS
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. . inventory . x . Q . x+Q . Q . x+2Q . Q . x+3Q . Q . x+4Q . d(D;x+4Q)=2 . D . u(D;x)=3 FigureC-1. Illustrationofd(D;x+lQ)andu(D;x)withl=4 InordertoproveV(x+lQ)+culQ>V(x)forl2fl2N:u(S;x)+1lu(D)]TJ /F4 11.955 Tf 11.95 0 Td[(0;x))]TJ /F4 11.955 Tf 11.96 0 Td[(1g;itsufcestoshowqu(x+iQ)>0fori2[u(S;x);l)]TJ /F4 11.955 Tf 11.96 0 Td[(1]\N.FromLemma 4.9 ,wehavequ(y)>0forS0fori2[u(S;x);l)]TJ /F4 11.955 Tf 11.95 0 Td[(1]andso[V(x+lQ)+culQ])]TJ /F3 11.955 Tf 11.95 0 Td[(V(x)>0.Hence, from( 4 ), wehave Lu2V(x)=minl2N;u(S;x)+1lu(D)]TJ /F7 7.97 Tf 6.58 0 Td[(0;x))]TJ /F7 7.97 Tf 6.59 0 Td[(1fV(x+lQ)+culQg>V(x):(C)(d)Lu3V(x)>V(x)forxS Weproveintwocases:case1wherexSandx6=D)]TJ /F3 11.955 Tf 11.43 0 Td[(iQfori=1;2;:::,andcase2wherexSandx=D)]TJ /F3 11.955 Tf 11.96 0 Td[(iQfori=1;2;:::. Case1:xSandx6=D)]TJ /F3 11.955 Tf 12.16 0 Td[(iQfori=1;2;:::. Forlu(D)]TJ /F4 11.955 Tf 11.96 0 Td[(0;x)andl2N,fromthedenitionofu(y;x)in( 4 ),wehavex+lQDandd(D;x+lQ)=l)]TJ /F3 11.955 Tf 9.48 0 Td[(u(D;x)+1,wheretheintuitioncanbeunderstoodbyFigure C-1 .Accordingly,wehave lQ)]TJ /F3 11.955 Tf 11.95 0 Td[(d(D;x+lQ)Q=u(D;x)Q)]TJ /F3 11.955 Tf 11.96 0 Td[(QforxSandl2N:(C) 210

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From( 4 ),wehaveV(x+lQ)( 4\0008 )=(x+lQ)]TJ /F3 11.955 Tf 11.96 0 Td[(d(D;x+lQ)Q)+cdd(D;x+lQ)Q( C\00055 )=(x+u(D;x)Q)]TJ /F3 11.955 Tf 11.95 0 Td[(Q)+cd(l)]TJ /F3 11.955 Tf 11.96 0 Td[(u(D;x)+1)Q( 4\0008 )=V(x+u(D;x)Q)]TJ /F3 11.955 Tf 11.95 0 Td[(Q)+cd(l)]TJ /F3 11.955 Tf 11.95 0 Td[(u(D;x)+1)Q:BeforeweproveLu3V(x)>V(x),werstprove(x+u(D)]TJ /F3 11.955 Tf 11.95 0 Td[(Q;x)Q)+cuu(D)]TJ /F3 11.955 Tf 11.96 0 Td[(Q;x)Q(x+u(S;x)Q)+cuu(S;x)Q.NotethatD)]TJ /F3 11.955 Tf 11.95 0 Td[(Q>SbasedonTheorem 4.4 ,wehaveu(D)]TJ /F3 11.955 Tf 11.95 0 Td[(Q;x)u(S;x).Ifu(D)]TJ /F3 11.955 Tf 11.95 0 Td[(Q;x)=u(S;x),theclaimobviouslyholds.Ifu(D)]TJ /F3 11.955 Tf 11.95 0 Td[(Q;x)>u(S;x),wehave (x+u(D)]TJ /F3 11.955 Tf 11.96 0 Td[(Q;x)Q)+cuu(D)]TJ /F3 11.955 Tf 11.96 0 Td[(Q;x)Q)]TJ /F3 11.955 Tf 11.95 0 Td[((x+u(S;x)Q))]TJ /F3 11.955 Tf 11.95 0 Td[(cuu(S;x)Q=(x+u(S;x)Q+(u(D)]TJ /F3 11.955 Tf 11.96 0 Td[(Q;x))]TJ /F3 11.955 Tf 11.96 0 Td[(u(S;x))Q))]TJ /F3 11.955 Tf 9.3 0 Td[((x+u(S;x)Q)+cu(u(D)]TJ /F3 11.955 Tf 11.96 0 Td[(Q;x))]TJ /F3 11.955 Tf 11.95 0 Td[(u(S;x))Q( 4\00071 )=Pu(D)]TJ /F5 7.97 Tf 6.59 0 Td[(Q;x))]TJ /F5 7.97 Tf 6.58 0 Td[(u(S;x))]TJ /F7 7.97 Tf 6.58 0 Td[(1i=0qu(x+u(S;x)Q+iQ)>0;(C)wheretheinequalityholdsbecausequ(x)>0forS
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. . inventory . D)]TJ /F4 11.955 Tf 11.95 0 Td[(2Q . Q . D)]TJ /F3 11.955 Tf 11.95 0 Td[(Q . Q . D . Q . D+Q . Q . D+2Q . d(D;D+2Q)=3 . u(D;D)]TJ /F4 11.955 Tf 11.96 0 Td[(2Q)=3 FigureC-2. Illustrationofd(D;x+lQ)andu(D;x)withx=D)]TJ /F4 11.955 Tf 11.96 0 Td[(2Qandl=4 Lu3V(x)( 4\00079 )=minl2N;lu(D;x)fV(x+lQ)+culQg( 4\0008 )=minl2N;lu(D;x)f(x+u(D;x)Q)]TJ /F3 11.955 Tf 11.96 0 Td[(Q)+cd(l)]TJ /F3 11.955 Tf 11.95 0 Td[(u(D;x)+1)Q+culQg=(x+u(D;x)Q)]TJ /F3 11.955 Tf 11.96 0 Td[(Q)+cdQ+cuu(D;x)Q=(x+u(D)]TJ /F3 11.955 Tf 11.95 0 Td[(Q;x)Q)+cdQ+cuu(D;x)Q=(x+u(D)]TJ /F3 11.955 Tf 11.95 0 Td[(Q;x)Q)+cuu(D)]TJ /F3 11.955 Tf 11.96 0 Td[(Q;x)Q+(cd+cu)Q( C\00056 )(x+u(S;x)Q)+cuu(S;x)Q+(cd+cu)Q>(x+u(S;x)Q)+cuu(S;x)Q( 4\0008 )=V(x);(C)wherethefourthandfthequalitiesholdbecauseofthedenitionofu(y;x)in( 4 )andthefactthatxS
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V(x+lQ)( 4\0008 )=(x+lQ)]TJ /F3 11.955 Tf 11.95 0 Td[(d(D;x+lQ)Q)+cdd(D;x+lQ)Q( C\00058 )=(x+u(D;x)Q)]TJ /F4 11.955 Tf 11.95 0 Td[(2Q)+cd(l)]TJ /F3 11.955 Tf 11.96 0 Td[(u(D;x)+2)Q( 4\0008 )=V(x+u(D;x)Q)]TJ /F4 11.955 Tf 11.96 0 Td[(2Q)+cd(l)]TJ /F3 11.955 Tf 11.95 0 Td[(u(D;x)+2)Q: BeforeweproveLu3V(x)>V(x),werstprove(x+u(D)]TJ /F4 11.955 Tf 11.95 0 Td[(2Q;x)Q)+cuu(D)]TJ /F4 11.955 Tf 11.96 0 Td[(2Q;x)Q(x+u(S;x)Q)+cuu(S;x)Qforx=D)]TJ /F3 11.955 Tf 12.51 0 Td[(iQfori=1;2;:::.NotethatD)]TJ /F3 11.955 Tf 13.07 0 Td[(Q>SbasedonTheorem 4.4 .IfD)]TJ /F4 11.955 Tf 13.08 0 Td[(2QS,wecertainlyhaveu(D)]TJ /F4 11.955 Tf 11.96 0 Td[(2Q;x)u(S;x).IfD)]TJ /F4 11.955 Tf 12.3 0 Td[(2Qu(S;x),wehave (x+u(D)]TJ /F4 11.955 Tf 11.96 0 Td[(2Q;x)Q)+cuu(D)]TJ /F4 11.955 Tf 11.96 0 Td[(2Q;x)Q)]TJ /F3 11.955 Tf 11.95 0 Td[((x+u(S;x)Q))]TJ /F3 11.955 Tf 11.95 0 Td[(cuu(S;x)Q=(x+u(S;x)Q+(u(D)]TJ /F4 11.955 Tf 11.95 0 Td[(2Q;x))]TJ /F3 11.955 Tf 11.96 0 Td[(u(S;x))Q))]TJ /F3 11.955 Tf 9.29 0 Td[((x+u(S;x)Q)+cu(u(D)]TJ /F4 11.955 Tf 11.95 0 Td[(2Q;x))]TJ /F3 11.955 Tf 11.96 0 Td[(u(S;x))Q( 4\00071 )=Pu(D)]TJ /F7 7.97 Tf 6.59 0 Td[(2Q;x))]TJ /F5 7.97 Tf 6.58 0 Td[(u(S;x))]TJ /F7 7.97 Tf 6.58 0 Td[(1i=0qu(x+u(S;x)Q+iQ)>0;(C)wheretheinequalityholdsbecausequ(y)>0forS
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Lu3V(x)( 4\00079 )=minl2N;lu(D;x)fV(x+lQ)+culQg( 4\0008 )=minl2N;lu(D;x)f(x+u(D;x)Q)]TJ /F4 11.955 Tf 11.95 0 Td[(2Q)+cd(l)]TJ /F3 11.955 Tf 11.96 0 Td[(u(D;x)+2)Q+culQg=(x+u(D;x)Q)]TJ /F4 11.955 Tf 11.96 0 Td[(2Q)+2cdQ+cuu(D;x)Q=(x+u(D)]TJ /F4 11.955 Tf 11.96 0 Td[(2Q;x)Q)+2cdQ+cuu(D;x)Q=(x+u(D)]TJ /F4 11.955 Tf 11.96 0 Td[(2Q;x)Q)+cuu(D)]TJ /F4 11.955 Tf 11.95 0 Td[(2Q;x)Q+2(cd+cu)Q( C\00059 )(x+u(S;x)Q)+cuu(S;x)Q+2(cd+cu)Q>(x+u(S;x)Q)+cuu(S;x)Q( 4\0008 )=V(x);(C)wherethefourthandfthequalitiesholdbecauseofthedenitionofu(y;x)in( 4 )andthefactthatx=D)]TJ /F3 11.955 Tf 11.95 0 Td[(iQfori=1;2;:::incase2. C.10ContinuedProofofProposition 4.2 Proof. Inthispart,weproveLu2V(x)>V(x)forSDbasedonlu(D;x)andthedenitionofu(y;x)in( 4 ),andthesecondequalityholdsbecauselQ)]TJ /F3 11.955 Tf -425.36 -23.9 Td[(d(D;x+lQ)Q=u(D;x)Q)]TJ /F3 11.955 Tf 11.95 0 Td[(Q,wheretheintuitioncanbeunderstoodbyFigure C-1 .BeforeweproveLu3V(x)>V(x),werstprove(x+u(D)]TJ /F3 11.955 Tf 11.95 0 Td[(Q;x)Q)+cuu(D)]TJ /F3 11.955 Tf 11.96 0 Td[(Q;x)Q(x).Basedonthedenitionofu(y;x)in( 4 ),wehaveu(D)]TJ /F3 11.955 Tf 11.96 0 Td[(Q;x)0.Ifu(D)]TJ /F3 11.955 Tf 11.96 0 Td[(Q;x)=0,theclaimobviouslyholds.Ifu(D)]TJ /F3 11.955 Tf 11.95 0 Td[(Q;x)1,wehave 214

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(x+u(D)]TJ /F3 11.955 Tf 11.95 0 Td[(Q;x)Q)+cuu(D)]TJ /F3 11.955 Tf 11.95 0 Td[(Q;x)Q)]TJ /F3 11.955 Tf 11.96 0 Td[((x)( 4\00071 )=Pu(D)]TJ /F5 7.97 Tf 6.59 0 Td[(Q;x))]TJ /F7 7.97 Tf 6.59 0 Td[(1i=0qu(x+iQ)>0;(C)wheretheinequalityholdsbecausequ(x)>0forSV(x)inPropositon 4.1 ,wehaveLu2V(x)( 4\00090 )=minl2N;lu(D;x)fV(x+lQ)+culQg( 4\0008 )=minl2N;lu(D;x)f(x+u(D;x)Q)]TJ /F3 11.955 Tf 11.96 0 Td[(Q)+cd(l)]TJ /F3 11.955 Tf 11.95 0 Td[(u(D;x)+1)Q+culQg=(x+u(D;x)Q)]TJ /F3 11.955 Tf 11.96 0 Td[(Q)+cdQ+cuu(D;x)Q=(x+u(D)]TJ /F3 11.955 Tf 11.95 0 Td[(Q;x)Q)+cuu(D)]TJ /F3 11.955 Tf 11.96 0 Td[(Q;x)Q+(cu+cd)Q( C\00061 )(x)+(cu+cd)Q>(x)( 4\0008 )=V(x);wherethefourthequalityholdsbecauseofthedenitionofu(y;x)in( 4 ). Case2:SDbasedonlu(D;x)andthedenitionofu(y;x)in( 4 ),andthesecondequalityholdsbecauselQ)]TJ /F3 11.955 Tf -425.36 -23.9 Td[(d(D;x+lQ)Q=u(D;x)Q)]TJ /F4 11.955 Tf 12.19 0 Td[(2Qwhenx=D)]TJ /F3 11.955 Tf 12.2 0 Td[(iQfori=1;2;:::,wheretheintuitioncanbeunderstoodbyFigure C-2 . 215

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BeforeweproveLu2V(x)>V(x),werstprove(x+u(D)]TJ /F4 11.955 Tf 11.95 0 Td[(2Q;x)Q)+cuu(D)]TJ /F4 11.955 Tf 11.96 0 Td[(2Q;x)Q(x)forx>Sandx=D)]TJ /F3 11.955 Tf 12.04 0 Td[(iQfori=1;2;:::.Ifu(D)]TJ /F4 11.955 Tf 11.95 0 Td[(2Q;x)=0,theclaimobviouslyholds.Ifu(D)]TJ /F4 11.955 Tf 11.96 0 Td[(2Q;x)1,wehave (x+u(D)]TJ /F4 11.955 Tf 11.95 0 Td[(2Q;x)Q)+cuu(D)]TJ /F4 11.955 Tf 11.96 0 Td[(2Q;x)Q)]TJ /F3 11.955 Tf 11.96 0 Td[((x)( 4\00071 )=Pu(D)]TJ /F7 7.97 Tf 6.59 0 Td[(2Q;x))]TJ /F7 7.97 Tf 6.58 0 Td[(1i=0qu(x+iQ)>0;(C)wheretheinequalityholdsbecausequ(x)>0forSV(x)inPropositon 4.1 ,wehaveLu2V(x)( 4\00090 )=minl2N;lu(D;x)fV(x+lQ)+culQg( 4\0008 )=minl2N;lu(D;x)f(x+u(D;x)Q)]TJ /F4 11.955 Tf 11.95 0 Td[(2Q)+cd(l)]TJ /F3 11.955 Tf 11.96 0 Td[(u(D;x)+2)Q+culQg=(x+u(D;x)Q)]TJ /F4 11.955 Tf 11.96 0 Td[(2Q)+2cdQ+cuu(D;x)Q=(x+u(D)]TJ /F4 11.955 Tf 11.96 0 Td[(2Q;x)Q)+cuu(D)]TJ /F4 11.955 Tf 11.95 0 Td[(2Q;x)Q+2(cu+cd)Q( C\00062 )(x)+2(cu+cd)Q>(x)( 4\0008 )=V(x);wherethefourthequalityholdsbecauseofthedenitionofu(y;x)in( 4 )andthefactthatx=D)]TJ /F3 11.955 Tf 11.96 0 Td[(iQfori=1;2;:::incase2. 216

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APPENDIXDPROOFS D.1ProofofProposition 5.1 Proof. Let Q(x):=[ w0 (x)+c]=)]TJ ET BT /F3 11.955 Tf 265.95 -98.75 Td[( 2e)]TJ ET BT /F5 7.97 Tf 289.52 -93.81 Td[( 2x:(D)Intherstpartofthisproof,weshowthatforeachKbelongstoacertainrange,thereexistsauniquepairs(K)andS(K)suchthatQ(s(K))=)]TJ /F3 11.955 Tf 9.3 0 Td[(KandQ(S(K))=)]TJ /F3 11.955 Tf 9.3 0 Td[(K,i.e.,( 5 )and( 5 )aresatised.First,notethat Q0 (x)=( w00 (x)= 2+ w0 (x)+c)=e)]TJ ET BT /F5 7.97 Tf 284.92 -196.52 Td[( 2x= f0 (x)=e)]TJ ET BT /F5 7.97 Tf 359.66 -196.52 Td[( 2xbasedon( D )and( 5 ).FromLemma 5.4 ,wehavethatthereexistsazf>>><>>>>:<0x0x>zf:(D)Inaddition,Q(x)reachesitsminimumatx=zf.Second,weprovelimx!Q(x)=0.From( D ),wehavelimx!Q(x)=limx![ w0 (x)+c]=)]TJ ET BT /F3 11.955 Tf 195.46 -418.39 Td[( 2e)]TJ ET BT /F5 7.97 Tf 219.03 -413.45 Td[( 2x (D)=limx! w0 (x)=)]TJ ET BT /F3 11.955 Tf 167.51 -448.31 Td[( 2e)]TJ ET BT /F5 7.97 Tf 191.08 -443.37 Td[( 2x=limx!e1xZ1xh0(y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1ydy+e)]TJ /F5 7.97 Tf 6.59 0 Td[(2xZxh0(y)e2ydy=)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(2e)]TJ /F5 7.97 Tf 6.58 0 Td[(2x=limx!e(1+2)xZ1xh0(y)e)]TJ /F5 7.97 Tf 6.58 0 Td[(1ydy=2+limx!Zxh0(y)e2ydy=2=limx!h0(x)e)]TJ /F5 7.97 Tf 6.59 0 Td[(1x=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(2(1+2)e)]TJ /F7 7.97 Tf 6.59 0 Td[((1+2)x=limx!h0(x)=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(2(1+2)e)]TJ /F5 7.97 Tf 6.58 0 Td[(2x=0; 217

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wherethethirdequalityholdsbecauseoftheexpressionof w0 (x)in( 5 ),thefthequalityholdsbecauseofL'Hospital'sRule,andthelastequalityholdsbecauseofAssumption 5.2 (e).Third,weshowthatthereexistsauniquex0suchthatQ(x0)=0.Fromtheexpressionof w0 (x)in( 5 ),wehavethat limx!+1( w0 (x)+c)=limx!+1he 1xR1x h0 (y)e)]TJ ET BT /F5 7.97 Tf 221.8 -168.8 Td[( 1ydy+e)]TJ ET BT /F5 7.97 Tf 274.51 -168.8 Td[( 2xRx h0 (y)e 2ydyi+c=limx!+1( h0 (x)= 1+ h0 (x)= 2)+cc>0;(D)whereandthesecondequalityholdsbecauseofL'Hospital'sRuleandtherstinequalityholdsbecauseh0(x)0forxabasedonAssumption 5.2 (c).Therefore,from( D ),( D )and( D ),wehavethatthereexistsauniquex0suchthat x0>zfandQ(x0)=0:(D)Finally,basedon( D ),( D ),and( D ),wehavethatforeachKsatisfyingQ(zf)<)]TJ /F3 11.955 Tf 9.3 0 Td[(K<0,thereexistsauniquepair(s(K);S(K))suchthat s(K)
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. . x . Q(x) . x0 . zf . )]TJ /F3 11.955 Tf 9.3 0 Td[(K . s(K) . S(K) FigureD-1. IllustrationofQ(x),s(K),andS(K) Q(x)=[ w0 (x)+c]=)]TJ ET BT /F3 11.955 Tf 175.99 -238.77 Td[( 2e)]TJ ET BT /F5 7.97 Tf 199.56 -233.83 Td[( 2x8>>>>>>>>>><>>>>>>>>>>:>)]TJ /F3 11.955 Tf 9.3 0 Td[(Kforx)]TJ /F3 11.955 Tf 9.3 0 Td[(Kforx>S(K):Inaddition,basedon( D ),( D ),( D ),and( D ),wehave limK!)]TJ /F5 7.97 Tf 15.05 0 Td[(Q(zf)s(K)=limK!)]TJ /F5 7.97 Tf 15.06 0 Td[(Q(zf)S(K)=zf;limK!0s(K)=;andlimK!0S(K)=x0:(D)Intheremainingpartofthisproof,weprovethatthereexistsauniqueKsuchthat ( 5 ),( 5 ),and( 5 ) aresatised.Dene (K)=ZS(K)s(K) w0 (x)+c+K 2 e)]TJ ET BT /F5 7.97 Tf 304 -486.13 Td[(2 xdx:(D)Notingthat( 5 )isequivalentto(K)=)]TJ /F4 11.955 Tf 9.54 0 Td[(^c0basedon( D ),weprovetheclaimbyproving(i)limK!)]TJ /F5 7.97 Tf 15.05 0 Td[(Q(zf)(K)=0,(ii)limK!0(K)=,and(iii) 0 (K)>0.For(i),sincelimK!)]TJ /F5 7.97 Tf 15.06 0 Td[(Q(zf)s(K)=limK!)]TJ /F5 7.97 Tf 15.06 0 Td[(Q(zf)S(K)=zfbasedon( D ),wehave limK!)]TJ /F5 7.97 Tf 15.05 0 Td[(Q(zf)(K)=0(D)For(ii),from( D ),wehave 219

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(K)=w(S(K))+cS(K))]TJ /F3 11.955 Tf 11.96 0 Td[(Ke)]TJ ET BT /F5 7.97 Tf 216.71 -42.88 Td[(2 S(K))]TJ /F9 11.955 Tf 11.96 9.69 Td[(w(s(K))+cs(K))]TJ /F3 11.955 Tf 11.96 0 Td[(Ke)]TJ ET BT /F5 7.97 Tf 402.3 -42.88 Td[(2 s(K)=[w(S(K))+cS(K)+( w0 (S(K))+c)= 2])]TJ /F4 11.955 Tf 11.95 0 Td[([w(s(K))+cs(K)+( w0 (s(K))+c)= 2]( 5\00023 )=f(S(K)))]TJ /F3 11.955 Tf 11.95 0 Td[(f(s(K)); (D)wherethesecondequalityholdsbecauseof( D )andQ(s(K))=)]TJ /F3 11.955 Tf 9.3 0 Td[(KandQ(S(K))=)]TJ /F3 11.955 Tf 9.3 0 Td[(Kbasedon( D ).SincelimK!0s(K)=andlimK!0S(K)=x0basedon( D ),wehave limK!0(K)( D\00010 )=f(x0))]TJ /F4 11.955 Tf 17.97 0 Td[(limx!f(x)=;(D)wherethelastequalityholdsbecauseoflimx!f(x)=+1basedon( 5 ).For(iii),from( D ),wehave 0 (K)=ZS(K)s(K) 2 e)]TJ ET BT /F5 7.97 Tf 203.8 -357.17 Td[(2 xdx=e)]TJ ET BT /F5 7.97 Tf 258.65 -357.17 Td[(2 s(K))]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ ET BT /F5 7.97 Tf 313.07 -357.17 Td[(2 S(K)>0;(D)wheretheinequalityholdsbecauses(K)zfbasedon( D ),wehave w00 (s(K))= 2+ w0 (s(K))+c<0and w00 (S(K))= 2+ w0 (S(K))+c>0.Lettings=s(K)andS=S(K),wehavetheresults.Finally,weprovesomeby-productsthatwouldusedlater.From( D ),( D ),and( D ),wehavethatlim^c0!+1K(c2;^c0)=0. 220

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D.2ProofofProposition 5.2 Proof. First,fromtheproofofProposition 5.1 ,wehavethats(c),S(c)andK(c)aredeterminedbythefollowingequations:8>>>><>>>>:^F(s;S;K;c):= w0 (s)+K 2e)]TJ ET BT /F5 7.97 Tf 221.85 -109.66 Td[( 2s+c=0^G(s;S;K;c):= w0 (S)+K 2e)]TJ ET BT /F5 7.97 Tf 224.24 -133.56 Td[( 2S+c=0^H(s;S;K;c):=w(S))]TJ /F3 11.955 Tf 11.95 0 Td[(w(s)+Ke)]TJ ET BT /F5 7.97 Tf 247.8 -157.47 Td[( 2s)]TJ /F3 11.955 Tf 11.96 0 Td[(Ke)]TJ ET BT /F5 7.97 Tf 298.75 -157.47 Td[( 2S+^c0+c(S)]TJ /F3 11.955 Tf 11.96 0 Td[(s)=0:Next,substitutings(c),S(c),andK(c)intotheaboveequationsandgettingtherst-orderderivativetoc,wehave 266664^Fs^FS^FK^Gs^GS^GK^Hs^HS^HK377775266664s0(c)S0(c)K0(c)377775=266664)]TJ /F4 11.955 Tf 11.95 3.02 Td[(^Fc)]TJ /F4 11.955 Tf 11.96 3.02 Td[(^Gc)]TJ /F4 11.955 Tf 12.33 3.03 Td[(^Hc377775=266664)]TJ /F4 11.955 Tf 9.3 0 Td[(1)]TJ /F4 11.955 Tf 9.3 0 Td[(1)]TJ /F4 11.955 Tf 11.29 0 Td[((S)]TJ /F3 11.955 Tf 11.96 0 Td[(s)377775;(D)where266664^Fs^FS^FK^Gs^GS^GK^Hs^HS^HK377775 (D)=266664 w00 (s))]TJ ET BT /F3 11.955 Tf 108.31 -452.16 Td[( 22Ke)]TJ ET BT /F5 7.97 Tf 142.7 -447.82 Td[( 2s0 2e)]TJ ET BT /F5 7.97 Tf 433.57 -447.82 Td[( 2s0 w00 (S))]TJ ET BT /F3 11.955 Tf 295.07 -476.07 Td[( 22Ke)]TJ ET BT /F5 7.97 Tf 329.46 -471.73 Td[( 2S 2e)]TJ ET BT /F5 7.97 Tf 432.73 -471.73 Td[( 2S)]TJ ET BT /F3 11.955 Tf 33.35 -499.97 Td[(w0 (s))]TJ /F3 11.955 Tf 11.95 0 Td[(K 2e)]TJ ET BT /F5 7.97 Tf 108.45 -495.63 Td[( 2s)]TJ /F3 11.955 Tf 11.96 0 Td[(c=)]TJ /F3 11.955 Tf 9.3 0 Td[(F=0)]TJ ET BT /F3 11.955 Tf 216.72 -499.97 Td[(w0 (S))]TJ /F3 11.955 Tf 11.95 0 Td[(K 2e)]TJ ET BT /F5 7.97 Tf 294.2 -495.63 Td[( 2S)]TJ /F3 11.955 Tf 11.96 0 Td[(c=)]TJ /F3 11.955 Tf 9.3 0 Td[(G=0e)]TJ ET BT /F5 7.97 Tf 406.89 -495.63 Td[( 2s)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ ET BT /F5 7.97 Tf 447.02 -495.63 Td[( 2S377775:Furthermore,fromthethirdequationof( D ),wehave K0(c)=)]TJ /F4 11.955 Tf 12.34 3.02 Td[(^Hc=^HK=)]TJ /F4 11.955 Tf 11.29 0 Td[((S)]TJ /F3 11.955 Tf 11.96 0 Td[(s)=)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(e)]TJ ET BT /F5 7.97 Tf 282.13 -559.76 Td[( 2s)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ ET BT /F5 7.97 Tf 322.26 -559.76 Td[( 2S<0;(D)becauses
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Intherstpartoftheremainingproof,weproves0(c)<0.Substituting( D )intotherstequation,wehave ^Fss0(c)=)]TJ /F4 11.955 Tf 11.95 3.03 Td[(^Fc)]TJ /F4 11.955 Tf 14.61 3.03 Td[(^FKK0(c)=)]TJ /F4 11.955 Tf 9.3 0 Td[(1+ 2e)]TJ ET BT /F5 7.97 Tf 241.87 -66.79 Td[( 2s(S)]TJ /F3 11.955 Tf 11.96 0 Td[(s)=)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(e)]TJ ET BT /F5 7.97 Tf 321.82 -66.79 Td[( 2s)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ ET BT /F5 7.97 Tf 361.95 -66.79 Td[( 2S:(D)Inthefollowing,weproves0(c)<0byshowing(i)^Fs<0and(ii) )]TJ /F4 11.955 Tf 11.96 0 Td[(1+ 2e)]TJ ET BT /F5 7.97 Tf 180.43 -150.47 Td[( 2s(S)]TJ /F3 11.955 Tf 11.96 0 Td[(s)=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(e)]TJ ET BT /F5 7.97 Tf 260.38 -150.47 Td[( 2s)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ ET BT /F5 7.97 Tf 300.51 -150.47 Td[( 2S>0:(D)For(i),from( D ),wehavethat ^Fs= w00 (s))]TJ ET BT /F3 11.955 Tf 243.8 -233.6 Td[( 22Ke)]TJ ET BT /F5 7.97 Tf 278.19 -228.67 Td[( 2s:(D)Basedon( 5 ),weknow w0 (x)+c+K 2e)]TJ ET BT /F5 7.97 Tf 243.19 -302.48 Td[( 2x8>>>>>>>>>><>>>>>>>>>>:>0forx0forx>S:andsothederivativeof w0 (x)+c+K 2e)]TJ ET BT /F5 7.97 Tf 162.86 -370.13 Td[( 2xatx=sis w00 (s))]TJ ET BT /F3 11.955 Tf 219.13 -422.29 Td[( 22Ke)]TJ ET BT /F5 7.97 Tf 253.52 -417.35 Td[( 2s<0:(D)For(ii),let^f1(x):= 2e)]TJ ET BT /F5 7.97 Tf 151 -448.33 Td[( 2s(x)]TJ /F3 11.955 Tf 11.95 0 Td[(s))]TJ /F9 11.955 Tf 11.07 9.68 Td[()]TJ /F3 11.955 Tf 5.48 -9.68 Td[(e)]TJ ET BT /F5 7.97 Tf 232.72 -448.33 Td[( 2s)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ ET BT /F5 7.97 Tf 272.84 -448.33 Td[( 2x.Toprove( D )isequivalenttoshow^f1(S)0becausee)]TJ ET BT /F5 7.97 Tf 146.48 -472.24 Td[( 2s)]TJ /F3 11.955 Tf 12.01 0 Td[(e)]TJ ET BT /F5 7.97 Tf 186.73 -472.24 Td[( 2S>0basedons0forx>s.SinceS>sbasedon( 5 ),wehave^f1(S)>^f1(s)=0.Intheremainingpartofthisproof,weproveS0(c)<0.Substituting( D )intothesecondequation,wehave ^GSS0(c)=)]TJ /F4 11.955 Tf 11.96 3.02 Td[(^Gc)]TJ /F4 11.955 Tf 14.62 3.02 Td[(^GKK0(c)=)]TJ /F4 11.955 Tf 9.3 0 Td[(1+ 2e)]TJ ET BT /F5 7.97 Tf 245.55 -603.14 Td[( 2S(S)]TJ /F3 11.955 Tf 11.95 0 Td[(s)=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(e)]TJ ET BT /F5 7.97 Tf 327.18 -603.14 Td[( 2s)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ ET BT /F5 7.97 Tf 367.31 -603.14 Td[( 2S:(D)Inthefollowing,weproveS0(c)<0byshowing(i)^GS>0and(ii) 222

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)]TJ /F4 11.955 Tf 11.95 0 Td[(1+ 2e)]TJ ET BT /F5 7.97 Tf 181.22 -30.93 Td[( 2S(S)]TJ /F3 11.955 Tf 11.95 0 Td[(s)=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(e)]TJ ET BT /F5 7.97 Tf 262.85 -30.93 Td[( 2s)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ ET BT /F5 7.97 Tf 302.98 -30.93 Td[( 2S<0(D)For(i),from( D ),wehavethat ^GS= w00 (S))]TJ ET BT /F3 11.955 Tf 246.82 -114.06 Td[( 22Ke)]TJ ET BT /F5 7.97 Tf 281.21 -109.13 Td[( 2S:(D)Basedon( 5 ),weknow w0 (x)+c+K 2e)]TJ ET BT /F5 7.97 Tf 243.19 -182.94 Td[( 2x8>>>>>>>>>><>>>>>>>>>>:>0forx0forx>S:andsothederivativeof w0 (x)+c+K 2e)]TJ ET BT /F5 7.97 Tf 162.86 -250.59 Td[( 2xatx=Sis w00 (S))]TJ ET BT /F3 11.955 Tf 220.48 -302.75 Td[( 22Ke)]TJ ET BT /F5 7.97 Tf 254.87 -297.81 Td[( 2S>0:(D)For(ii),let^f2(x):= 2e)]TJ ET BT /F5 7.97 Tf 149.92 -328.79 Td[( 2S(S)]TJ /F3 11.955 Tf 11.95 0 Td[(x))]TJ /F9 11.955 Tf 12.14 9.68 Td[()]TJ /F3 11.955 Tf 5.48 -9.68 Td[(e)]TJ ET BT /F5 7.97 Tf 237.84 -328.79 Td[( 2x)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ ET BT /F5 7.97 Tf 278.82 -328.79 Td[( 2S.Toprove( D )isequivalenttoshow^f2(s)0becausee)]TJ ET BT /F5 7.97 Tf 159.37 -352.7 Td[( 2s)]TJ /F3 11.955 Tf 12.81 0 Td[(e)]TJ ET BT /F5 7.97 Tf 201.2 -352.7 Td[( 2S>0basedons0forx>>><>>>>:^F(s;S;K;^c0):= w0 (s)+K 2e)]TJ ET BT /F5 7.97 Tf 224.22 -532.44 Td[( 2s+c=0^G(s;S;K;^c0):= w0 (S)+K 2e)]TJ ET BT /F5 7.97 Tf 226.6 -556.35 Td[( 2S+c=0^H(s;S;K;^c0):=w(S))]TJ /F3 11.955 Tf 11.95 -.01 Td[(w(s)+Ke)]TJ ET BT /F5 7.97 Tf 250.17 -580.26 Td[( 2s)]TJ /F3 11.955 Tf 11.95 .01 Td[(Ke)]TJ ET BT /F5 7.97 Tf 301.11 -580.26 Td[( 2S+^c0+c(S)]TJ /F3 11.955 Tf 11.95 .01 Td[(s)=0: 223

PAGE 224

Next,substitutings(^c0),S(^c0),andK(^c0)intotheaboveequationsandgettingtherst-orderderivativeto^c0,wehave266664^Fs^FS^FK^Gs^GS^GK^Hs^HS^HK377775266664s0(^c0)S0(^c0)K0(^c0)377775=266664)]TJ /F4 11.955 Tf 11.95 3.02 Td[(^F^c0)]TJ /F4 11.955 Tf 11.97 3.03 Td[(^G^c0)]TJ /F4 11.955 Tf 12.33 3.02 Td[(^H^c0377775=26666400)]TJ /F4 11.955 Tf 9.3 0 Td[(1377775;where266664^Fs^FS^FK^Gs^GS^GK^Hs^HS^HK377775isgivenin( D ).Notethatthethirdequationgives K0(^c0)=)]TJ /F4 11.955 Tf 12.33 3.02 Td[(^H^c0=^HK=)]TJ /F4 11.955 Tf 9.3 0 Td[(1=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(e)]TJ ET BT /F5 7.97 Tf 269.94 -247.58 Td[( 2s)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ ET BT /F5 7.97 Tf 310.07 -247.58 Td[( 2S<0:(D)Intheremainingpartofthisproof,weshows0(^c0)<0andS0(^c0)>0.Fors0(^c0),substituting( D )intotherstequation,wehave^Fss0(^c0)=)]TJ /F4 11.955 Tf 11.95 3.02 Td[(^F^c0)]TJ /F4 11.955 Tf 14.61 3.02 Td[(^FKK0(^c0)= 2e)]TJ ET BT /F5 7.97 Tf 264.91 -343.22 Td[( 2s=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(e)]TJ ET BT /F5 7.97 Tf 303.75 -343.22 Td[( 2s)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ ET BT /F5 7.97 Tf 343.88 -343.22 Td[( 2S>0;wheretheinequalityfollowsfroms0;wheretheinequalityfollowsfroms0because^GS>0basedon( D )and( D ). D.4ProofofProposition 5.5 Proof. First,( 5 )holdsbecauseofthedenitionofQ1(x)in( 5 ),thedenitionofQ2(x)in( 5 ),and( 5 )intheproofofProposition 5.1 withcireplacingcandci;0denedin( 5 )replacingc0fori=1;2. 224

PAGE 225

Second,from( 5 ),wehave Q0i (x)( 5\00059 )=f w00 (x)+ 2[ w0 (x)+ci]g=)]TJ ET BT /F3 11.955 Tf 220.55 -87.67 Td[( 2e)]TJ ET BT /F5 7.97 Tf 244.12 -82.73 Td[( 2x= f0i (x)=e)]TJ ET BT /F5 7.97 Tf 323.17 -82.73 Td[( 2x8>>>><>>>>:<0forx0forx>zf;i::Finally, Q0i (si)<0holdsbecauseof( 5 )and( 5 ). D.5ProofofProposition 5.5 Proof. (i)From( 5 ),( 5 ),andthedenitionofgi(x)in( 5 ),wehaveg1(s1)=g1(S1)+c0andg2(s2)=g2(S2)+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d.(ii)From( 5 ),wehave g0i (x)= w0 (x)+ci+Ki 2e)]TJ ET BT /F5 7.97 Tf 304.56 -264.27 Td[( 2x.Hence,inordertoprove g0i (x)= w0 (x)+ci+Ki 2e)]TJ ET BT /F5 7.97 Tf 142.63 -331.01 Td[( 2x8>>>>>>>>>><>>>>>>>>>>:>0forx0forx>Siasdescribedin( 5 ),itsufcestoprove[ w0 (x)+ci]=)]TJ ET BT /F3 11.955 Tf 172.29 -488.79 Td[( 2e)]TJ ET BT /F5 7.97 Tf 195.86 -483.85 Td[( 2x8>>>>>>>>>><>>>>>>>>>>:>)]TJ /F3 11.955 Tf 9.29 0 Td[(Kiforx)]TJ /F3 11.955 Tf 9.29 0 Td[(Kiforx>Si;whichholdsbecauseof( 5 ).Inaddition,from( 5 ),wehave (bettertoaddtwographsaboutg0i(x)andgi(x)here) 225

PAGE 226

maxxSigi(x)=gi(si)andminxsigi(x)=gi(Si):(iii)Equation( 5 )holdsbecauseofthedenitionoffi(x)in( 5 ),thedenitionofgi(x)in( 5 ),andtherelationshipdescribedin( 5 ).Similarly,( 5 )holdsbecauseofthedenitionoffi(x)in( 5 ),thedenitionofgi(x)in( 5 ),andtherelationshipdescribedin( 5 ).(iv)Equation( 5 )followsfrom( 5 ),( 5 ),and( 5 ).Similarly,( 5 )followsfrom( 5 ),( 5 ),and( 5 ).(v)Werstprove f01 (s1)<0bycontradiction.Notethatthereexistsa12[s1;S1]suchthatf1(S1))]TJ /F3 11.955 Tf 12.96 0 Td[(f1(s1)= f01 (1) f01 (s1),wheretheinequalityholdsbecausef001(x)0basedon( 5 )and1s1.If f01 (s1)0,wehavef1(S1)f1(s1),whichcontradictswithf1(s1)=f1(S1)+c0basedon( 5 )sincec0>0.Thus,wehave f01 (s1)<0.Next,from( 5 ),wehave f00i (x)0andso f01 (x) f01 (s1)<0forxs1.Similarly,wecanprove f02 (x)<0forxs2. D.6ProofofProposition 5.6 Proof. First,weprove( 5 )byproving(i)s2
PAGE 227

g02 (s1)= g01 (s1)+c2)]TJ /F3 11.955 Tf 11.95 0 Td[(c1=c2)]TJ /F3 11.955 Tf 11.95 0 Td[(c1<0;(D)wherethesecondequalityholdsbecause g01 (s1)=0basedon( 5 )andtheinequalityholdsbecausec1>c2basedonAssumption 5.2 .From( 5 )and( D ),wehave s4K2andCase(b):K1=K2.ForCase(a),from( 5 ),wehave g02 (s4)= w0 (s4)+c2+K2 2e)]TJ ET BT /F5 7.97 Tf 302.74 -451.31 Td[( 2s4( D\00030 )=(K2)]TJ /F3 11.955 Tf 11.95 0 Td[(K1)e)]TJ ET BT /F5 7.97 Tf 288.44 -478.21 Td[( 2s4<0; (D)wheretheinequalityholdsbecauseK1>K2underCase(a).Hence,basedon( 5 )and( D ),wehave s2K2:(D) 227

PAGE 228

ForCase(b),from( 5 ),wehave g02 (s4)= w0 (s4)+c2+K2 2e)]TJ ET BT /F5 7.97 Tf 302.74 -66.79 Td[( 2s4( D\00030 )=(K2)]TJ /F3 11.955 Tf 11.95 0 Td[(K1)e)]TJ ET BT /F5 7.97 Tf 288.44 -93.69 Td[( 2s4=0; (D)wheretheinequalityholdsbecauseK1=K2underCase(b).Hence,s4isoneofthezeropointsofg02(x).From( 5 ),weknowthatg02(x)hasonlytwozeropointss2andS2.Sinces4
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Similarly,from( 5 )and( 5 ),wehaveG2(s2)=G2(S2)+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(s2),whichisequivalentto s2=S2+[c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d]=c2+[G2(S2))]TJ /F3 11.955 Tf 11.95 0 Td[(G2(s2)]=c2:(D)Third,wehavec2x+G1(x)( 5\00063 )=c2x+w(x))]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 289.69 -168.9 Td[( 2x=c2x+w(x))]TJ /F3 11.955 Tf 11.95 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 266.34 -195.79 Td[( 2x+(K2)]TJ /F3 11.955 Tf 11.95 0 Td[(K1)e)]TJ ET BT /F5 7.97 Tf 362.21 -195.79 Td[( 2x( 5\00062 )=g2(x)+(K2)]TJ /F3 11.955 Tf 11.95 0 Td[(K1)e)]TJ ET BT /F5 7.97 Tf 302.7 -222.69 Td[( 2x: (D)Finally,wehave c2x+G1(x)( 5\00063 )=c2x+w(x))]TJ /F3 11.955 Tf 11.95 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 289.74 -306.37 Td[( 2x( 5\00077 )=g2(x):(D)Nowwestarttoprovetheclaims.(i)Toprovex2;1
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(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)(x2;1)]TJ /F3 11.955 Tf 11.95 0 Td[(s1)=)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(S1)]TJ /F4 11.955 Tf 13.95 3.02 Td[(S2+G1(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(G1)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(c0(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)=c1)]TJ /F4 11.955 Tf 11.96 0 Td[([G1(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(G1(s1)](c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)=c1=)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(S1)]TJ /F4 11.955 Tf 13.95 3.02 Td[(S2+G1(S1)c2=c1)]TJ /F3 11.955 Tf 11.95 0 Td[(G1)]TJ /F4 11.955 Tf 7.48 -6.67 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0+c0c2=c1+G1(s1))]TJ /F3 11.955 Tf 11.95 0 Td[(G1(s1)c2=c1 (D)=)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(S1)]TJ /F4 11.955 Tf 13.95 3.02 Td[(S2+G1(s1))]TJ /F3 11.955 Tf 11.95 0 Td[(G1)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(c0+[G1(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(G1(s1)+c0]c2=c1( D\00035 )=)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(S1)]TJ /F4 11.955 Tf 13.95 3.02 Td[(S2+G1(s1))]TJ /F3 11.955 Tf 11.95 0 Td[(G1)]TJ /F4 11.955 Tf 7.48 -6.67 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(c0+c2(s1)]TJ /F3 11.955 Tf 11.96 0 Td[(S1)=)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(s1)]TJ /F4 11.955 Tf 13.96 3.02 Td[(S2+G1(s1))]TJ /F3 11.955 Tf 11.96 0 Td[(G1)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2( D\00037 )=)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+g2(s1))]TJ /F3 11.955 Tf 11.95 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2+(K2)]TJ /F3 11.955 Tf 11.96 0 Td[(K1)e)]TJ ET BT /F5 7.97 Tf 312.35 -258.05 Td[( 2s1)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ ET BT /F5 7.97 Tf 356.63 -258.05 Td[( 2S2)]TJ /F3 11.955 Tf 23.91 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+g2(s1))]TJ /F3 11.955 Tf 11.95 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2 (D)<)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+g2(s2))]TJ /F3 11.955 Tf 11.95 0 Td[(g2(S2)( 5\00065 )=0;wheretherstequalityholdsbecauseof( D )and( D ),therstinequalityholdsbecauseK1K2underCase1ands1c2basedonAssumption 5.2 ,wehavex2;1S1)]TJ /F3 11.955 Tf 11.96 0 Td[(d.From( D ),wehavex2;1)]TJ /F4 11.955 Tf 11.95 0 Td[((S1)]TJ /F3 11.955 Tf 11.96 0 Td[(d)=)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(c1S1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2S2=(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)+G1(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(G1)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2=(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2))]TJ /F3 11.955 Tf 11.95 0 Td[(S1=c2S1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2S2+G1(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(G1)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2=(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)=g2(S1))]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2=(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)>0; 230

PAGE 231

wherethelastequalityholdsbecauseof( 5 )and( 5 ),andtheinequalityfollowsfromg2(S1)>g2)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2becauseof( 5 )andS1K2andCase(b):K1=K2.Case(a):fors2K2,wehave 231

PAGE 232

g2(x))]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.47 -6.67 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d=g2(x))]TJ /F3 11.955 Tf 11.95 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+(K2)]TJ /F3 11.955 Tf 11.96 0 Td[(K1)e)]TJ ET BT /F5 7.97 Tf 344.18 -69.78 Td[( 2x)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ ET BT /F5 7.97 Tf 385.16 -69.78 Td[( 2S2K2andxs2whenK1=K2.First,from( 5 ),wehave 232

PAGE 233

G1(x)=w(x))]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 209.15 -30.93 Td[( 2x=w(x))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 305.01 -30.93 Td[( 2x=G2(x):(D)Next,from( D )and( D ),wehave(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)(x2;1)]TJ /F3 11.955 Tf 11.95 0 Td[(s2)=)]TJ /F4 11.955 Tf 11.29 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+(c1S1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2S2)+[G1(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(G1(S2)])]TJ /F4 11.955 Tf 11.29 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)S2)]TJ /F4 11.955 Tf 11.95 0 Td[([c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d](c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)=c2)]TJ /F4 11.955 Tf 11.95 0 Td[([G2(S2))]TJ /F3 11.955 Tf 11.96 0 Td[(G2(s2)](c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)=c2( D\00041 )=)]TJ /F4 11.955 Tf 11.29 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+(c1S1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2S2)+[G2(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(G2(S2)])]TJ /F4 11.955 Tf 11.95 0 Td[((c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)S2)]TJ /F4 11.955 Tf 11.29 0 Td[([c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d]c1=c2+[c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d])]TJ /F4 11.955 Tf 11.95 0 Td[([G2(S2))]TJ /F3 11.955 Tf 11.96 0 Td[(G2(s2)]c1=c2+[G2(S2))]TJ /F3 11.955 Tf 11.95 0 Td[(G2(s2)]=c0+[c1S1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2S2)]TJ /F4 11.955 Tf 11.95 0 Td[((c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)S2]+[G2(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(G2(S2)]+[G2(S2))]TJ /F3 11.955 Tf 11.96 0 Td[(G2(s2)])]TJ /F4 11.955 Tf 11.29 0 Td[([c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d]c1=c2)]TJ /F4 11.955 Tf 11.96 0 Td[([G2(S2))]TJ /F3 11.955 Tf 11.96 0 Td[(G2(s2)]c1=c2=c0+c1(S1)]TJ /F3 11.955 Tf 11.95 0 Td[(S2)+G2(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(G2(s2))]TJ /F4 11.955 Tf 11.95 0 Td[([c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d+G2(S2))]TJ /F3 11.955 Tf 11.96 0 Td[(G2(s2)]c1=c2( D\00036 )=c0+c1(S1)]TJ /F3 11.955 Tf 11.95 0 Td[(S2)+G2(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(G2(s2))]TJ /F3 11.955 Tf 11.95 0 Td[(c2(s2)]TJ /F3 11.955 Tf 11.96 0 Td[(S2)c1=c2=c0+c1(S1)]TJ /F3 11.955 Tf 11.95 0 Td[(s2)+G2(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(G2(s2)( D\00041 )=c0+c1(S1)]TJ /F3 11.955 Tf 11.95 0 Td[(s2)+G1(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(G1(s2)=c0+g1(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(g1(s2)>c0+g1(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(g1(s1)( 5\00064 )=0;wherethelastsecondequalityholdsbecauseof( 5 )and( 5 ),andtheinequalityfollowsfromg1(s2)c2basedonAssumption 5.2 ,wehavex2;1>s2. D.8ProofofProposition 5.8 Proof. Werstnotethat,basedon( 5 ),f2(x)isstrictlydecreasingforxzf;2and minx2Rf2(x)=f2(zf;2):(D) 233

PAGE 234

Thus,inordertoprove( 5 )and( 5 ),itsufcestoshowf2(zf;2)d.ForCase(i),whend=d,basedon( 5 ),wehave K1( 5\00038 )=K2whend=d:(D)Then,from( 5 ),( 5 ),and( D ),wehave g2(x)=g2(x)whend=d:(D)Moreover,from( 5 ),( 5 ),( D ),and( D ),wehave S2=S2whend=d:(D)Therefore,wehavef2(zf;2)( D\00042 )f2(S2)( 5\00069 )=g2(S2)0andc1>c2>0basedonAssumption 5.2 ,d>0basedon( 5 ),andg2(S2)=g2)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2basedon( D )and( D ).ForCase(ii),werstnotethatg2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2doesnotchangewithdbecasueofthedenitionofS2in( 5 ),thedenitionofg2(x)in( 5 ),andthefactthatK1doesnotchangewithdbasedon( 5 ).Inaddition,f2(zf;2)doesnotchangewithdbasedon( 5 )and( 5 ).Thus,wehavef2(zf;2)( D\00046 )d, andwehaveprovedtheexistenceofs2suchthat( 5 )and( 5 )hold. 234

PAGE 235

Finally,basedon( 5 ),wenotethats2)]TJ /F3 11.955 Tf 9.3 0 Td[(K1 2e)]TJ ET BT /F5 7.97 Tf 363.48 -97.9 Td[( 2s2whenK1>K2.Basedon( 5 ),itsufcestoproves2K2.Fromthedenitionofs2in( 5 )and( 5 ),wehaves2K2,itsufcestoshowf2(s4)K2,whichweproveasfollows. bettertoaddagraphhere. First,basedon( 5 ),wehave w0 (s4)+c2=)]TJ ET BT /F3 11.955 Tf 244.45 -293.5 Td[( 2K1e)]TJ ET BT /F5 7.97 Tf 282.71 -288.56 Td[( 2s4:(D)Next,from( 5 ),wehavef2(s4)=[ w0 (s4)+c2]= 2+w(s4)+c2s4( D\00047 )=)]TJ /F3 11.955 Tf 9.3 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 241.19 -405.61 Td[( 2s4+w(s4)+c2s4( 5\00077 )=g2(s4)K2basedon( D ),s4
PAGE 236

g2(x)=g2(x)whenK1=K2:(D)Next,from( 5 ),( 5 ),and( D ),wehave S2=S2whenK1=K2:(D)Thus,wehavef2(s2)( 5\00092 )=g2)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d=g2(S2)+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d( 5\00069 )=f2(S2)+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d( 5\00071 )=f2(s2);wherethesecondequalityholdsbecauseof( D )and( D ). D.10ProofofLemma 5.6 Proof. Intherstpartofthisproof,weshow( 5 )holds.Dene Fw(x)= w0 (x))]TJ ET BT /F3 11.955 Tf 226.25 -434.96 Td[( 1Ae 1x+ 2Be)]TJ ET BT /F5 7.97 Tf 313.86 -430.02 Td[( 2x:(D)From( 5 ),wehaveFw(z1)=)]TJ /F3 11.955 Tf 9.3 0 Td[(c1andFw(z2)=)]TJ /F3 11.955 Tf 9.3 0 Td[(c2.Soinordertoprove )]TJ /F3 11.955 Tf 11.96 0 Td[(c1
PAGE 237

F0w (x)( D\00050 )= w00 (x))]TJ ET BT /F3 11.955 Tf 155.69 -47.82 Td[( 21Ae 1x)]TJ ET BT /F3 11.955 Tf 210.42 -47.82 Td[( 22Be)]TJ ET BT /F5 7.97 Tf 243.49 -42.88 Td[( 2x= w00 (x))]TJ ET BT /F3 11.955 Tf 129.39 -74.71 Td[( 21Ae 1x)]TJ ET BT /F3 11.955 Tf 184.11 -74.71 Td[( 1 2Ae( 1+ 2)z1e)]TJ ET BT /F5 7.97 Tf 273.39 -69.78 Td[( 2x+ 2e)]TJ ET BT /F5 7.97 Tf 325.74 -69.78 Td[( 2x[ w0 (z1)+c1]e 2z1< w00 (x)+ 2e)]TJ ET BT /F5 7.97 Tf 152.76 -96.68 Td[( 2x[ w0 (z1)+c1]e 2z1; (D)wherethesecondequaltiyholdsbecauseB= 1Ae 1z1e 2z1= 2)]TJ /F4 11.955 Tf 12.33 0 Td[([ w0 (z1)+c1]e 2z1= 2basedon( 5 ),andtheinequalityholdsbecauseA>0basedonTheorem 5.6 .Next,weprove e)]TJ ET BT /F5 7.97 Tf 148.1 -228.17 Td[( 2x[ w0 (z1)+c1]e 2z1<[ w0 (x)+c1];(D)whichisequivalentto[ w0 (z1)+c1]=)]TJ ET BT /F3 11.955 Tf 195.41 -263.49 Td[( 2e)]TJ ET BT /F5 7.97 Tf 218.98 -259.15 Td[( 2z1<[ w0 (x)+c1]=)]TJ ET BT /F3 11.955 Tf 331.13 -263.49 Td[( 2e)]TJ ET BT /F5 7.97 Tf 354.71 -259.15 Td[( 2x.Since[ w0 (x)+c1]=)]TJ ET BT /F3 11.955 Tf 226.17 -311.3 Td[( 2e)]TJ ET BT /F5 7.97 Tf 249.74 -306.37 Td[( 2x=Q1(x)basedon( 5 ),weonlyneedtoproveQ1(z1)
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and Gw;2(x):=w(x))]TJ /F3 11.955 Tf 11.96 0 Td[(Ae 1x)]TJ /F3 11.955 Tf 11.95 0 Td[(Be)]TJ ET BT /F5 7.97 Tf 164.77 -66.79 Td[( 2x)]TJ /F9 11.955 Tf 11.96 13.27 Td[(hw)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 274.27 -66.79 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(xi:(D)Weprove( 5 )and( 5 )holdbyprovingGw;i(x)<0forz20forz2s1,weprove L V(x)=minf L1 V(x); L2 V(x)g>V(x)byproving(i) L1 V(x)>V(x)and(ii) L2 V(x)>V(x)separately.For(i),notingthatS1>s1basedon( 5 ),weprove L1 V(x)>V(x)forx>s1bydiscussingtwocases:s1
PAGE 239

L1 V(x)( 5\00019 )=minyxfV(y)+c0+c1(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x)g=minyxw(y))]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 253.86 -73.67 Td[( 2y+c0+c1(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x)( 5\00062 )=minyxfg1(y)+c0)]TJ /F3 11.955 Tf 11.95 0 Td[(c1xg=g1(S1)+c0)]TJ /F3 11.955 Tf 11.95 0 Td[(c1x>g1(x))]TJ /F3 11.955 Tf 11.96 0 Td[(c1x( 5\00062 )=w(x))]TJ /F3 11.955 Tf 11.95 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 249.26 -189.04 Td[( 2x=V(x);wherethesecondandthelastequalitiesholdbecauseof( 5 )andyx>s1,thefourthequalityholdsbecauseof( 5 ),yx,andx2(s1;S1),andtheinequalityholdsbecauseg1(x)w(x))]TJ /F3 11.955 Tf 11.95 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 219.8 -529.68 Td[( 2x=V(x);wherethesecondandthelastequalitiesholdbecauseof( 5 ),yx>S1,andS1>s1basedon( 5 ),andthefourthequalityholdsbecauseg01(z)>0forz>S1basedon( 5 ). 239

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Thus,wehave L1 V(x)>V(x)forx>s1.For(ii),notingthatS2>s1basedon( 5 ),weprove L2 V(x)>V(x)forx>s1bydiscussingtwocases:s1g2(x))]TJ /F3 11.955 Tf 11.95 0 Td[(c2x( 5\00077 )=w(x))]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 211.54 -244.71 Td[( 2x=V(x);wherethesecondandthelastequalitiesholdbecauseof( 5 )andyx>s1,thefourthequalityholdsbecauseof( 5 )andx2)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(s1;S2,andtheinequalityholdsbecauseg2(x)w(x))]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 188.18 -530.37 Td[( 2x=V(x);wherethesecondandthelastequalitiesholdbecauseof( 5 ),yx>S2,andS2>s1basedon( 5 ),andthefourthequalityholdsbecauseg02(z)>0forz>S2basedon( 5 )andyxS2. 240

PAGE 241

Thus,wehave L2 V(x)>V(x)forx>s1.Insum,wehave L V(x)=minf L1 V(x); L2 V(x)g>V(x)forx>s1. D.12ProofofProposition 5.11 Proof. Basedon( 5 ),wehave L V(x)=minf L1 V(x); L2 V(x)g,whereL1V(x)isdescribedin( 5 )andL2V(x)isdescribedin( 5 ).Forx2[z1;s1],weprove L V(x)=V(x)=V(S1)+(S1)]TJ /F3 11.955 Tf 11.96 0 Td[(x)byproving(i) L1 V(x)=V(x)=V(S1)+(S1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)and(ii) L2 V(x)V(x)forz1xs1:Notethatforz1xs1,from( 5 ),wecanwrite L1 V(x)as L1 V(x)=minf L1;l V(x); L1;4 V(x)g;(D)where L1;l V(x)=minz1xys1fV(y)+c0+c1(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x)g;(D) and L1;4 V(x)=miny>s1fV(y)+c0+c1(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x)g:(D)Similarly,from( 5 ),wecanwrite L2 V(x)as and L2 V(x)=minf L2;l V(x); L2;4 V(x)g;(D)where L2;l V(x)=minz1xys1fV(y)+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x)g;(D) and L2;4 V(x)=miny>s1fV(y)+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d+c2(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x)g:(D)Inrstpartoftheremainingproof,weprove(i) L1 V(x)=V(x)byproving(a) L1;l V(x)>V(x)and(b) L1;4 V(x)=V(x)=V(S1)+(S1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)forz1xs1. 241

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For(a),wehave L1;l V(x)( D\00057 )=minz1xys1fV(y)+c0+c1(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x)g( 5\0006 )=minz1xys1w(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 247.46 -85.12 Td[( 2S1+c0+c1(S1)]TJ /F3 11.955 Tf 11.95 0 Td[(y)+c0+c1(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x)=minz1xys1w(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 228.34 -109.03 Td[( 2S1+c0+c1(S1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)+c0=w(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 154.21 -132.93 Td[( 2S1+c0+c1(S1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)+c0;=V(x)+c0>V(x);(D)wherethelastequalityholdsbecauseof( 5 )andz1xs1.For(b),wehave L1;4 V(x)( D\00058 )=miny>s1fV(y)+c0+c1(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x)g;( 5\0006 )=miny>s1w(y))]TJ /F3 11.955 Tf 11.95 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 279.85 -311.02 Td[( 2y+c0+c1(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x)( 5\00062 )=miny>s1fg1(y)+c0)]TJ /F3 11.955 Tf 11.96 0 Td[(c1xg( 5\00066 )=g1(S1)+c0)]TJ /F3 11.955 Tf 11.95 0 Td[(c1x( 5\00062 )=w(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 243.77 -382.74 Td[( 2S1+c0+c1(S1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)=V(x);(D)wherethelastequalityholdsbecauseof( 5 )andz1xs1.Inaddition,wehave L1;4 V(x)=w(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 242.36 -513.1 Td[( 2S1+c0+c1(S1)]TJ /F3 11.955 Tf 11.96 0 Td[(x);=V(S1)+c0+c1(S1)]TJ /F3 11.955 Tf 11.96 0 Td[(x)=V(S1)+(S1)]TJ /F3 11.955 Tf 11.96 0 Td[(x) (D)wheretherstequalityholdsbecauseofthelastsecondequationin( D ),thesecondequalityholdsbecauseof( 5 ),S1>s1basedon( 5 ),andthelastequalityfollows 242

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fromthedenitionof(x)in( 5 ),S1)]TJ /F3 11.955 Tf 12.41 0 Td[(xS1)]TJ /F3 11.955 Tf 12.41 0 Td[(z1basedonz1xs1,z1>x2;1basedonbasedonTheorem 5.6 ,andS1)]TJ /F3 11.955 Tf 11.95 0 Td[(x2;1x2;1basedonTheorem 5.6 ,andthelastinequalityholdsbecausec0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)(x2;1)]TJ /F3 11.955 Tf 11.95 0 Td[(s1+d)0basedon( 5 ).For(d)withz1xs1,wehave 243

PAGE 244

L2;4 V(x)( D\00061 )=minys1fV(y)+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x)g( 5\0006 )=minys1w(y))]TJ /F3 11.955 Tf 11.95 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 228.71 -61.21 Td[( 2y+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d+c2(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x)( 5\00077 )=minys1fg2(y)+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)dg=g2)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d( 5\00077 )=w)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2+c2S2)]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 230.6 -132.93 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d( 5\00083 )=U2(x)U1(x)( 5\00082 )=w(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 192.63 -204.66 Td[( 2S1+c0+c1(S1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)=V(x);(D)wherethefourthequalityholdsbecauseof( 5 )ands10forz>x2;1basedon( 5 )and( 5 ),x>z1,andz1>x2;1basedonTheorem 5.6 ,andthelastequalityholdsbecauseof( 5 )andz1xs1.Thus,basedon( D ),( D ),and( D ),wehave L2 V(x)=minf L2;l V(x); L2;4 V(x)gV(x)forz1xs1:(D)Insum,from( 5 ),( D ),and( D ),wehave L V(x)=minf L1 V(x); L2 V(x)g=V(x)=V(S1)+(S1)]TJ /F3 11.955 Tf 11.96 0 Td[(x)forz1xs1. D.13ProofofProposition 5.12 Proof. Basedon( 5 ),wehave L V(x)=minf L1 V(x); L2 V(x)g,whereL1V(x)isdescribedin( 5 )andL2V(x)isdescribedin( 5 ).Forx2(z2;z1),weprove L V(x)>V(x)byproving(i) L1 V(x)>V(x)and(ii) L2 V(x)>V(x)forz2
PAGE 245

L1 V(x)=minn ~L1;2 V(x); L1;3 V(x); L1;4 V(x)o;(D)where ~L1;2 V(x):=minz2V(x)byproving(a) ~L1;2 V(x)>V(x),(b) L1;3 V(x)>V(x),and(c) L1;4 V(x)>V(x)forz2w(x))]TJ /F3 11.955 Tf 11.95 0 Td[(Ae 1x)]TJ /F3 11.955 Tf 11.95 0 Td[(Be)]TJ ET BT /F5 7.97 Tf 207.69 -607.61 Td[( 2x=V(x);(D) 245

PAGE 246

wherethethirdequalityholdsbecause w0 (y))]TJ ET BT /F3 11.955 Tf 251.42 -11.96 Td[( 1Ae 1y+ 2Be)]TJ ET BT /F5 7.97 Tf 339.49 -7.62 Td[( 2y+c1>0basedon( 5 )andyx,andthelastequalityholdsbecauseof( 5 )andz2w(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 160.69 -204.66 Td[( 2S1+c0+c1(S1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)( 5\00099 )>w(x))]TJ /F3 11.955 Tf 11.95 0 Td[(Ae 1x)]TJ /F3 11.955 Tf 11.95 0 Td[(Be)]TJ ET BT /F5 7.97 Tf 216.74 -228.62 Td[( 2x=V(x);(D)wherethelastequalityholdsbecauseof( 5 )andz2s1fV(y)+c0+c1(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x)g( 5\0006 )=miny>s1w(y))]TJ /F3 11.955 Tf 11.95 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 279.85 -382.79 Td[( 2y+c0+c1(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x)( 5\00062 )=miny>s1fg1(y)+c0)]TJ /F3 11.955 Tf 11.96 0 Td[(c1xg( 5\00066 )=g1(S1)+c0)]TJ /F3 11.955 Tf 11.95 0 Td[(c1x( 5\00062 )=w(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 243.77 -454.52 Td[( 2S1+c0+c1(S1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)( 5\00099 )>w(x))]TJ /F3 11.955 Tf 11.96 0 Td[(Ae 1x)]TJ /F3 11.955 Tf 11.96 0 Td[(Be)]TJ ET BT /F5 7.97 Tf 276.46 -478.48 Td[( 2x=V(x);(D)wherethelastequalityholdsbecauseof( 5 )andz2V(x)forz2
PAGE 247

Insecondpartoftheremainingproof,weprove(ii) L2 V(x)>V(x)byproving(d) ~L2;2 V(x)>V(x),(e)L2;3V(x)>V(x),and(f)L2;4V(x)>V(x)forz2w(x))]TJ /F3 11.955 Tf 11.96 0 Td[(Ae 1x)]TJ /F3 11.955 Tf 11.95 0 Td[(Be)]TJ ET BT /F5 7.97 Tf 156.08 -469.06 Td[( 2x=V(x);(D)wherethethirdequalityholdsbecause w0 (x))]TJ ET BT /F3 11.955 Tf 251.89 -538.12 Td[( 1Ae 1x+ 2Be)]TJ ET BT /F5 7.97 Tf 340.21 -533.78 Td[( 2x+c2<0basedon( 5 )andxy
PAGE 248

L2;3 V(x)( D\00074 )=minz1ys1fV(y)+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x)g( 5\0006 )=minz1ys1w(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 188.27 -85.12 Td[( 2S1+c0+c1(S1)]TJ /F3 11.955 Tf 11.96 0 Td[(y)+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d+c2(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x)=w(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 127.49 -109.03 Td[( 2S1+c0+c1(S1)]TJ /F3 11.955 Tf 11.96 0 Td[(s1)+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2(s1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)( 5\00062 )=g1(S1)+c0)]TJ /F3 11.955 Tf 11.95 0 Td[(c1s1+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2(s1)]TJ /F3 11.955 Tf 11.96 0 Td[(x)( 5\00064 )=g1(s1))]TJ /F3 11.955 Tf 11.96 0 Td[(c1s1+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d+c2(s1)]TJ /F3 11.955 Tf 11.96 0 Td[(x)( 5\00062 )=w(s1))]TJ /F3 11.955 Tf 11.95 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 125.8 -180.75 Td[( 2s1+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2(s1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)( 5\00077 )=g2(s1)+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d)]TJ /F3 11.955 Tf 11.96 0 Td[(c2x>g2)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d)]TJ /F3 11.955 Tf 11.96 0 Td[(c2x( 5\00077 )=w)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 129.34 -252.47 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d+c2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(x( 5\00098 )>w(x))]TJ /F3 11.955 Tf 11.96 0 Td[(Ae 1x)]TJ /F3 11.955 Tf 11.96 0 Td[(Be)]TJ ET BT /F5 7.97 Tf 160.18 -276.44 Td[( 2x=V(x);(D)wheretherstinequalityholdsbecauseof( 5 )ands1s1fV(y)+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x)g( 5\0006 )=miny>s1w(y))]TJ /F3 11.955 Tf 11.95 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 228.71 -466.47 Td[( 2y+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d+c2(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x)( 5\00077 )=miny>s1fg2(y)+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d)]TJ /F3 11.955 Tf 11.96 0 Td[(c2xg=g2)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d)]TJ /F3 11.955 Tf 11.96 0 Td[(c2x( 5\00077 )=w)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 194.48 -538.2 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(x( 5\00098 )>w(x))]TJ /F3 11.955 Tf 11.95 0 Td[(Ae 1x)]TJ /F3 11.955 Tf 11.95 0 Td[(Be)]TJ ET BT /F5 7.97 Tf 225.32 -562.16 Td[( 2x=V(x);(D)wherethefourthequalityholdsbecauseof( 5 )ands1
PAGE 249

Thus,from( D ),( D )( D ),wehave L2 V(x)=minn ~L2;2 V(x); L2;3 V(x); L2;4 V(x)o>V(x)forz2V(x)forz2V(x)and(ii) L2 V(x)=V(x)=V)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2+)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(xforxz2:Notethatforxz2,from( 5 ),wecanwrite L1 V(x) L1 V(x)=minf L1;1 V(x); L1;2 V(x); L1;3 V(x); L1;4 V(x)g;(D)where L1;1 V(x):=minxyz2fV(y)+c0+c1(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x)g(D) L1;2 V(x):=minz2yz1fV(y)+c0+c1(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x)g(D) L1;3 V(x)isdenedin( D ),and L1;4 V(x)isdenedin( D ).Similarly,from( 5 ),wecanwriteL2V(x)as L2 V(x)=minf L2;1 V(x); L2;2 V(x); L2;3 V(x); L2;4 V(x)g;(D)where L2;1 V(x):=minxyz2fV(y)+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x)g(D) 249

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L2;2 V(x):=minz2yz1fV(y)+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x)g(D) L2;3 V(x)isdenedin( D ),and L2;4 V(x)isdenedin( D ).Inrstpartoftheremainingproof,weprove(i) L1 V(x)>V(x)byproving(a) L1;1 V(x)>V(x),(b) L1;2 V(x)>V(x),(c) L1;3 V(x)>V(x),and(d) L1;4 V(x)>V(x)forxz2.For(a),wehave L1;1 V(x)( D\00084 )=minxyz2fV(y)+c0+c1(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x)g( 5\0006 )=minxyz2w)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 185 -270.9 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d+c2)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(y+c0+c1(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x);=w)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 125.52 -294.81 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(x+c0;=V(x)+c0>V(x);(D)wherethelastequalityholdsbecauseof( 5 )andxz2.For(b),wehave L1;2 V(x)( D\00085 )=minz2V(x);(D) 250

PAGE 251

wherethethirdequalityholdsbecause w0 (x))]TJ ET BT /F3 11.955 Tf 251.89 -11.96 Td[( 1Ae 1x+ 2Be)]TJ ET BT /F5 7.97 Tf 340.21 -7.62 Td[( 2x+c1>0basedon( 5 ),thefourthequalityfollowsfromthethirdequationof( 5 ),andthelastequalityholdsbecauseof( 5 )andxz2.For(c),wehave L1;3 V(x)( D\00071 )=minz1ys1fV(y)+c0+c1(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x)g( 5\0006 )=minz1ys1w(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 242.59 -156.84 Td[( 2S1+c0+c1(S1)]TJ /F3 11.955 Tf 11.95 0 Td[(y)+c0+c1(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x)=minz1ys1w(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 223.46 -180.75 Td[( 2S1+c0+c1(S1)]TJ /F3 11.955 Tf 11.96 0 Td[(x)+c0=w(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 160.69 -204.66 Td[( 2S1+c0+c1(S1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)+c0( 5\00082 )=U1(x)+c0>U2(x)+c0( 5\00083 )=w)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 185.9 -276.38 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(x+c0=V(x)+c0>V(x);(D)wheretherstinequalityholdsbecausexz2,z2U2(z)forz
PAGE 252

L1;4 V(x)( D\00058 )=miny>s1fV(y)+c0+c1(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x)g( 5\0006 )=miny>s1w(y))]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 230.97 -61.21 Td[( 2y+c0+c1(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x)( 5\00062 )=miny>s1fg1(y)+c0)]TJ /F3 11.955 Tf 11.95 0 Td[(c1xg( 5\00066 )=g1(S1)+c0)]TJ /F3 11.955 Tf 11.96 0 Td[(c1x( 5\00062 )=w(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 194.89 -132.93 Td[( 2S1+c0+c1(S1)]TJ /F3 11.955 Tf 11.96 0 Td[(x)( 5\00082 )=U1(x)+c0>U2(x)+c0( 5\00083 )=w)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 196.75 -204.66 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d+c2)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(x+c0=V(x);(D)wheretheinequalityholdsbecausexz2,z2U2(z)forzV(x)forxz2:(D)Insecondpartoftheremainingproof,weprove(ii) L2 V(x)=V(x)=V)]TJ /F4 11.955 Tf 7.48 -6.67 Td[(S2+)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(xbyproving(e) L2;1 V(x)>V(x),(f) L2;2 V(x)V(x),(g)L2;3V(x)>V(x),and(h)L2;4V(x)=V(x)=V)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2+)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(xforz2
PAGE 253

For(e),wehave L2;1 V(x)( D\00087 )=minxyz2fV(y)+c0+c1(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x)g( 5\0006 )=minxyz2w)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 185 -87.61 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d+c2)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(y+c0+c1(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x);=w)]TJ /F4 11.955 Tf 7.47 -6.67 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 125.52 -111.52 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2)]TJ /F4 11.955 Tf 7.48 -6.67 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(x+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d=V(x)+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d>V(x);(D)wherethelastequalityholdsbecauseof( 5 )andxz2.For(f),wehave 253

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L2;2 V(x)( D\00088 )=minz2
PAGE 255

L2;3 V(x)( D\00074 )=minz1ys1fV(y)+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x)g( 5\0006 )=minz1ys1w(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 188.27 -85.12 Td[( 2S1+c0+c1(S1)]TJ /F3 11.955 Tf 11.96 0 Td[(y)+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d+c2(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x)=w(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 127.49 -109.03 Td[( 2S1+c0+c1(S1)]TJ /F3 11.955 Tf 11.96 0 Td[(s1)+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2(s1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)( 5\00062 )=g1(S1)+c0)]TJ /F3 11.955 Tf 11.95 0 Td[(c1s1+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2(s1)]TJ /F3 11.955 Tf 11.96 0 Td[(x)( 5\00064 )=g1(s1))]TJ /F3 11.955 Tf 11.96 0 Td[(c1s1+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d+c2(s1)]TJ /F3 11.955 Tf 11.96 0 Td[(x)( 5\00062 )=w(s1))]TJ /F3 11.955 Tf 11.95 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 125.8 -180.75 Td[( 2s1+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2(s1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)( 5\00077 )=g2(s1)+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d)]TJ /F3 11.955 Tf 11.96 0 Td[(c2x>g2)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d)]TJ /F3 11.955 Tf 11.96 0 Td[(c2x( 5\00077 )=w)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 129.34 -252.47 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d+c2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(x=V(x);(D)wheretheinequalityholdsbecauseof( 5 )ands1s1fV(y)+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x)g( 5\0006 )=miny>s1w(y))]TJ /F3 11.955 Tf 11.95 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 228.71 -442.51 Td[( 2y+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d+c2(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x)( 5\00077 )=miny>s1fg2(y)+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d)]TJ /F3 11.955 Tf 11.96 0 Td[(c2xg=g2)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d)]TJ /F3 11.955 Tf 11.96 0 Td[(c2x( 5\00077 )=w)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 194.48 -514.23 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(x=V(x);(D)wherethefourthequalityholdsbecauseof( 5 )ands1
PAGE 256

L2;4 V(x)=w)]TJ /F4 11.955 Tf 7.48 -6.67 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 209.49 -42.88 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d+c2)]TJ /F4 11.955 Tf 7.47 -6.67 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(x;=V)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d+c2)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(x=V)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2+)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(x (D)wheretherstequalityholdsbecauseofthelastsecondequationin( D ),thesecondequalityholdsbecauseof( 5 ),S2>s1basedon( 5 ),andthelastequalityfollowsfromthedenitionof(x)in( 5 ),S2)]TJ /F3 11.955 Tf 11.96 0 Td[(xS2)]TJ /F3 11.955 Tf 11.97 0 Td[(z2basedonxz2,z2dbasedon( 5 ).Thus,from( D ),( D ),( D ),( D ),and( D ),wehave L2 V(x)=minf L2;1 V(x); L2;2 V(x); L2;3 V(x); L2;4 V(x)g (D)=V(x)=V)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2+)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(xforxz2:Insum,from( 5 ),( D ),and( D ),wehave L V(x)=minf L1 V(x); L2 V(x)g=V(x)=V)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2+)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(xforxz2. D.15ProofofTheorem 5.1 Werststatesomelemmas. LemmaD.1. Let f0( )=)]TJ /F3 11.955 Tf 9.3 0 Td[(r+ +2 2=2;(D)and 2=h+p 2+2r2i=2.Thenf0( 1)=f0()]TJ ET BT /F3 11.955 Tf 281.16 -556.99 Td[( 2)=0, 2>0andf0( )0for 2[)]TJ ET BT /F3 11.955 Tf 33.99 -580.9 Td[( 2;0]. 256

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Proof. Theconclusionimmediatelyfollowsthefactthatf0()isaquadraticfunctionofand1=h)]TJ /F3 11.955 Tf 9.3 0 Td[(+p 2+2r2i=2and)]TJ /F3 11.955 Tf 9.3 0 Td[(2=)]TJ /F9 11.955 Tf 11.29 13.27 Td[(h+p 2+2r2i=2aretworootsoff0()=0. LemmaD.2. Forthefunctionse 1xande)]TJ ET BT /F5 7.97 Tf 222.39 -103.25 Td[( 2x,where1=h)]TJ /F3 11.955 Tf 9.3 0 Td[(+p 2+2r2i=2and 2=h+p 2+2r2i=2,wehaveA)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(e1x=A)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2x=0,whereAisdenedin( 5 ). Proof. From( 5 ),wehaveA)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(e1x=e1x)]TJ /F3 11.955 Tf 9.3 0 Td[(r+1+1 2221=e1xf0(1)=0,wherethesecondequalityfollows( D )andthelastequalityholdsbecausef0(1)=0basedonLemma D.1 .Similarly,wehaveA)]TJ /F3 11.955 Tf 5.47 -9.69 Td[(e)]TJ /F5 7.97 Tf 6.59 0 Td[(2x=e)]TJ /F5 7.97 Tf 6.59 0 Td[(2x)]TJ /F3 11.955 Tf 9.3 0 Td[(r)]TJ /F3 11.955 Tf 11.95 0 Td[(2+1 2222=e)]TJ /F5 7.97 Tf 6.58 0 Td[(2xf0()]TJ /F3 11.955 Tf 9.29 0 Td[(2)=0,wherethesecondequalityfollows( D )andthelastequalityholdsbecausef0()]TJ /F3 11.955 Tf 9.3 0 Td[(2)=0basedonLemma D.1 . LemmaD.3. Forthefunctionw(x) denedin( 5 ) ,wehaveAw(x)+h(x)=0 ,whereAisdenedin( 5 ). Proof. Thisconclusionfollowsfrompage45in[ 31 ]. NowwestarttoproveProposition 5.14 . Proof. Intherstpartofthisproof,weshowAV(x)+h(x)=0forx>s1.From( 5 ),wehaveV(x)=w(x))]TJ /F3 11.955 Tf 11.95 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 153.88 -473.83 Td[( 2xforx>s1.Thus,AV(x)+h(x)= Aw(x))]TJ /F3 11.955 Tf 11.95 0 Td[(K1A)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(e)]TJ ET BT /F5 7.97 Tf 298.8 -515.47 Td[( 2x+h(x) = Aw(x)+h(x) =0;wherethesecondequalityholdsbecauseA)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(e)]TJ ET BT /F5 7.97 Tf 251.64 -588.09 Td[( 2x=0basedonLemma D.2 inAppendix D.15 ,andthelastequalityholdsbecauseAw(x)+h(x)=0basedonLemma D.3 inAppendix D.15 . 257

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Inthesecondpartofthisproof,weshowAV(x)+h(x)=0forz20forz10forxs1basedon( 5 ),wehaveg1(S1)+c0)]TJ /F3 11.955 Tf 11.95 0 Td[(f1(x)g1(S1)+c0)]TJ /F3 11.955 Tf 11.96 0 Td[(f1(s1)=0forxs1.Nowweareabletoprovetherstclaim.Forz1
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AV(x)+h(x)( 5\0006 )=A)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(w(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 208.19 -37.3 Td[( 2S1+c0+c1(S1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)+h(x)=A)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(w(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 189.06 -61.21 Td[( 2S1+c0+c1(S1)]TJ /F3 11.955 Tf 11.96 0 Td[(x))]TJ /F3 11.955 Tf 11.95 0 Td[(A0w(x)=A)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(w(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 189.06 -85.12 Td[( 2S1+c0+c1S1)]TJ /F3 11.955 Tf 11.95 0 Td[(A0(w(x)+c1x)( 5\00013 )=)]TJ /F3 11.955 Tf 9.3 0 Td[(r)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(w(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 217.77 -109.03 Td[( 2S1+c0+c1S1+r(w(x)+c1x))]TJ /F3 11.955 Tf 11.95 0 Td[(( w0 (x)+c1))]TJ /F4 11.955 Tf 11.29 0 Td[((2=2) w00 (x)( D\000101 ))]TJ /F3 11.955 Tf 37.09 0 Td[(r[ w0 (x)+c1]= 2)]TJ /F3 11.955 Tf 11.95 0 Td[(( w0 (x)+c1))]TJ /F4 11.955 Tf 11.96 0 Td[((2=2) w00 (x)=[ w0 (x)+c1]()]TJ /F3 11.955 Tf 9.3 0 Td[(r= 2)]TJ /F3 11.955 Tf 11.95 0 Td[())]TJ /F4 11.955 Tf 11.95 0 Td[((2=2) w00 (x)=)]TJ /F4 11.955 Tf 11.29 0 Td[((2=2)[ 2( w0 (x)+c1)+ w00 (x)]( 5\00050 )=)]TJ /F4 11.955 Tf 11.29 0 Td[((2=2) 2 f01 (x)0;wherethesecondequalityholdsbecauseAw(x)+h(x)=0basedonLemma D.3 ,thesixthequalityholdsbecauseof)]TJ /F3 11.955 Tf 9.3 0 Td[(r)]TJ /F3 11.955 Tf 11.31 0 Td[( 2+2 22=2=0basedonLemma D.1 ,andthelastinequalityholdsbasedon( 5 ).Inthefourthpartofthisproof,weshowAV(x)+h(x)>0forx
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Togettheseresults,notethatg2)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d)]TJ /F3 11.955 Tf 11.98 0 Td[(f2(s2)=0basedon( 5 )and)]TJ /F4 11.955 Tf 5.9 -9.69 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.67 Td[(S2+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d)]TJ /F3 11.955 Tf 11.95 0 Td[(f2(x)0=)]TJ ET BT /F3 11.955 Tf 286.14 -83.68 Td[(f02 (x)>0forxs2becausex
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D.16ProofofLemma 5.7 Proof. Weprovetheclaimbyprovingthatforz2[x2;1;s1],wehave(i)^F2(s2)[f1(z))]TJ /F3 11.955 Tf 11.95 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(c0]e)]TJ ET BT /F5 7.97 Tf 206.28 -56.06 Td[( 1z,(ii) ^F02 (x)<0forxs2,(iii)limx!^F2(x)=+1,and(iv)^F2(z)<[f1(z))]TJ /F3 11.955 Tf 11.95 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(c0]e)]TJ ET BT /F5 7.97 Tf 241.14 -79.97 Td[( 1z.Thenweareabletoreachtheconclusionthatforanyz2[x2;1;s1]thereexistsaunique(z)suchthat (z)s2;(z)
PAGE 262

Inthefollowing,weprove ^F02 (x)<0forxs2byproving f02 (x)<0forxs2andf2(x))]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d0forxs2:First,basedon( 5 ),wehave f02 (x)<0forxs2.Then,notingthatf2(s2))]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.67 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d=0basedon( 5 ),wehave f2(x))]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.95 0 Td[((c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d0forxs2:(D)Thus,asdiscussedabove,wehavetheconclusionof(ii),i.e., ^F02 (x)= f02 (x)e)]TJ ET BT /F5 7.97 Tf 89.86 -192.8 Td[( 1x)]TJ ET BT /F3 11.955 Tf 118.83 -197.74 Td[( 1f2(x))]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)de)]TJ ET BT /F5 7.97 Tf 324.76 -192.8 Td[( 1x<0forxs2:(D)For(iii),fromlimx!f2(x)=+1basedon( 5 ),wehavethatlimx!^F2(x)=+1.For(iv),basedon( 5 ),weprove^F2(z)=f2(z))]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)de)]TJ ET BT /F5 7.97 Tf 269.46 -336.25 Td[( 1z<[f1(z))]TJ /F3 11.955 Tf 11.95 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(c0]e)]TJ ET BT /F5 7.97 Tf 419.51 -336.25 Td[( 1zbyprovingf2(z))]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.95 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d
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4U2;1(z)basedon( 5 ),andtheinequalityholdsbecause4U2;1(z)>0forz>x2;1basedon( 5 )andc1>c2basedonAssumption 5.2 . D.17ProofofLemma 5.8 Proof. Intherstpartofthisproof,wereformulateandsimplifyg(z).Dene A(z)=[ 2=( 1+ 2)][f1(z))]TJ /F3 11.955 Tf 11.96 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(c0]e)]TJ ET BT /F5 7.97 Tf 342.73 -133.16 Td[( 1z;(D)wehaveA=A(z1)basedon( 5 ).From( D )and( D ),wehave A(z)=[ 2=( 1+ 2)]f2((z)))]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)de)]TJ ET BT /F5 7.97 Tf 358.9 -228.8 Td[( 1(z):(D)Nowwestarttoexploreg(z).From( 5 ),wehave g(z)=fw(z)+c1z1)]TJ /F3 11.955 Tf 11.95 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(c0)]TJ /F4 11.955 Tf 11.95 0 Td[([ w0 (z)+c1]= 1ge 2z)]TJ /F9 11.955 Tf 11.29 9.68 Td[(w((z))+c2z2)]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d)]TJ /F4 11.955 Tf 11.96 0 Td[([ w0 ((z))+c2]= 1e 2(z)=fw(z)+c1z1)]TJ /F3 11.955 Tf 11.95 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(c0+[ w0 (z)+c1]= 2ge 2z)]TJ /F4 11.955 Tf 11.29 0 Td[([ w0 (z)+c1](1= 1+1= 2)e 2z)]TJ /F9 11.955 Tf 11.29 9.68 Td[(w((z))+c2z2)]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+[ w0 ((z))+c2]= 2e 2(z))]TJ /F4 11.955 Tf 11.29 0 Td[([ w0 ((z))+c2](1= 1+1= 2)e 2(z)=[f1(z))]TJ /F3 11.955 Tf 11.95 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(c0]e)]TJ ET BT /F5 7.97 Tf 168.92 -462.3 Td[( 1z)]TJ /F4 11.955 Tf 11.95 0 Td[([ w0 (z)+c1](1= 1+1= 2)e 2zf2((z)))]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)de)]TJ ET BT /F5 7.97 Tf 254.82 -486.21 Td[( 1(z))]TJ /F4 11.955 Tf 11.96 0 Td[([ w0 ((z))+c2](1= 1+1= 2)e 2(z)=[( 1+ 2)= 2]A(z)e 1z)]TJ /F4 11.955 Tf 11.95 0 Td[([ w0 (z)+c1](1= 1+1= 2)e 2z)]TJ /F9 11.955 Tf 11.29 9.68 Td[([( 1+ 2)= 2]A(z)e 1(z))]TJ /F4 11.955 Tf 11.96 0 Td[([ w0 ((z))+c2](1= 1+1= 2)e 2(z)=[( 1+ 2)=( 1 2)] 1A(z)e( 1+ 2)z)]TJ /F4 11.955 Tf 11.96 0 Td[([ w0 (z)+c1]e 2z)]TJ /F9 11.955 Tf 11.29 9.69 Td[( 1A(z)e( 1+ 2)(z))]TJ /F4 11.955 Tf 11.95 0 Td[([ w0 ((z))+c2]e 2(z)=[( 1+ 2)=( 1 2)]~g(z);(D) 263

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wherethethridequalityholdsbecauseof( 5 ),thefourthequalityholdsbecauseof( D )and( D ),and ~g(z):= 1A(z)e( 1+ 2)z)]TJ /F3 11.955 Tf 11.96 0 Td[(e( 1+ 2)(z))]TJ /F4 11.955 Tf 11.33 0 Td[([ w0 (z)+c1]e 2z+[ w0 ((z))+c2]e 2(z):(D)Basedon( D ),inordertoproveg0(z)>0,itsufcestoprove ~g0 (z)>0.Toprove ~g0 (z)>0,basedon( D ),weneedtocompute A0 (z)and 0 (z)rst.Inthesecondpartofthisproof,wecompute A0 (z)and 0 (z),andprove 0 (z)>0.For A0 (z),from( D ),wehave A0 (z)=[ 2=( 1+ 2)]f f01 (z))]TJ ET BT /F3 11.955 Tf 239.63 -251.04 Td[( 1[f1(z))]TJ /F3 11.955 Tf 11.96 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(c0]ge)]TJ ET BT /F5 7.97 Tf 379.6 -246.1 Td[( 1z=[ 2=( 1+ 2)] f01 (z)e)]TJ ET BT /F5 7.97 Tf 238.03 -273 Td[( 1z)]TJ ET BT /F3 11.955 Tf 266.5 -277.94 Td[( 1A(z)( 1+ 2)= 2; (D)wherethelastequalityholdsbecauseof( D ).For 0 (z),from( D ),wehave 0 (z)=f f01 (z))]TJ ET BT /F3 11.955 Tf 152.36 -389.39 Td[( 1[f1(z))]TJ /F3 11.955 Tf 11.96 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(c0]ge)]TJ ET BT /F5 7.97 Tf 292.33 -385.05 Td[( 1z f02 ((z)))]TJ ET BT /F3 11.955 Tf 111.31 -406.32 Td[( 1f2((z)))]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)de)]TJ ET BT /F5 7.97 Tf 338.65 -402.87 Td[( 1(z)= f01 (z)e)]TJ ET BT /F5 7.97 Tf 94.38 -426.29 Td[( 1z)]TJ ET BT /F3 11.955 Tf 122.86 -431.23 Td[( 1A(z)( 1+ 2)= 2= f02 ((z))e)]TJ ET BT /F5 7.97 Tf 306.79 -426.29 Td[( 1(z))]TJ ET BT /F3 11.955 Tf 346.13 -431.23 Td[( 1A(z)( 1+ 2)= 2; (D)wherethelastequalityholdsbecauseof( D )and( D ).Next,weprovethat 0 (z)>0byproving(i) f01 (z)<0forz0.For(i),theclaimfollowsfrom( 5 ).For(ii),theclaimholdsbecausef02(x)<0forxs2and(z)s2forz
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f1(z))]TJ /F3 11.955 Tf 11.96 0 Td[(f1(s1)= f01 (1)(z)]TJ /F3 11.955 Tf 11.96 0 Td[(s1) f01 (s1)(z)]TJ /F3 11.955 Tf 11.96 0 Td[(s1)>0forz[f1(s1))]TJ /F3 11.955 Tf 11.95 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(c0]=0forz0:(D)Insum,from( D ),(i),(ii),and(iii),wehave 0 (z)>0forzs1:(D)Intheremainingpartofthisproof,wecompute ~g0 (z)andprovethatitispositive.From( D ),wehave ~g0 (z)= 2[ w0 ((z))+c2]e 2(z) 0 (z))]TJ /F4 11.955 Tf 11.96 0 Td[([ w0 (z)+c1]e 2z+ w00 ((z))e 2(z) 0 (z))]TJ ET BT /F3 11.955 Tf 423.7 -468.2 Td[(w00 (z)e 2z+ 1 A0 (z)e( 1+ 2)z)]TJ /F3 11.955 Tf 11.96 0 Td[(e( 1+ 2)(z)+ 1A(z)( 1+ 2)e( 1+ 2)z)]TJ ET BT /F3 11.955 Tf 358.67 -495.1 Td[(0 (z)e( 1+ 2)(z) (D)= 2 f02 ((z))e 2(z) 0 (z))]TJ ET BT /F3 11.955 Tf 164.54 -546.9 Td[( 2 f01 (z)e 2z+ 1 A0 (z)e( 1+ 2)z)]TJ /F3 11.955 Tf 11.96 0 Td[(e( 1+ 2)(z)+ 1A(z)( 1+ 2)e( 1+ 2)z)]TJ ET BT /F3 11.955 Tf 185.51 -573.8 Td[(0 (z)e( 1+ 2)(z);wherethelastequalityholdsbecauseof( 5 ).Notethatthesumoftherstandlasttermsof( D )is 265

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1A(z)( 1+ 2)e( 1+ 2)z)]TJ ET BT /F3 11.955 Tf 171.45 -47.82 Td[(0 (z)e( 1+ 2)(z)+ 2 f02 ((z))e 2(z) 0 (z))]TJ ET BT /F3 11.955 Tf 399.01 -47.82 Td[(f01 (z)e 2z= 2 f02 ((z))e 2(z))]TJ ET BT /F3 11.955 Tf 134.23 -74.71 Td[( 1A(z)( 1+ 2)e( 1+ 2)(z) 0 (z)+ 1A(z)( 1+ 2)e( 1+ 2)z)]TJ ET BT /F3 11.955 Tf 180.89 -101.61 Td[( 2 f01 (z)e 2z (D)= 2 f02 ((z))e)]TJ ET BT /F5 7.97 Tf 101.49 -123.57 Td[( 1(z))]TJ ET BT /F3 11.955 Tf 140.82 -128.51 Td[( 1A(z)( 1+ 2)= 2e( 1+ 2)(z) 0 (z)+ 2e( 1+ 2)z 1A(z)( 1+ 2)= 2)]TJ ET BT /F3 11.955 Tf 218.83 -155.4 Td[(f01 (z)e)]TJ ET BT /F5 7.97 Tf 260.4 -150.47 Td[( 1z= 2 f01 (z)e)]TJ ET BT /F5 7.97 Tf 84.37 -177.37 Td[( 1z)]TJ ET BT /F3 11.955 Tf 112.85 -182.3 Td[( 1A(z)( 1+ 2)= 2e( 1+ 2)(z)+ 2e( 1+ 2)z 1A(z)( 1+ 2)= 2)]TJ ET BT /F3 11.955 Tf 216.84 -209.2 Td[(f01 (z)e)]TJ ET BT /F5 7.97 Tf 258.41 -204.26 Td[( 1z (D)= 2 f01 (z)e)]TJ ET BT /F5 7.97 Tf 84.37 -231.16 Td[( 1z)]TJ ET BT /F3 11.955 Tf 112.85 -236.09 Td[( 1A(z)( 1+ 2)= 2e( 1+ 2)(z))]TJ /F3 11.955 Tf 11.95 0 Td[(e( 1+ 2)z; (D)wherethethirdequalityholdsbecauseof( D ).Thus,from( D )and( D ),wehave ~g0 (z)=e( 1+ 2)z)]TJ /F3 11.955 Tf 11.95 0 Td[(e( 1+ 2)(z) 1 A0 (z))]TJ ET BT /F3 11.955 Tf 223.39 -355.64 Td[( 2 f01 (z)e)]TJ ET BT /F5 7.97 Tf 283.5 -350.7 Td[( 1z)]TJ ET BT /F3 11.955 Tf 311.97 -355.64 Td[( 1A(z)( 1+ 2)= 2=e( 1+ 2)z)]TJ /F3 11.955 Tf 11.95 0 Td[(e( 1+ 2)(z)[ 1 2=( 1+ 2))]TJ ET BT /F3 11.955 Tf 259.07 -382.53 Td[( 2] f01 (z)e)]TJ ET BT /F5 7.97 Tf 322.43 -377.6 Td[( 1z)]TJ ET BT /F3 11.955 Tf 350.91 -382.53 Td[( 1A(z)( 1+ 2)= 2; (D)wherethelastequalityholdsbecauseof( D ).Inthefollowing,weanalyzethesignofthethreetermsseparately.Fortherstterm,from( D ),wehave(z)0:(D)Forthesecondterm,itiseasytoseethat 1 2=( 1+ 2))]TJ ET BT /F3 11.955 Tf 200.75 -609.13 Td[( 2= 2[ 1=( 1+ 2))]TJ /F4 11.955 Tf 11.95 0 Td[(1]<0:(D) 266

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Forthethirdterm,from f01 (z)<0forz0basedon( D ),wehave f01 (z)e)]TJ ET BT /F5 7.97 Tf 176.4 -78.74 Td[( 1z)]TJ ET BT /F3 11.955 Tf 204.87 -83.68 Td[( 1A(z)( 1+ 2)= 2<0:(D)Finally,from( D ),( D ),( D ),( D ),and( D ),wehave g0 (z)>0andnishtheproof. D.18ProofofLemma 5.9 Proof. Wewanttoprovethatg(x2;1)<0where g(x2;1)=fw(x2;1)+c1x2;1)]TJ /F3 11.955 Tf 11.95 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[([ w0 (x2;1)+c1]= 1ge 2x2;1)]TJ /F9 11.955 Tf 11.29 9.68 Td[(w((x2;1))+c2(x2;1))]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d)]TJ /F4 11.955 Tf 11.95 0 Td[([ w0 ((x2;1))+c2]= 1e 2(x2;1):(D)basedon( 5 ).Fromthedenitionsthatf1(x)=[ w0 (x)+c1]= 2+w(x)+c1xandf2(x)=[ w0 (x)+c2]= 2+w(x)+c2xbasedon( 5 )and( D ),weknowthat(x2;1)isdecidedbyfw(x2;1)+c1x2;1)]TJ /F3 11.955 Tf 11.96 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(c0+[w0(x2;1)+c1]= 2ge)]TJ ET BT /F5 7.97 Tf 304.36 -425.31 Td[( 1x2;1 (D)=w((x2;1))+c2(x2;1))]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+[ w0 ((x2;1))+c2]= 2e)]TJ ET BT /F5 7.97 Tf 419.51 -452.2 Td[( 1(x2;1):Notethatbasedon( 5 ),wehavethat( D )isequivalentto f2((x2;1)))]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)de)]TJ ET BT /F5 7.97 Tf 240.46 -547.84 Td[( 1(x2;1)=[f1(x2;1))]TJ /F3 11.955 Tf 11.96 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(c0]e)]TJ ET BT /F5 7.97 Tf 423.92 -547.84 Td[( 1x2;1(D)Intherstpartofthisproof,wereformulateandsimplifytheproblem.Let 267

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= 2= 1>0;u=)]TJ /F3 11.955 Tf 9.3 0 Td[(c0;;a=w(x2;1)+c1x2;1)]TJ /F3 11.955 Tf 11.95 0 Td[(g1(S1); (D)b=)]TJ /F4 11.955 Tf 9.71 0 Td[(g2)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F4 11.955 Tf 11.95 0 Td[((c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d;=x2;1)]TJ /F3 11.955 Tf 11.95 0 Td[((x2;1); (D)F1(x)=w(x)+c1x;F2(x)=w(x)+c2x: (D)e1= F01 (x2;1);ande2()=F2(x2;1)]TJ /F3 11.955 Tf 11.95 0 Td[(): (D)Notethatusingnewnotations,( D )becomes f2(x2;1)]TJ /F3 11.955 Tf 11.96 0 Td[())]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)de 1=[f1(x2;1))]TJ /F3 11.955 Tf 11.96 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(c0];(D)whichisequivalentto (u+a+e1= 2)e)]TJ ET BT /F5 7.97 Tf 193.87 -302.88 Td[( 1=[u+b+e2())]TJ ET BT /F3 11.955 Tf 311.76 -307.82 Td[(e02 ()= 2]:(D)And( D)-222()]TJ /F4 11.955 Tf 21.26 0 Td[(129 )becomesg(x2;1)=(u+a)]TJ /F3 11.955 Tf 11.96 0 Td[(e1= 1)e 2x2;1 (D))]TJ /F4 11.955 Tf 11.96 0 Td[([u+b+e2()+ e02 ()= 1]e 2x2;1e)]TJ ET BT /F5 7.97 Tf 341.49 -419.94 Td[( 2:Nowwediscussthebasicideaofourproof.Notethatforagiven^u>0,whichis^c0>0,wehaveunique^S1,^S2,^x2;1,and(^x2;1)decidedbyProposition 5.1 ,( 5 ),( 5 ),and( 5 )repectively.Thecorresponding^a,^b,^e1,^,and^e2(^)satisfy( D ).Wewanttoproveg(^x2;1)=g^u;^a;^b;^e1;^;^e2(^)>0.Weproveinthisway.Fixa=^a;b=^b,e1=^e1,x2;1=^x2;1.Onlylet(u)tochangewithuinu^uande2((u))tochangewith(u)suchthat( D )satised.Ifgu;^a;^b;^e1;(u);e2((u))>0forallu^u,thencertainlywehaveg^u;^a;^b;^e1;^;^e2(^)>0. 268

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Inthefollowing,fornotationsimplicity,wewritegu;^a;^b;^e1;(u);e2((u))asg(u;(u)),(u)as,ande2((u))ase2().Fornotationsimplicity,wewrite^a,^b,and^e1asa,b,ande1,butrememberthattheyaredecidedby^uanddonotchangewithu.Next,werstprove:(i)u+a+e1= 2>0for u^u,(ii)foranyu^u,thereexistsaunique(u)>0suchthat D ,whichisequivalentto( D )satised,(iii)(u)^forallu^u(iv)u+b+e2())]TJ ET BT /F3 11.955 Tf 168.56 -145.24 Td[(e02 ()= 2>0,and(v)e1<0,For(i),sincex2;1f1(s1))]TJ /F3 11.955 Tf 11.96 0 Td[(g1(S1))]TJ /F4 11.955 Tf 12.2 0 Td[(^c0( D\000106 )=0:(D)Foru^u,wehaveu+a+e1= 2^u+a+e1= 2 (D)=w(x2;1)+c1x2;1+[w0(x2;1)+c1]= 2)]TJ /F3 11.955 Tf 11.95 0 Td[(g1(S1))]TJ /F4 11.955 Tf 12.2 0 Td[(^c0( 5\00045 )=f1(x2;1))]TJ /F3 11.955 Tf 11.96 0 Td[(g1(S1))]TJ /F4 11.955 Tf 12.2 0 Td[(^c0( D\000139 )>0;wheretherstequalityholdsbecauseof( D ),( D ),and( D ),For(ii),werstdene ~F():=f2(x2;1)]TJ /F3 11.955 Tf 11.96 0 Td[())]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.95 0 Td[((c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)de 1:(D)Inthefollowing,weprovethereexistsaunique (u)>0(D) 269

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suchthat( D )satisedbyproving(a)~F(maxf0;x2;1)]TJ /F4 11.955 Tf 12.44 0 Td[(s2g)<[f1(x2;1))]TJ /F3 11.955 Tf 11.95 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(c0];(b)lim!+1~F()=+1,and(c)~F0()>0for>maxf0;x2;1)]TJ /F4 11.955 Tf 12.44 0 Td[(s2g.Wediscussintwocasesinwhichx2;1)]TJ /F4 11.955 Tf 12.44 0 Td[(s2<0andx2;1)]TJ /F4 11.955 Tf 12.44 0 Td[(s20respectively.UnderCasex2;1)]TJ /F4 11.955 Tf 12.44 0 Td[(s2<0,for(a),from( D ),wehave~F(0)=f2(x2;1))]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d( D\000110 )<[f1(x2;1))]TJ /F3 11.955 Tf 11.95 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(c0]:For(b),fromlimx!f2(x)=+1basedon( 5 ),wehavethatlimx!~F()=+1.For(c),from( D ),wehave~F0()=)]TJ /F3 11.955 Tf 9.3 0 Td[(f02(x2;1)]TJ /F3 11.955 Tf 11.96 0 Td[()e 1+ 1f2(x2;1)]TJ /F3 11.955 Tf 11.95 0 Td[())]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.95 0 Td[((c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)de 1: (D)=)]TJ /F3 11.955 Tf 9.3 0 Td[(f02(x2;1)]TJ /F3 11.955 Tf 11.96 0 Td[()e 1+ 1~F():Weprove ~F0()>0for>0(D)byprovingf02(x2;1)]TJ /F3 11.955 Tf 11.95 0 Td[()<0for>0and~F()>0for>0.Forf02(x2;1)]TJ /F3 11.955 Tf 11.96 0 Td[(),wehavef02(x2;1)]TJ /F3 11.955 Tf 11.96 0 Td[()<0for>0becausef02(x)<0forxs2basedon( 5 )andx2;1)]TJ /F3 11.955 Tf 12.51 0 Td[(
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~F()( D\000141 )=f2(x2;1)]TJ /F3 11.955 Tf 11.95 0 Td[())]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.67 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(c0)]TJ /F4 11.955 Tf 11.95 0 Td[((c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d>f2(s2)]TJ /F3 11.955 Tf 11.96 0 Td[())]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d>f2(s2))]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d( 5\00092 )=0;wheretheinequalitiesholdbecauseof( 5 ),x2;1maxf0;x2;1)]TJ /F4 11.955 Tf 12.43 0 Td[(s2g0.UnderCasex2;1)]TJ /F4 11.955 Tf 12.82 0 Td[(s20,notethatfrom( D ),wehavethatforagivenu>^u,whichcorrespondstoc0<^c0, f1(x2;1))]TJ /F3 11.955 Tf 11.96 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(c0>f1(x2;1))]TJ /F3 11.955 Tf 11.96 0 Td[(g1(S1))]TJ /F4 11.955 Tf 12.2 0 Td[(^c0( D\000140 )>0(D)For(a),from( D ),wehave~F(x2;1)]TJ /F4 11.955 Tf 12.43 0 Td[(s2)=f2(s2))]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d( 5\00092 )=0( D\000145 )<[f1(x2;1))]TJ /F3 11.955 Tf 11.96 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(c0]:For(b),fromlimx!f2(x)=+1basedon( 5 ),wehavethatlimx!~F()=+1.For(c),basedon( D ),weprove ~F0()>0for>x2;1)]TJ /F4 11.955 Tf 12.43 0 Td[(s2(D)byprovingf02(x2;1)]TJ /F3 11.955 Tf 11.95 0 Td[()<0for>x2;1)]TJ /F4 11.955 Tf 12.44 0 Td[(s2and~F()>0for>x2;1)]TJ /F4 11.955 Tf 12.43 0 Td[(s2.Forf02(x2;1)]TJ /F3 11.955 Tf 11.96 0 Td[(),wehavef02(x2;1)]TJ /F3 11.955 Tf 11.96 0 Td[()<0for>x2;1)]TJ /F4 11.955 Tf 12.81 0 Td[(s2becausef02(x)<0forxs2basedon( 5 )andx2;1)]TJ /F3 11.955 Tf 11.96 0 Td[(
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~F()( D\000141 )=f2(x2;1)]TJ /F3 11.955 Tf 11.95 0 Td[())]TJ /F4 11.955 Tf 12.38 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(c0)]TJ /F4 11.955 Tf 11.95 0 Td[((c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d>f2(s2))]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d( 5\00092 )=0;wheretheinequalitiesholdbecauseof( 5 )and>x2;1)]TJ /F4 11.955 Tf 12.44 0 Td[(s2.For(iii),takingderivativetou=)]TJ /F3 11.955 Tf 9.3 0 Td[(c0in( D ),wehave[)]TJ /F3 11.955 Tf 9.3 0 Td[(f02(x2;1)]TJ /F3 11.955 Tf 11.96 0 Td[()0(u)+1]e 1+ 1f2(x2;1)]TJ /F3 11.955 Tf 11.95 0 Td[())]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.67 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(c0)]TJ /F4 11.955 Tf 11.95 0 Td[((c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)de 10(u)=1:Fromtheaboveequationand( D ),wehave 0(u)=1)]TJ /F3 11.955 Tf 11.96 0 Td[(e 1 ~F0()<0foru^u;(D)wheretheinequalityholdsbecause~F0()>0basedon( D )and( D ),and10by( D ).Thus,basedon( D ),wehave (u)(^u)=^:(D)For(iv),wehave u+b+e2())]TJ ET BT /F3 11.955 Tf 125.74 -492.6 Td[(e02 ()= 2( D\000137 )=(u+a+e1= 2)e)]TJ ET BT /F5 7.97 Tf 312.53 -487.66 Td[( 1( D\000140 )>0:(D)For(v),wehave e1( D\000135 )= F01 (x2;1)( D\000134 )= w0 (x2;1)+c1 w0 (s1)+c1( 5\00054 )=)]TJ /F3 11.955 Tf 9.3 0 Td[(K1 2e)]TJ ET BT /F5 7.97 Tf 375.39 -583.29 Td[( 2s1<0;(D)wheretherstinequalityholdsbecause w00 (x)0basedon( 5 )andx2;1
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Finally,wesimplifytheexpressionofg(x2;1)in( D ).Beforewedothis,werstgiveanexpressionofe)]TJ /F5 7.97 Tf 6.59 0 Td[(.Dividing(u+a+e1= 2)andtakingpowerof1= 1inbothsidesof( D ),wehave e)]TJ /F5 7.97 Tf 6.59 0 Td[(=[u+b+e2())]TJ ET BT /F3 11.955 Tf 258.03 -99.5 Td[(e02 ()= 2]1= 1 [u+a+e1= 2]1= 1:(D)From( D ),wehaveg((u);u)( D\000138 )=(u+a)]TJ /F3 11.955 Tf 11.95 0 Td[(e1= 1)e 2x2;1)]TJ /F4 11.955 Tf 11.29 0 Td[([u+b+e2()+ e02 ()= 1]e 2x2;1e)]TJ ET BT /F5 7.97 Tf 322.51 -199.18 Td[( 2=e 2x2;1(u+a)]TJ /F3 11.955 Tf 11.96 0 Td[(e1= 1)1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ ET BT /F5 7.97 Tf 291.9 -223.69 Td[( 2[u+b+e2()+ e02 ()= 1] (u+a)]TJ /F5 7.97 Tf 6.59 0 Td[(e1= 1)=e 2x2;1fu+a)]TJ /F3 11.955 Tf 11.95 0 Td[(e1= 1g[1)]TJ /F4 11.955 Tf 12.37 0 Td[(~g(;u)];where ~g((u);u):=e)]TJ ET BT /F5 7.97 Tf 107.77 -328.66 Td[( 2[u+b+e2()+ e02 ()= 1] (u+a)]TJ /F5 7.97 Tf 6.58 0 Td[(e1= 1)( D\000186 )=[u+b+e2())]TJ ET BT /F5 7.97 Tf 175.89 -350.16 Td[(e02 ()= 2] 2= 1 (u+a+e1= 2) 2= 1[u+b+e2()+ e02 ()= 1] (u+a)]TJ /F5 7.97 Tf 6.58 0 Td[(e1= 1)=1+b)]TJ /F5 7.97 Tf 6.59 0 Td[(a+e2())]TJ /F4 11.955 Tf 6.59 -1 Td[([ e02 ()+e1]= 2 u+a+e1= 2 2= 11+b)]TJ /F5 7.97 Tf 6.59 0 Td[(a+e2()+[ e02 ()+e1]= 1 u+a)]TJ /F5 7.97 Tf 6.58 0 Td[(e1= 1:(D)Sinceu+a)]TJ /F3 11.955 Tf 11.84 0 Td[(e1= 1>0basedon( D ),wehavethatg(;u)and1)]TJ /F4 11.955 Tf 12.26 0 Td[(~g(;u)havethesamesign.Inthesecondpartofthisproof,inordertoprovethatg(x2;1)<0,wewillprovethat~g((u);u)>1byproving(a)limu!+1~g((u);u)=1and(b)@~g @u<0.Werstprovelimu!+1~g((u);u)=1.From( D )and( D ),weknowthat0<<.Sowhenu!+1,wehave1+b)]TJ /F3 11.955 Tf 11.96 0 Td[(a+e2())]TJ /F4 11.955 Tf 11.96 0 Td[([ e02 ()+e1]= 2 u+a+e1= 2!1and 273

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1+b)]TJ /F3 11.955 Tf 11.96 0 Td[(a+e2()+[ e02 ()+e1]= 1 u+a)]TJ /F3 11.955 Tf 11.96 0 Td[(e1= 1!1:Thus,wehavelimu!+1~g((u);u)=1:Thenweprove@~g @u<0:Tocompute@~g @u,weneedtocompute 0 (u)rst.Takingderivativetouin( D ),wehavee)]TJ ET BT /F5 7.97 Tf 67.38 -156.94 Td[( 1)]TJ ET BT /F3 11.955 Tf 97.02 -161.88 Td[( 1fu+a+e1= 2ge)]TJ ET BT /F5 7.97 Tf 205.74 -156.94 Td[( 1 0 (u)=1+[ e02 ())]TJ ET BT /F3 11.955 Tf 329.51 -161.88 Td[(e002 ()= 2] 0 (u):Afterrearrangement,wehave 0 (u)=e)]TJ ET BT /F5 7.97 Tf 241.55 -227.65 Td[( 1)]TJ /F4 11.955 Tf 11.96 0 Td[(1 [ e02 ())]TJ ET BT /F3 11.955 Tf 183.05 -248.28 Td[(e002 ()= 2]+ 1(u+a+e1= 2)e)]TJ ET BT /F5 7.97 Tf 355.14 -244.82 Td[( 1:(D)Substituting( D )into( D )anddeletinge)]TJ ET BT /F5 7.97 Tf 261.86 -266.12 Td[( 1,wehave 0 (u)=nu+b+e2())]TJ ET BT /F5 7.97 Tf 146.03 -317.88 Td[(e02 ()= 2 u+a+e1= 2)]TJ /F4 11.955 Tf 11.96 0 Td[(1o1 [ e02 ())]TJ ET BT /F5 7.97 Tf 242.22 -329.75 Td[(e002 ()= 2]+ 1[u+b+e2())]TJ ET BT /F5 7.97 Tf 346.94 -329.75 Td[(e02 ()= 2]=b)]TJ /F5 7.97 Tf 6.59 0 Td[(a+e2())]TJ /F4 11.955 Tf 6.58 -1 Td[([ e02 ()+e1]= 2 (u+a+e1= 2)1 [ e02 ())]TJ ET BT /F5 7.97 Tf 224.59 -356.17 Td[(e002 ()= 2]+ 1[u+b+e2())]TJ ET BT /F5 7.97 Tf 329.31 -356.17 Td[(e02 ()= 2]:(D)Nowwecompute@~g @uandproveitisnegative.From( D ),wehave 274

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@~g @u= 2 11+b)]TJ /F5 7.97 Tf 6.59 0 Td[(a+e2())]TJ /F4 11.955 Tf 6.59 -1 Td[([ e02 ()+e1]= 2 u+a+e1= 2 2= 1)]TJ /F7 7.97 Tf 6.59 0 Td[(1f[ e02 ())]TJ ET BT /F3 11.955 Tf 102.34 -73.16 Td[(e002 ()= 2] 0 (u)(u+a+e1= 2))]TJ /F4 11.955 Tf 11.96 0 Td[([b)]TJ /F3 11.955 Tf 11.96 0 Td[(a+e2())]TJ /F4 11.955 Tf 11.96 0 Td[([ e02 ()+e1]= 2]g1 (u+a+e1= 2)21+b)]TJ /F5 7.97 Tf 6.59 0 Td[(a+e2()+[ e02 ()+e1]= 1 u+a)]TJ /F5 7.97 Tf 6.58 0 Td[(e1= 1+1+b)]TJ /F5 7.97 Tf 6.58 0 Td[(a+e2())]TJ /F4 11.955 Tf 6.59 -1 Td[([ e02 ()+e1]= 2 u+a+e1= 2 2= 11 (u+a)]TJ /F5 7.97 Tf 6.59 0 Td[(e1= 1)2f[ e02 ()+ e002 ()= 1] 0 (u)(u+a)]TJ /F3 11.955 Tf 11.95 0 Td[(e1= 1))]TJ /F4 11.955 Tf 11.96 0 Td[([b)]TJ /F3 11.955 Tf 11.96 0 Td[(a+e2()+[ e02 ()+e1]= 1]g= 2 1[u+b+e2())]TJ ET BT /F5 7.97 Tf 114.51 -174.44 Td[(e02 ()= 2] 2= 1)]TJ /F12 5.978 Tf 5.75 0 Td[(1 (u+a+e1= 2) 2= 1)]TJ /F12 5.978 Tf 5.75 0 Td[(1f[ e02 ())]TJ ET BT /F3 11.955 Tf 102.34 -206.58 Td[(e002 ()= 2] 0 (u)(u+a+e1= 2))]TJ /F4 11.955 Tf 11.96 0 Td[([b)]TJ /F3 11.955 Tf 11.96 0 Td[(a+e2())]TJ /F4 11.955 Tf 11.96 0 Td[([ e02 ()+e1]= 2]g1 (u+a+e1= 2)2[u+b+e2()+ e02 ()= 1] (u+a)]TJ /F5 7.97 Tf 6.58 0 Td[(e1= 1)+[u+b+e2())]TJ ET BT /F5 7.97 Tf 111.67 -247.65 Td[(e02 ()= 2] 2= 1 (u+a+e1= 2) 2= 11 (u+a)]TJ /F5 7.97 Tf 6.59 0 Td[(e1= 1)2f[ e02 ()+ e002 ()= 1] 0 (u)(u+a)]TJ /F3 11.955 Tf 11.95 0 Td[(e1= 1))]TJ /F4 11.955 Tf 11.96 0 Td[([b)]TJ /F3 11.955 Tf 11.96 0 Td[(a+e2()+[ e02 ()+e1]= 1]g=[u+b+e2())]TJ ET BT /F5 7.97 Tf 103.03 -296.96 Td[(e02 ()= 2] 2= 1)]TJ /F12 5.978 Tf 5.76 0 Td[(1 (u+a+e1= 2) 2= 1)]TJ /F12 5.978 Tf 5.75 0 Td[(11 (u+a)]TJ /F5 7.97 Tf 6.59 0 Td[(e1= 1)1 (u+a+e1= 2)g1(;u);whereg1((u);u)= 2 1f[ e02 ())]TJ ET BT /F3 11.955 Tf 97.79 -420.63 Td[(e002 ()= 2] 0 (u)(u+a+e1= 2))]TJ /F4 11.955 Tf 11.96 0 Td[([b)]TJ /F3 11.955 Tf 11.96 0 Td[(a+e2())]TJ /F4 11.955 Tf 11.95 0 Td[([ e02 ()+e1]= 2]g[u+b+e2()+ e02 ()= 1] (u+a)]TJ /F3 11.955 Tf 11.96 0 Td[(e1= 1) (D)+f[ e02 ()+ e002 ()= 1] 0 (u)(u+a)]TJ /F3 11.955 Tf 11.96 0 Td[(e1= 1))]TJ /F4 11.955 Tf 11.95 0 Td[([b)]TJ /F3 11.955 Tf 11.95 0 Td[(a+e2()+[ e02 ()+e1]= 1]g[u+b+e2())]TJ ET BT /F3 11.955 Tf 117.54 -503.69 Td[(e02 ()= 2] (u+a)]TJ /F3 11.955 Tf 11.96 0 Td[(e1= 1):Notethat u+a)]TJ /F3 11.955 Tf 11.95 0 Td[(e1= 1>u+a+e1= 2>0(D) 275

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basedon( D )ande1<0in( D ),andu+b+e2())]TJ ET BT /F3 11.955 Tf 330.97 -11.96 Td[(e02 ()= 2>0basedon( D ).Wehave[u+b+e2())]TJ ET BT /F5 7.97 Tf 153.06 -28.17 Td[(e02 ()= 2] 2= 1)]TJ /F12 5.978 Tf 5.76 0 Td[(1 (u+a+e1= 2) 2= 1)]TJ /F12 5.978 Tf 5.76 0 Td[(11 (u+a)]TJ /F5 7.97 Tf 6.58 0 Td[(e1= 1)1 (u+a+e1= 2)>0,andso@~g((u);u) @uandg1((u);u)havethesamesign.Thus,inordertoprove@~g((u);u) @u<0,itissufcenttoproveg1((u);u)<0.Inthethirdpartofthisproof,weproveg1((u);u)<0.Afterrearrangementof( D ),wehaveg1((u);u)= 0 (u)[ e02 ()+ e002 ()= 1][u+b+e2())]TJ ET BT /F3 11.955 Tf 312.02 -203.22 Td[(e02 ()= 2]+ 0 (u) 2 1[ e02 ())]TJ ET BT /F3 11.955 Tf 199.68 -230.12 Td[(e002 ()= 2][u+b+e2()+ e02 ()= 1] (D))]TJ /F4 11.955 Tf 11.96 0 Td[([b)]TJ /F3 11.955 Tf 11.96 0 Td[(a+e2()+[ e02 ()+e1]= 1][u+b+e2())]TJ ET BT /F3 11.955 Tf 370.62 -253.06 Td[(e02 ()= 2] (u+a)]TJ /F3 11.955 Tf 11.96 0 Td[(e1= 1))]TJ ET BT /F3 11.955 Tf 112.4 -285.29 Td[( 2 1[b)]TJ /F3 11.955 Tf 11.95 0 Td[(a+e2())]TJ /F4 11.955 Tf 11.96 0 Td[([ e02 ()+e1]= 2][u+b+e2()+ e02 ()= 1] (u+a+e1= 2):Notethatthesumofthersttwotermsoftherightsideof( D )equalstothefollowingterm 0 (u)[ e02 ()+ e002 ()= 1]f1[u+b+e2())]TJ ET BT /F3 11.955 Tf 246.91 -412.92 Td[(e02 ()= 2]+[ e02 ())]TJ ET BT /F3 11.955 Tf 359.4 -412.92 Td[(e002 ()= 2]g1 1 (D)+ 0 (u) 2 1[ e02 ())]TJ ET BT /F3 11.955 Tf 103.09 -470.5 Td[(e002 ()= 2]1[u+b+e2()+ e02 ()= 1])]TJ ET BT /F3 11.955 Tf 333.56 -462.41 Td[( 1 2[ e02 ()+ e002 ()= 1]1 1= 0 (u)[ e02 ()+ e002 ()= 1]f1[u+b+e2())]TJ ET BT /F3 11.955 Tf 250.7 -502.58 Td[(e02 ()= 2]+[ e02 ())]TJ ET BT /F3 11.955 Tf 363.19 -502.58 Td[(e002 ()= 2]g1 1+ 0 (u) 2 1[ e02 ())]TJ ET BT /F3 11.955 Tf 103.09 -533.29 Td[(e002 ()= 2]f1[u+b+e2())]TJ ET BT /F3 11.955 Tf 263.52 -533.29 Td[(e02 ()= 2]+[ e02 ())]TJ ET BT /F3 11.955 Tf 376.01 -533.29 Td[(e002 ()= 2]g1 1wheretherstequalityholdsbecause 276

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[ e02 ()+ e002 ()= 1][ e02 ())]TJ ET BT /F3 11.955 Tf 247.78 -29.69 Td[(e002 ()= 2])]TJ /F4 11.955 Tf 11.29 0 Td[(( 2= 1)[ e02 ())]TJ ET BT /F3 11.955 Tf 186.95 -53.59 Td[(e002 ()= 2]( 1= 2)[ e02 ()+ e002 ()= 1]=0:Substituting( D )into( D )andnotingthatf1[u+b+e2())]TJ ET BT /F3 11.955 Tf 202.19 -148.77 Td[(e02 ()= 2]+[ e02 ())]TJ ET BT /F3 11.955 Tf 314.68 -148.77 Td[(e002 ()= 2]gisthedenominatorof( D ),wehave( D )(whichisthesumofthersttwotermsoftherightsideof( D ))equals=b)]TJ /F3 11.955 Tf 11.96 0 Td[(a+e2())]TJ /F4 11.955 Tf 11.95 0 Td[([ e02 ()+e1]= 2 (u+a+e1= 2)[ e02 ()+ e002 ()= 1]+ 2 1[ e02 ())]TJ ET BT /F3 11.955 Tf 374.37 -262.84 Td[(e002 ()= 2]1 1= 2 1[b)]TJ /F3 11.955 Tf 11.96 0 Td[(a+e2())]TJ /F4 11.955 Tf 11.95 0 Td[([ e02 ()+e1]= 2][(1= 2+1= 1) e02 ()]=(u+a+e1= 2): (D)Thus,from( D )and( D ),wehave g1((u);u)= 2 1[b)]TJ /F3 11.955 Tf 11.95 0 Td[(a+e2())]TJ /F4 11.955 Tf 11.96 0 Td[([ e02 ()+e1]= 2][(1= 2+1= 1) e02 ()]=(u+a+e1= 2))]TJ /F4 11.955 Tf 11.29 0 Td[([b)]TJ /F3 11.955 Tf 11.95 0 Td[(a+e2()+[ e02 ()+e1]= 1][u+b+e2())]TJ ET BT /F3 11.955 Tf 300.13 -432.19 Td[(e02 ()= 2]=(u+a)]TJ /F3 11.955 Tf 11.95 0 Td[(e1= 1))]TJ ET BT /F5 7.97 Tf 41.11 -451.28 Td[( 2 1[b)]TJ /F3 11.955 Tf 11.95 0 Td[(a+e2())]TJ /F4 11.955 Tf 11.95 0 Td[([ e02 ()+e1]= 2][u+b+e2()+ e02 ()= 1]=(u+a+e1= 2):(D)Finally,byaddingtherstandthelasttermof( D ),wehave 277

PAGE 278

g1((u);u)=)]TJ ET BT /F5 7.97 Tf 43.64 -60.74 Td[( 2 1[b)]TJ /F3 11.955 Tf 11.96 0 Td[(a+e2())]TJ /F4 11.955 Tf 11.96 0 Td[([ e02 ()+e1]= 2]f[u+b+e2()+ e02 ()= 1])]TJ /F4 11.955 Tf 11.96 0 Td[((1= 2+1= 1) e02 ()g=(u+a+e1= 2))]TJ /F4 11.955 Tf 11.3 0 Td[([b)]TJ /F3 11.955 Tf 11.96 0 Td[(a+e2()+[ e02 ()+e1]= 1][u+b+e2())]TJ ET BT /F3 11.955 Tf 302.66 -113.37 Td[(e02 ()= 2]=(u+a)]TJ /F3 11.955 Tf 11.96 0 Td[(e1= 1)=)]TJ ET BT /F5 7.97 Tf 43.64 -132.46 Td[( 2 1[b)]TJ /F3 11.955 Tf 11.96 0 Td[(a+e2())]TJ /F4 11.955 Tf 11.96 0 Td[([ e02 ()+e1]= 2][u+b+e2())]TJ ET BT /F3 11.955 Tf 314.33 -137.27 Td[(e02 ()= 2]=(u+a+e1= 2))]TJ /F4 11.955 Tf 11.3 0 Td[([b)]TJ /F3 11.955 Tf 11.96 0 Td[(a+e2()+[ e02 ()+e1]= 1][u+b+e2())]TJ ET BT /F3 11.955 Tf 302.66 -161.18 Td[(e02 ()= 2]=(u+a)]TJ /F3 11.955 Tf 11.96 0 Td[(e1= 1)=)]TJ /F4 11.955 Tf 11.3 0 Td[([u+b+e2())]TJ ET BT /F3 11.955 Tf 131.55 -185.09 Td[(e02 ()= 2] 2 1[b)]TJ /F5 7.97 Tf 6.58 0 Td[(a+e2())]TJ /F4 11.955 Tf 6.59 -.99 Td[([ e02 ()+e1]= 2] (u+a+e1= 2)+[b)]TJ /F5 7.97 Tf 6.59 0 Td[(a+e2()+[ e02 ()+e1]= 1] (u+a)]TJ /F5 7.97 Tf 6.59 0 Td[(e1= 1)=)]TJ /F4 11.955 Tf 11.3 0 Td[([u+b+e2())]TJ ET BT /F3 11.955 Tf 131.55 -237.69 Td[(e02 ()= 2]=[(u+a+e1= 2)(u+a)]TJ /F3 11.955 Tf 11.96 0 Td[(e1= 1)]g2(;u);(D)where g2((u);u):= 2 1[b)]TJ /F3 11.955 Tf 11.95 0 Td[(a+e2())]TJ /F4 11.955 Tf 11.95 0 Td[([ e02 ()+e1]= 2](u+a)]TJ /F3 11.955 Tf 11.96 0 Td[(e1= 1)+[b)]TJ /F3 11.955 Tf 11.95 0 Td[(a+e2()+[ e02 ()+e1]= 1](u+a+e1= 2)= 2 1f[b)]TJ /F3 11.955 Tf 11.96 0 Td[(a+e2()](u+a))]TJ /F4 11.955 Tf 11.95 0 Td[([b)]TJ /F3 11.955 Tf 11.95 0 Td[(a+e2()]e1= 1+[ e02 ()+e1]e1=( 1 2)g+[b)]TJ /F3 11.955 Tf 11.95 0 Td[(a+e2()](u+a)+[b)]TJ /F3 11.955 Tf 11.96 0 Td[(a+e2()]e1= 2+[ e02 ()+e1]e1=( 1 2)=1+ 2 1[b)]TJ /F3 11.955 Tf 11.95 0 Td[(a+e2()](u+a)+[b)]TJ /F3 11.955 Tf 11.95 0 Td[(a+e2()]e11)]TJ /F4 11.955 Tf 11.95 0 Td[(( 2= 1)2= 2+1+ 2 1[ e02 ()+e1]e1=( 1 2)=1+ 2 1g3(;u);(D)wherethesecondequalityholdsbecause)]TJ /F4 11.955 Tf 11.29 0 Td[(( 2= 1)[ e02 ()+e1](u+a)= 2+[ e02 ()+e1]= 1(u+a)=0;andthelastequalityholdsbecause 278

PAGE 279

g3((u);u):=[b)]TJ /F3 11.955 Tf 11.95 0 Td[(a+e2()](u+a)+[b)]TJ /F3 11.955 Tf 11.96 0 Td[(a+e2()]e1[1)]TJ /F4 11.955 Tf 11.96 0 Td[(( 2= 1)]= 2+[ e02 ()+e1]e1=( 1 2)=[b)]TJ /F3 11.955 Tf 11.95 0 Td[(a+e2()](u+a)+[b)]TJ /F3 11.955 Tf 11.95 0 Td[(a+e2()]e1= 2)]TJ /F4 11.955 Tf 11.29 0 Td[([b)]TJ /F3 11.955 Tf 11.95 0 Td[(a+e2()]e1= 1+[ e02 ()+e1]e1=( 1 2)=[b)]TJ /F3 11.955 Tf 11.95 0 Td[(a+e2()](u+a+e1= 2))-222(fb)]TJ /F3 11.955 Tf 11.95 0 Td[(a+e2())]TJ /F4 11.955 Tf 11.96 0 Td[([ e02 ()+e1]= 2ge1= 1:(D)Fromu+a)]TJ /F3 11.955 Tf 10.68 0 Td[(e1= 1>u+a+e1= 2>0basedon( D )andu+b+e2())]TJ ET BT /F3 11.955 Tf 398.22 -178.08 Td[(e02 ()= 2>0basedon( D ),wehave)]TJ /F4 11.955 Tf 11.29 0 Td[([u+b+e2())]TJ ET BT /F3 11.955 Tf 157.48 -249.81 Td[(e02 ()= 2]=[(u+a+e1= 2)(u+a)]TJ /F3 11.955 Tf 11.95 0 Td[(e1= 1)]<0:Thus,g1(;u)hasdifferentsignwithg2((u);u)andg3((u);u)basedon( D )and( D ).Hence,inordertoproveg1((u);u)<0,itsufcestoproveg3((u);u)>0.Intheremainingpartofthisproof,weshowg3((u);u)>0.Werstproveaninequality.From( D )andthefactthate)]TJ /F5 7.97 Tf 6.59 0 Td[(x>1)]TJ /F3 11.955 Tf 11.95 0 Td[(xforx>0,wehave[u+b+e2())]TJ ET BT /F3 11.955 Tf 100.04 -447.55 Td[(e02 ()= 2]( D\000137 )=(u+a+e1= 2)e)]TJ ET BT /F5 7.97 Tf 290.09 -442.61 Td[( 1>(u+a+e1= 2)(1)]TJ ET BT /F3 11.955 Tf 426.19 -447.55 Td[( 1):Movingu+a+e1= 2totheleftside,wehaveb)]TJ /F3 11.955 Tf 11.96 0 Td[(a+e2())]TJ /F4 11.955 Tf 11.96 0 Td[([ e02 ()+e1]= 2>)]TJ /F4 11.955 Tf 11.29 0 Td[((u+a+e1= 2) 1:Sincee1<0basedon( D ),wehave )-222(fb)]TJ /F3 11.955 Tf 11.96 0 Td[(a+e2())]TJ /F4 11.955 Tf 11.96 0 Td[([ e02 ()+e1]= 2ge1= 1>(u+a+e1= 2)e1:(D) 279

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Nowwestarttoproveg3((u);u)>0.From( D )and( D ),wehave g3((u);u)>[b)]TJ /F3 11.955 Tf 11.95 0 Td[(a+e2()+e1](u+a+e1= 2):(D)Thus,inordertoproveg3((u);u)>0,itsufcestoprove[b)]TJ /F3 11.955 Tf 11.95 0 Td[(a+e2()+e1](u+a+e1= 2)>0:Sinceu+a+e1= 2>0basedon( D ),weonlyneedtoproveb)]TJ /F3 11.955 Tf 11.22 0 Td[(a+e2()+e1>0.From( D ),( D ),and( D ),wehaveb)]TJ /F3 11.955 Tf 11.96 0 Td[(a+e2()+e1=)]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F4 11.955 Tf 11.95 0 Td[((c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d)]TJ /F3 11.955 Tf 11.96 0 Td[(w(x2;1))]TJ /F3 11.955 Tf 11.96 0 Td[(c1x2;1+g1(S1)+w(x2;1)]TJ /F3 11.955 Tf 11.96 0 Td[()+c2(x2;1)]TJ /F3 11.955 Tf 11.96 0 Td[()+[ w0 (x2;1)+c1]=)]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2+g1(S1))]TJ /F4 11.955 Tf 11.96 0 Td[([w(x2;1)+c1x2;1)]TJ /F3 11.955 Tf 11.96 0 Td[(w(x2;1)]TJ /F3 11.955 Tf 11.96 0 Td[()])]TJ /F4 11.955 Tf 11.95 0 Td[((c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d+c2(x2;1)]TJ /F3 11.955 Tf 11.95 0 Td[()+[ w0 (x2;1)+c1]=)]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2+g1(S1))]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)(x2;1)]TJ /F3 11.955 Tf 11.95 0 Td[(+d))]TJ /F4 11.955 Tf 11.95 0 Td[([w(x2;1)+c1x2;1)]TJ /F3 11.955 Tf 11.95 0 Td[(w(x2;1)]TJ /F3 11.955 Tf 11.96 0 Td[()+c1(x2;1)]TJ /F3 11.955 Tf 11.96 0 Td[()]+[ w0 (x2;1)+c1]=(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)(x2;1+d))]TJ /F4 11.955 Tf 11.95 0 Td[((c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)(x2;1)]TJ /F3 11.955 Tf 11.96 0 Td[(+d)+[ w0 (x2;1)+c1])]TJ /F3 11.955 Tf 11.96 0 Td[([ w0 (2;1)+c1]; (D)where2;12[x2;1)]TJ /F3 11.955 Tf 11.96 0 Td[(;x2;1]andthelastequalityholdsbecauseof( 5 ).Finally,from( D ),wehave b)]TJ /F3 11.955 Tf 11.95 0 Td[(a+e2()+e1=(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)+ w00 (2;1)(x2;1)]TJ /F3 11.955 Tf 11.96 0 Td[(2;1)>0;(D)where2;12[2;1;x2;1],theinequalityholdsbecauseof>0basedon( D )and w00 (x)0basedon( 5 ). 280

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From( D ),( D ),and( D ),wehaveg3((u);u)>0andthisnishestheproof. D.19ProofofLemma 5.10 Proof. Westateanequivalentexpressionsof( 5 ),whichwewilluselater.g(z)=fw(z)+c1z)]TJ /F3 11.955 Tf 11.96 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(c0+[ w0 (z)+c1]= 2ge 2z)]TJ /F4 11.955 Tf 11.96 0 Td[((1= 1+1= 2)[ w0 (z)+c1]e 2z)]TJ /F9 11.955 Tf 11.95 9.68 Td[(w((z))+c2(z))]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+[ w0 ((z))+c2]= 2e 2(z) (D)+(1= 1+1= 2)[ w0 ((z))+c2]e 2(z)=[f1(z))]TJ /F3 11.955 Tf 11.95 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(c0]e 2z)]TJ /F4 11.955 Tf 11.96 0 Td[((1= 1+1= 2)[ w0 (z)+c1]e 2z)]TJ /F9 11.955 Tf 11.95 9.69 Td[(f2((z)))]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.67 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)de 2(z)+(1= 1+1= 2)[ w0 ((z))+c2]e 2(z);wherethelastequalityholdsbecauseof( 5 ).Weprovethetheoreminthreecases.UnderCase1.0whereK1=K2,weproveg(s1)=0.UnderCase1.1whereK1>K2ands2x2;1,weproveg(s1)>0.UnderCase1.2whereK1>K2andx2;10withzr2(x2;1;s1]denedsuchthat(zr)=x2;1.ForCase1.1,from( 5 )and( 5 ),wehavethatwhenz=s1,f1(z))]TJ /F3 11.955 Tf 12.26 0 Td[(g1(S1))]TJ /F3 11.955 Tf -447.97 -23.91 Td[(c0=f1(s1))]TJ /F3 11.955 Tf 12.42 0 Td[(f1(S1))]TJ /F3 11.955 Tf 12.43 0 Td[(c0( 5\00070 )=0.From( 5 ),wehavethatf2(s2))]TJ /F4 11.955 Tf 12.84 0 Td[(g2)]TJ /F4 11.955 Tf 7.47 -6.67 Td[(S2)]TJ /F3 11.955 Tf 12.43 0 Td[(c0)]TJ /F4 11.955 Tf -442.21 -23.91 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d=0andso(s1)=s2,becauseof( 5 ),( D ),( D ),and( D ).NoteunderCase1.1wehaves2x2;1.Thus,wehave(s1)=s2x2;1.Thus,whenz=s1,from( D )andtheabovediscussion,wehave 281

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g(s1)=(1= 1+1= 2)[ w0 (s2)+c2]e 2s2)]TJ /F4 11.955 Tf 11.95 0 Td[([ w0 (s1)+c1]e 2s1 (D)( 5\00054 )=(1= 1+1= 2)[ w0 (s2)+c2]e 2s2+ 2K1( 5\00094 )>(1= 1+1= 2)[ 2K1)]TJ ET BT /F3 11.955 Tf 234.6 -102.66 Td[( 2K1]=0:ForCase1.0,wehave[ w0 (s2)+c2]e 2s2=[ w0 (s)+c2]e 2s2=)]TJ /F3 11.955 Tf 9.3 0 Td[(K2=)]TJ /F3 11.955 Tf 9.3 0 Td[(K1basedonfromK1=K2and( 5 ).Sosimilarlywith( D ),wehaveg(s1)=0.ForCase1.2,x2;10basedon( D ),wehavethatthereexistaunique zr0.From( 5 ),wehave g(zr)=fw(zr)+c1zr)]TJ /F3 11.955 Tf 11.96 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[([ w0 (zr)+c1]= 1ge 2zr)]TJ /F9 11.955 Tf 11.29 9.69 Td[(w(x2;1)+c2x2;1)]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.67 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d)]TJ /F4 11.955 Tf 11.96 0 Td[([ w0 (x2;1)+c2]= 1e 2x2;1:(D)From( 5 ),( D )and( D ),weknowthat(zr)isdecidedbyfw(zr)+c1zr)]TJ /F3 11.955 Tf 11.96 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(c0+[w0(zr)+c1]= 2ge)]TJ ET BT /F5 7.97 Tf 308.53 -488.49 Td[( 1zr (D)=w(x2;1)+c2x2;1)]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.67 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+[ w0 (x2;1)+c2]= 2e)]TJ ET BT /F5 7.97 Tf 402.26 -515.39 Td[( 1x2;1:Notethatbasedon( 5 ),wehavethat( D )isequivalentto [f1(zr))]TJ /F3 11.955 Tf 11.96 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(c0]e)]TJ ET BT /F5 7.97 Tf 143.23 -611.02 Td[( 1zr=f2(x2;1))]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.95 0 Td[((c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)de)]TJ ET BT /F5 7.97 Tf 380.36 -611.02 Td[( 1x2;1:(D) 282

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Intherstpartofthisproof,wereformulateandsimplifytheproblem.Let u=)]TJ /F3 11.955 Tf 9.3 0 Td[(c0;b=w(x2;1)+c2x2;1)]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d;a=)]TJ /F3 11.955 Tf 9.3 0 Td[(g1(S1);=zr)]TJ /F3 11.955 Tf 11.95 0 Td[((zr)=zr)]TJ /F3 11.955 Tf 11.95 0 Td[(x2;1;e1()=w(x2;1+)+c1(x2;1+);ande2= w0 (x2;1)+c2:(D)Notethatusingnewnotations,( D )becomes [f1(x2;1+))]TJ /F3 11.955 Tf 11.96 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(c0]e)]TJ ET BT /F5 7.97 Tf 177.75 -179.62 Td[( 1=f2(x2;1))]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.67 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d;(D)whichisequivalentto [u+a+e1()+ e01 ()= 2]e)]TJ ET BT /F5 7.97 Tf 256.36 -263.3 Td[( 1=[u+b+e2= 2]:(D)And( D)-222()]TJ /F4 11.955 Tf 21.26 0 Td[(171 )becomes g(zr)=[u+a+e1())]TJ ET BT /F3 11.955 Tf 251.17 -340.26 Td[(e01 ()= 1]e 2x2;1e 2)-167(fu+b)]TJ /F3 11.955 Tf 11.95 0 Td[(e2= 1ge 2x2;1:(D)WefollowthesimilarlogicwiththatintheproofofLemma D.18 .Notethatforagiven^u>0,whichis^c0>0,wehaveunique^S1,^S2,^x2;1,and^zr=(^x2;1)decidedbyProposition 5.1 ,( 5 ),( 5 ),and( D )repectively.Thecorresponding^a,^b,^e2,^,and^e1^satisfy( D ).Wewanttoproveg(^zr)=g^u;^a;^b;^e2;^;^e1^>0.Weproveinthisway.Fixa=^a;b=^b,e2=^e2,x2;1=^x2;1,andzr=^zr.Onlylet(u)tochangewithuinu^uande1((u))tochangewith(u)suchthat( D )satised.Ifgu;^a;^b;^e2;(u);e1((u))>0forallu^u,thencertainlywehaveg^u;^a;^b;^e2;^;^e1^>0.Inthefollowing,fornotationsimplicity,wewritegu;^a;^b;^e2;(u);e1((u))asg((u);u),(u)as,ande1((u))ase1().Fornotationsimplicity,wewrite^a,^b,and^e2asa,b,ande2,butrememberthattheyaredecidedby^uanddonotchangewithu. 283

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Next,westudythesignsofseveralterms:(i)u+b+e2= 2>0,(ii)foranyu^u,thereexistsaunique(u)2(0;s2)]TJ /F3 11.955 Tf 11.95 0 Td[(x2;1)suchthat D ,whichisequivalentto( D )satised,(iii)u+a+e1()+ e01 ()= 2>0,(vi)e2<0,and(v) e01 ()<0.For(i),sincex2;1f2(s2))]TJ /F4 11.955 Tf 12.13 0 Td[(g2)]TJ /F4 11.955 Tf 7.47 -6.67 Td[(S2)]TJ /F4 11.955 Tf 11.96 0 Td[(^c0)]TJ /F4 11.955 Tf 11.72 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d( 5\00092 )=0:(D)Foru^u,wehaveu+b+e2= 2^u+b+e2= 2 (D)( D\000135 )=w(x2;1)+c2x2;1)]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.67 Td[(S2)]TJ /F4 11.955 Tf 12.2 0 Td[(^c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+[ w0 (x2;1)+c2]= 2( 5\00045 )=f2(x2;1))]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F4 11.955 Tf 12.2 0 Td[(^c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d( D\000178 )>0:For(ii),werstdene ~G():=[f1(x2;1+))]TJ /F3 11.955 Tf 11.96 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(c0]e)]TJ ET BT /F5 7.97 Tf 325.36 -435.43 Td[( 1:(D)Inthefollowing,weprovethereexistsaunique 2(0;s1)]TJ /F3 11.955 Tf 11.96 0 Td[(x2;1)(D)suchthat( D )satisedbyproving(a)~G(0)>f2(x2;1))]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d;(b)~G(s1)]TJ /F3 11.955 Tf 11.96 0 Td[(x2;1)
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~G(0)=[f1(x2;1))]TJ /F3 11.955 Tf 11.95 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(c0]( D\000110 )>f2(x2;1))]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.48 -6.67 Td[(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(c0)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d:For(b),from( D ),wehave~G(s1)]TJ /F3 11.955 Tf 11.96 0 Td[(x2;1)=[f1(s1))]TJ /F3 11.955 Tf 11.95 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(c0]e)]TJ ET BT /F5 7.97 Tf 285.43 -138.52 Td[( 1(s1)]TJ /F5 7.97 Tf 6.59 0 Td[(x2;1)( 5\00068 )=[g1(s1))]TJ /F3 11.955 Tf 11.96 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(c0]e)]TJ ET BT /F5 7.97 Tf 308.62 -165.41 Td[( 1(s1)]TJ /F5 7.97 Tf 6.59 0 Td[(x2;1)( 5\00064 )=0( D\000178 )0for2(0;s1)]TJ /F3 11.955 Tf 11.95 0 Td[(x2;1).Forf01(x2;1+),wehavef01(x2;1+)<0for2(0;s1)]TJ /F3 11.955 Tf 11.95 0 Td[(x2;1)becausef01(x)<0forxs1basedon( 5 )andx2;1+
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~G()( D\000180 )=[f1(x2;1+))]TJ /F3 11.955 Tf 11.96 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(c0]e)]TJ ET BT /F5 7.97 Tf 340.63 -42.88 Td[( 1>[f1(s1))]TJ /F3 11.955 Tf 11.95 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(c0]e)]TJ ET BT /F5 7.97 Tf 280.68 -69.78 Td[( 1( 5\00068 )=[g1(s1))]TJ /F3 11.955 Tf 11.96 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.96 0 Td[(c0]e)]TJ ET BT /F5 7.97 Tf 303.87 -96.68 Td[( 1(s1)]TJ /F5 7.97 Tf 6.59 0 Td[(x2;1)( 5\00064 )=0;wheretheinequalityholdsbecausef01(x)<0forxs1basedon( 5 )andx2;1+0:(D)For(vi),wehave e2= w0 (x2;1)+c2< w0 (s2)+c2<)]TJ ET BT /F3 11.955 Tf 276.86 -379.55 Td[(w00 (s2)= 2( 5\00021 )<0;(D)wheretherstinequalityholdsbecause w00 (x)>0basedon( 5 )andx2;10basedon( 5 )andzr
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Finally,wesimplifytheexpressionofg(zr)in( D ).Beforewedothis,werstgiveanexpressionofe.Dividing(u+b+e2= 2)andtakingpowerof1= 1inbothsidesof( D ),wehave e=[u+a+e1()+ e01 ()= 2]1= 1 [u+b+e2= 2]1= 1:(D)From( D ),wehave g((u);u)( D\000177 )=[u+a+e1())]TJ ET BT /F3 11.955 Tf 244.06 -191.56 Td[(e01 ()= 1]e 2x2;1e 2)-166(fu+b)]TJ /F3 11.955 Tf 11.95 0 Td[(e2= 1ge 2x2;1=e 2x2;1(u+b)]TJ /F3 11.955 Tf 11.95 0 Td[(e2= 1)nu+a+e1())]TJ ET BT /F5 7.97 Tf 294.82 -233.2 Td[(e01 ()= 1 u+b)]TJ /F5 7.97 Tf 6.58 0 Td[(e2= 1e 2)]TJ /F4 11.955 Tf 11.96 0 Td[(1o=e 2x2;1fu+b)]TJ /F3 11.955 Tf 11.96 0 Td[(e2= 1g[~g((u);u))]TJ /F4 11.955 Tf 11.96 0 Td[(1];(D)where ~g((u);u):=u+a+e1())]TJ ET BT /F5 7.97 Tf 147.37 -340.06 Td[(e01 ()= 1 u+b)]TJ /F5 7.97 Tf 6.59 0 Td[(e2= 1e 2( D\000186 )=u+a+e1())]TJ ET BT /F5 7.97 Tf 174.66 -364.92 Td[(e01 ()= 1 u+b)]TJ /F5 7.97 Tf 6.59 0 Td[(e2= 1[u+a+e1()+ e01 ()= 2] 2= 1 (u+b+e2= 2) 2= 1=1+a)]TJ /F5 7.97 Tf 6.58 0 Td[(b+e1())]TJ /F4 11.955 Tf 6.59 -1 Td[([ e01 ())]TJ /F5 7.97 Tf 6.59 0 Td[(e2]= 1 u+b)]TJ /F5 7.97 Tf 6.59 0 Td[(e2= 11+a)]TJ /F5 7.97 Tf 6.59 0 Td[(b+e1()+[ e01 ())]TJ /F5 7.97 Tf 6.59 0 Td[(e2]= 2 u+b+e2= 2 2= 1:(D)Sincee2<0basedon( D ),wehaveu+b)]TJ /F3 11.955 Tf 12.35 0 Td[(e2= 1>u+b+e2= 2( D\000179 )>0.Thus,basedon( D ),wehavethatg((u);u)and~g((u);u))]TJ /F4 11.955 Tf 12.34 0 Td[(1havethesamesign.Inthesecondpartofthisproof,inordertoprovethatg(zr)>0,wewillprovethat~g((u);u)>1byproving(a)limu!+1~g((u);u)=1and(b)@~g @u<0.Werstprovelimu!+1~g((u);u)=1.From( D ),weknowthatx 1)]TJ /F3 11.955 Tf 12.02 0 Td[(x2;1<
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and1+a)]TJ /F3 11.955 Tf 11.95 0 Td[(b+e1())]TJ /F4 11.955 Tf 11.96 0 Td[([ e01 ())]TJ /F3 11.955 Tf 11.96 0 Td[(e2]= 1 u+b)]TJ /F3 11.955 Tf 11.95 0 Td[(e2= 1!1:Thus,wehavelimu!+1~g((u);u)=1:Thenweprove@~g @u<0:Tocompute@~g @u,weneedtocompute 0 (u)rst.Takingderivativetouin( D ),wehave1+[ e01 ()+ e001 ()= 2] 0 (u)=e 1+ 1[u+b+e2= 2]e 1 0 (u):Afterrearrangement,wehave 0 (u)=e 1)]TJ /F4 11.955 Tf 11.95 0 Td[(1 [ e01 ()+ e001 ()= 2])]TJ ET BT /F3 11.955 Tf 254.2 -272.19 Td[( 1[u+b+e2= 2]e 1:(D)Substituting( D )into( D )anddeletinge 1,wehave 0 (u)=nu+a+e1()+ e01 ()= 2 u+b+e2= 2)]TJ /F4 11.955 Tf 11.96 0 Td[(1o1 [ e01 ()+ e001 ()= 2])]TJ ET BT /F5 7.97 Tf 285.8 -353.66 Td[( 1[u+a+e1()+ e01 ()= 2]=a)]TJ /F5 7.97 Tf 6.59 0 Td[(b+e1()+[ e01 ())]TJ /F5 7.97 Tf 6.58 0 Td[(e2]= 2 (u+b+e2= 2)1 [ e01 ()+ e001 ()= 2])]TJ ET BT /F5 7.97 Tf 267.3 -380.08 Td[( 1[u+a+e1()+ e01 ()= 2](D)Nowwecompute@~g @uandproveitisnegative.From( D ),wehave@~g @u=f[ e01 ())]TJ ET BT /F3 11.955 Tf 94.02 -485.31 Td[(e001 ()= 1] 0 (u)(u+b)]TJ /F3 11.955 Tf 11.96 0 Td[(e2= 1))]TJ /F4 11.955 Tf 11.96 0 Td[([a)]TJ /F3 11.955 Tf 11.95 0 Td[(b+e1())]TJ /F4 11.955 Tf 11.96 0 Td[([ e01 ())]TJ /F3 11.955 Tf 11.96 0 Td[(e2]= 1]g1 (u+b)]TJ /F5 7.97 Tf 6.59 0 Td[(e2= 1)21+a)]TJ /F5 7.97 Tf 6.59 0 Td[(b+e1()+[ e01 ())]TJ /F5 7.97 Tf 6.59 0 Td[(e2]= 2 u+b+e2= 2 2= 1+1+a)]TJ /F5 7.97 Tf 6.58 0 Td[(b+e1())]TJ /F4 11.955 Tf 6.59 -1 Td[([ e01 ())]TJ /F5 7.97 Tf 6.59 0 Td[(e2]= 1 u+b)]TJ /F5 7.97 Tf 6.59 0 Td[(e2= 12 11+a)]TJ /F5 7.97 Tf 6.58 0 Td[(b+e1()+[ e01 ())]TJ /F5 7.97 Tf 6.58 0 Td[(e2]= 2 u+b+e2= 2 2= 1)]TJ /F7 7.97 Tf 6.58 0 Td[(11 (u+b+e2= 2)2f[ e01 ()+ e001 ()= 2] 0 (u)(u+b+e2= 2))]TJ /F4 11.955 Tf 11.96 0 Td[([a)]TJ /F3 11.955 Tf 11.96 0 Td[(b+e1()+[ e01 ())]TJ /F3 11.955 Tf 11.96 0 Td[(e2]= 2]g=1+a)]TJ /F5 7.97 Tf 6.59 0 Td[(b+e1()+[ e01 ())]TJ /F5 7.97 Tf 6.58 0 Td[(e2]= 2 u+b+e2= 2 2= 1)]TJ /F7 7.97 Tf 6.58 0 Td[(11 (u+b)]TJ /F5 7.97 Tf 6.59 0 Td[(e2= 1)1 (u+b)]TJ /F5 7.97 Tf 6.59 0 Td[(e2= 1)gr;1((u);u);where 288

PAGE 289

gr;1((u);u)=f[ e01 ())]TJ ET BT /F3 11.955 Tf 77.69 -74.71 Td[(e001 ()= 1] 0 (u)(u+b)]TJ /F3 11.955 Tf 11.95 0 Td[(e2= 1))]TJ /F4 11.955 Tf 11.96 0 Td[([a)]TJ /F3 11.955 Tf 11.96 0 Td[(b+e1())]TJ /F4 11.955 Tf 11.95 0 Td[([ e01 ())]TJ /F3 11.955 Tf 11.95 0 Td[(e2]= 1]g[u+a+e1()+ e01 ()= 2]=(u+b)]TJ /F3 11.955 Tf 11.96 0 Td[(e2= 1) (D)+2 1f[ e01 ()+ e001 ()= 2] 0 (u)(u+b+e2= 2))]TJ /F4 11.955 Tf 11.95 0 Td[([a)]TJ /F3 11.955 Tf 11.95 0 Td[(b+e1()+[ e01 ())]TJ /F3 11.955 Tf 11.96 0 Td[(e2]= 2]g[u+a+e1())]TJ ET BT /F3 11.955 Tf 113.36 -159.22 Td[(e01 ()= 1]=(u+b+e2= 2):Notethat u+b)]TJ /F3 11.955 Tf 11.96 0 Td[(e2= 1( D\000185 )>u+b+e2= 2( D\000179 )>0;(D)andu+a+e1()+ e01 ()= 2( D\000184 )>0.Wehave1+a)]TJ /F3 11.955 Tf 11.96 0 Td[(b+e1()+[ e01 ())]TJ /F3 11.955 Tf 11.95 0 Td[(e2]= 2 u+b+e2= 2 2= 1)]TJ /F7 7.97 Tf 6.58 0 Td[(11 (u+b)]TJ /F3 11.955 Tf 11.96 0 Td[(e2= 1)1 (u+b)]TJ /F3 11.955 Tf 11.95 0 Td[(e2= 1)>0;andso@~g(;u) @uandgr;1(;u)havethesamesign.Thus,inordertoprove@~g((u);u) @u<0,itissufcenttoprovegr;1((u);u)<0.Inthethirdpartofthisproof,weprovegr;1((u);u)<0.Afterrearrangementof( D ),wehavegr;1((u);u)= 0 (u)f[ e01 ())]TJ ET BT /F3 11.955 Tf 180.45 -512.37 Td[(e001 ()= 1][u+a+e1()+ e01 ()= 2]g+ 0 (u)2 1[ e01 ()+ e001 ()= 2][u+a+e1())]TJ ET BT /F3 11.955 Tf 344.22 -539.26 Td[(e01 ()= 1] (D))]TJ /F4 11.955 Tf 11.96 0 Td[([a)]TJ /F3 11.955 Tf 11.96 0 Td[(b+e1())]TJ /F4 11.955 Tf 11.95 0 Td[([ e01 ())]TJ /F3 11.955 Tf 11.95 0 Td[(e2]= 1][u+a+e1()+ e01 ()= 2] (u+b)]TJ /F3 11.955 Tf 11.96 0 Td[(e2= 1))]TJ /F3 11.955 Tf 13.16 8.09 Td[(2 1[a)]TJ /F3 11.955 Tf 11.95 0 Td[(b+e1()+[ e01 ())]TJ /F3 11.955 Tf 11.95 0 Td[(e2]= 2][u+a+e1())]TJ ET BT /F3 11.955 Tf 390.84 -594.44 Td[(e01 ()= 1] (u+b+e2= 2): 289

PAGE 290

Notethatthesumofthersttwotermsoftherightsideof( D )equalstothefollowingterm 0 (u)f[ e01 ())]TJ ET BT /F3 11.955 Tf 104.24 -89.46 Td[(e001 ()= 1]f 1[u+a+e1()+ e01 ()= 2])]TJ /F4 11.955 Tf 11.95 0 Td[([ e01 ()+ e001 ()= 2]g+2 1[ e01 ()+ e001 ()= 2]n 1[u+a+e1())]TJ ET BT /F3 11.955 Tf 257.11 -113.37 Td[(e01 ()= 1]+1 2[ e01 ())]TJ ET BT /F3 11.955 Tf 382.58 -113.37 Td[(e001 ()= 1]oo1 1= 0 (u)f[ e01 ())]TJ ET BT /F3 11.955 Tf 104.24 -137.87 Td[(e001 ()= 1]f 1[u+a+e1()+ e01 ()= 2])]TJ /F4 11.955 Tf 11.95 0 Td[([ e01 ()+ e001 ()= 2]g+2 1[ e01 ()+ e001 ()= 2]f 1[u+a+e1()+ e01 ()= 2])]TJ /F4 11.955 Tf 11.95 0 Td[([ e01 ()+ e001 ()= 2]go1 1(D)wheretheequalityholdsbecause)]TJ /F4 11.955 Tf 11.29 0 Td[([ e01 ())]TJ ET BT /F3 11.955 Tf 186.78 -244.83 Td[(e001 ()= 1][ e01 ()+ e001 ()= 2]+2 1[ e01 ()+ e001 ()= 2]1 2[ e01 ())]TJ ET BT /F3 11.955 Tf 299.32 -268.74 Td[(e001 ()= 1]=0:Substituting( D )into( D )andnotethat[ e01 ()+ e001 ()= 2])]TJ ET BT /F3 11.955 Tf 213.89 -363.91 Td[( 1[u+a+e1()+ e01 ()= 2]isthedenominatorin( D ),wehave( D )(whichisthesumofthersttwotermsoftherightsideof( D ))equals=)]TJ /F3 11.955 Tf 13.15 8.09 Td[(a)]TJ /F3 11.955 Tf 11.95 0 Td[(b+e1()+[ e01 ())]TJ /F3 11.955 Tf 11.95 0 Td[(e2]= 2 (u+b+e2= 2)[ e01 ())]TJ ET BT /F3 11.955 Tf 253.28 -477.98 Td[(e001 ()= 1]+2 1[ e01 ()+ e001 ()= 2]1 1=)]TJ /F3 11.955 Tf 13.15 8.08 Td[(a)]TJ /F3 11.955 Tf 11.95 0 Td[(b+e1()+[ e01 ())]TJ /F3 11.955 Tf 11.95 0 Td[(e2]= 2 (u+b+e2= 2) e01 ()+2 1 e01 ()1 1=)]TJ /F3 11.955 Tf 13.15 8.09 Td[(2 1[a)]TJ /F3 11.955 Tf 11.95 0 Td[(b+e1()+[ e01 ())]TJ /F3 11.955 Tf 11.95 0 Td[(e2]= 2][(1= 2+1= 1) e01 ()]=(u+b+e2= 2): (D)Thus,from( D )and( D ),wehave 290

PAGE 291

gr;1((u);u)=)]TJ /F5 7.97 Tf 10.5 4.81 Td[(2 1[a)]TJ /F3 11.955 Tf 11.95 0 Td[(b+e1()+[ e01 ())]TJ /F3 11.955 Tf 11.95 0 Td[(e2]= 2][(1= 2+1= 1) e01 ()]=(u+b+e2= 2))]TJ /F4 11.955 Tf 11.29 0 Td[([a)]TJ /F3 11.955 Tf 11.95 0 Td[(b+e1())]TJ /F4 11.955 Tf 11.96 0 Td[([ e01 ())]TJ /F3 11.955 Tf 11.96 0 Td[(e2]= 1][u+a+e1()+ e01 ()= 2]=(u+b)]TJ /F3 11.955 Tf 11.96 0 Td[(e2= 1))]TJ /F5 7.97 Tf 10.5 4.82 Td[(2 1[a)]TJ /F3 11.955 Tf 11.95 0 Td[(b+e1()+[ e01 ())]TJ /F3 11.955 Tf 11.95 0 Td[(e2]= 2][u+a+e1())]TJ ET BT /F3 11.955 Tf 312.63 -113.37 Td[(e01 ()= 1]=(u+b+e2= 2):(D)Finally,byaddingtherstandthelasttermof( D ),wehavegr;1((u);u)=)]TJ /F5 7.97 Tf 10.5 4.82 Td[(2 1[a)]TJ /F3 11.955 Tf 11.95 0 Td[(b+e1()+[ e01 ())]TJ /F3 11.955 Tf 11.95 0 Td[(e2]= 2]f(1= 2+1= 1) e01 ()+[u+a+e1())]TJ ET BT /F3 11.955 Tf 242.75 -255.59 Td[(e01 ()= 1]g=(u+b+e2= 2))]TJ /F4 11.955 Tf 11.29 0 Td[([a)]TJ /F3 11.955 Tf 11.95 0 Td[(b+e1())]TJ /F4 11.955 Tf 11.95 0 Td[([ e01 ())]TJ /F3 11.955 Tf 11.95 0 Td[(e2]= 1][u+a+e1()+ e01 ()= 2]=(u+b)]TJ /F3 11.955 Tf 11.96 0 Td[(e2= 1)=)]TJ /F5 7.97 Tf 10.5 4.81 Td[(2 1[a)]TJ /F3 11.955 Tf 11.95 0 Td[(b+e1()+[ e01 ())]TJ /F3 11.955 Tf 11.95 0 Td[(e2]= 2][u+a+e1()+ e01 ()= 2]=(u+b+e2= 2))]TJ /F4 11.955 Tf 11.29 0 Td[([a)]TJ /F3 11.955 Tf 11.95 0 Td[(b+e1())]TJ /F4 11.955 Tf 11.95 0 Td[([ e01 ())]TJ /F3 11.955 Tf 11.95 0 Td[(e2]= 1][u+a+e1()+ e01 ()= 2]=(u+b)]TJ /F3 11.955 Tf 11.96 0 Td[(e2= 1)=)]TJ /F4 11.955 Tf 11.29 0 Td[([u+a+e1()+ e01 ()= 2]n2 1[a)]TJ /F3 11.955 Tf 11.96 0 Td[(b+e1()+[ e01 ())]TJ /F3 11.955 Tf 11.95 0 Td[(e2]= 2]=(u+b+e2= 2)+[a)]TJ /F3 11.955 Tf 11.95 0 Td[(b+e1())]TJ /F4 11.955 Tf 11.96 0 Td[([ e01 ())]TJ /F3 11.955 Tf 11.96 0 Td[(e2]= 1]=(u+b)]TJ /F3 11.955 Tf 11.95 0 Td[(e2= 1)g=)]TJ /F4 11.955 Tf 11.29 0 Td[([u+a+e1()+ e01 ()= 2]=[(u+b+e2= 2)(u+b)]TJ /F3 11.955 Tf 11.95 0 Td[(e2= 1)]gr;2(;u);where 291

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gr;2((u);u):=( 2= 1)[a)]TJ /F3 11.955 Tf 11.95 0 Td[(b+e1()+[ e01 ())]TJ /F3 11.955 Tf 11.95 0 Td[(e2]= 2](u+b)]TJ /F3 11.955 Tf 11.95 0 Td[(e2= 1)+[a)]TJ /F3 11.955 Tf 11.96 0 Td[(b+e1())]TJ /F4 11.955 Tf 11.95 0 Td[([ e01 ())]TJ /F3 11.955 Tf 11.95 0 Td[(e2]= 1](u+b+e2= 2)=( 2= 1)f[a)]TJ /F3 11.955 Tf 11.95 0 Td[(b+e1()](u+b))]TJ /F4 11.955 Tf 11.95 0 Td[([a)]TJ /F3 11.955 Tf 11.96 0 Td[(b+e1()]e2= 1)]TJ /F4 11.955 Tf 11.96 0 Td[([ e01 ())]TJ /F3 11.955 Tf 11.96 0 Td[(e2]e2=( 1 2)g+[a)]TJ /F3 11.955 Tf 11.96 0 Td[(b+e1()](u+b)+[a)]TJ /F3 11.955 Tf 11.95 0 Td[(b+e1()]e2= 2)]TJ /F4 11.955 Tf 11.95 0 Td[([ e01 ())]TJ /F3 11.955 Tf 11.95 0 Td[(e2]e2=( 1 2)=(1+ 2= 1)[a)]TJ /F3 11.955 Tf 11.95 0 Td[(b+e1()](u+b)+[a)]TJ /F3 11.955 Tf 11.96 0 Td[(b+e1()]e21)]TJ /F4 11.955 Tf 11.96 0 Td[(( 2= 1)2= 2)]TJ /F4 11.955 Tf 11.29 0 Td[((1+ 2= 1)[ e01 ())]TJ /F3 11.955 Tf 11.96 0 Td[(e2]e2=( 1 2);=(1+ 2= 1)gr;3(;u);wherethesecondequalityholdsbecause( 2= 1)[ e01 ())]TJ /F3 11.955 Tf 11.95 0 Td[(e2](u+b)= 2)]TJ /F4 11.955 Tf 11.96 0 Td[([ e01 ())]TJ /F3 11.955 Tf 11.96 0 Td[(e2](u+b)= 1=0;andthelastequalityholdsbecause gr;3((u);u):=[a)]TJ /F3 11.955 Tf 11.95 0 Td[(b+e1()](u+b)+[a)]TJ /F3 11.955 Tf 11.96 0 Td[(b+e1()]e2[1)]TJ /F4 11.955 Tf 11.96 0 Td[(( 2= 1)]= 2)]TJ /F4 11.955 Tf 11.29 0 Td[([ e01 ())]TJ /F3 11.955 Tf 11.95 0 Td[(e2]e2=( 1 2)=[a)]TJ /F3 11.955 Tf 11.95 0 Td[(b+e1()](u+b)+[a)]TJ /F3 11.955 Tf 11.96 0 Td[(b+e1()]e2= 2)]TJ /F4 11.955 Tf 11.29 0 Td[([a)]TJ /F3 11.955 Tf 11.95 0 Td[(b+e1()]e2= 1)]TJ /F4 11.955 Tf 11.95 0 Td[([ e01 ())]TJ /F3 11.955 Tf 11.95 0 Td[(e2]e2=( 1 2)=[a)]TJ /F3 11.955 Tf 11.95 0 Td[(b+e1()](u+b+e2= 2))-222(fa)]TJ /F3 11.955 Tf 11.95 0 Td[(b+e1()+[ e01 ())]TJ /F3 11.955 Tf 11.95 0 Td[(e2]= 2ge2= 1:(D)Fromu+b)]TJ /F3 11.955 Tf 10.58 0 Td[(e2= 1>u+b+e2= 2>0basedon( D ),andu+a+e1()+ e01 ()= 2>0basedon( D ),wehave)]TJ /F4 11.955 Tf 11.29 0 Td[([u+a+e1()+ e01 ()= 2]=[(u+b+e2= 2)(u+b)]TJ /F3 11.955 Tf 11.95 0 Td[(e2= 1)]<0:Hence,inordertoprovegr;1((u);u)<0,itsufcestoprovegr;3((u);u)>0. 292

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Intheremainingpartofthisproof,weshowgr;3((u);u)>0:Werstproveaninequality.From( D )andthefactthatex>1+xforx>0,wehave[u+a+e1()+ e01 ()= 2]=[u+b+e2= 2]e 1>[u+b+e2= 2](1+ 1);whichgivesa)]TJ /F3 11.955 Tf 11.96 0 Td[(b+e1()+[ e01 ())]TJ /F3 11.955 Tf 11.96 0 Td[(e2]= 2>[u+b+e2= 2] 1:Sincee2<0basedon( D ),wehave )-222(fa)]TJ /F3 11.955 Tf 11.96 0 Td[(b+e1()+[ e01 ())]TJ /F3 11.955 Tf 11.96 0 Td[(e2]= 2ge2= 1>)]TJ /F4 11.955 Tf 11.29 0 Td[([u+b+e2= 2]e2:(D)Nowwestarttoprovegr;3((u);u)>0.From( D )and( D ),wehavegr;3(;u)>[a)]TJ /F3 11.955 Tf 11.95 0 Td[(b+e1()][u+b+e2= 2])]TJ /F4 11.955 Tf 11.96 0 Td[([u+b+e2= 2]e2 (D)=[a)]TJ /F3 11.955 Tf 11.95 0 Td[(b+e1())]TJ /F3 11.955 Tf 11.96 0 Td[(e2][u+b+e2= 2]:From( D ),wehave 293

PAGE 294

a)]TJ /F3 11.955 Tf 11.95 0 Td[(b+e1())]TJ /F3 11.955 Tf 11.95 0 Td[(e2=)]TJ /F9 11.955 Tf 11.96 9.68 Td[(w(x2;1)+c2x2;1)]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d)]TJ /F3 11.955 Tf 11.95 0 Td[(g1(S1)+w(x2;1+)+c1(x2;1+))]TJ /F3 11.955 Tf 11.96 0 Td[([ w0 (x2;1)+c2]=g2)]TJ /F4 11.955 Tf 7.48 -6.67 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(g1(S1))]TJ /F4 11.955 Tf 11.95 0 Td[([w(x2;1)+c2x2;1)]TJ /F3 11.955 Tf 11.95 0 Td[(w(x2;1+)]+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d+c1(x2;1+))]TJ /F3 11.955 Tf 11.96 0 Td[([ w0 (x2;1)+c2]=g2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(g1(S1)+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)(x2;1+)]TJ /F3 11.955 Tf 11.95 0 Td[(d))]TJ /F4 11.955 Tf 11.96 0 Td[([w(x2;1)+c2x2;1)]TJ /F3 11.955 Tf 11.96 0 Td[(w(x2;1+)+c2(x2;1+)])]TJ /F3 11.955 Tf 11.95 0 Td[([ w0 (x2;1)+c2]=)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)(x2;1+d)+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)(x2;1+)]TJ /F3 11.955 Tf 11.95 0 Td[(d)+[ w0 (2;1)+c2])]TJ /F3 11.955 Tf 11.96 0 Td[([ w0 (x2;1)+c2]; (D)where2;12[x2;1;x2;1+]andthelastequalityholdsbecauseof( 5 ).From( D ),wehave a)]TJ /F3 11.955 Tf 11.95 0 Td[(b+e1())]TJ /F3 11.955 Tf 11.96 0 Td[(e2=(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)+ w00 (2;1)(2;1)]TJ /F3 11.955 Tf 11.96 0 Td[(x2;1)>0;(D)where2;12[x2;1;2;1],theinequalityholdsbecauseofc1>c2,>0basedon( D ),and w00 (x)>0basedon( 5 ).From( D ),( D ),and( D ),wehavegr;3(;u)>0andthisnishestheproof. D.20ProofofLemma 5.11 Proof. ForCase1.1basedon( 5 ),wehaves2x2;1.Soinordertoprovez20.From( 5 ),wehave A=[ 2=( 1+ 2)][f1(z1))]TJ /F3 11.955 Tf 11.95 0 Td[(g1(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(c0]e)]TJ ET BT /F5 7.97 Tf 324.59 -643.06 Td[( 1z1>0;(D) 294

PAGE 295

wheretheinequalityholdsbecausef1(z1))]TJ /F3 11.955 Tf 12.63 0 Td[(g1(S1))]TJ /F3 11.955 Tf 12.64 0 Td[(c0>0basedon( D )andz10( 5\00092 )=f2(s2))]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.95 0 Td[((c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d:Thus,z26=s2.ForCase1.2basedon( 5 ),wehavex2;10forzs1basedon( D )andz1
PAGE 296

f2(z2))]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.47 -6.67 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.95 0 Td[((c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d( 5\000108 )=[( 1+ 2)= 2]Ae 1z2( D\000203 )=0( 5\00092 )=f2(s2))]TJ /F4 11.955 Tf 12.37 0 Td[(g2)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)]TJ /F4 11.955 Tf 11.95 0 Td[((c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d:Thus,wehavez2=s2basedon( 5 )andz2( 5\000121 )=(z1)( 5\000113 )s2( 5\00091 )K1,from( 5 ),wehaveQ1)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S1=)]TJ /F3 11.955 Tf 9.3 0 Td[(K2<)]TJ /F3 11.955 Tf 9.3 0 Td[(K1.From( 5 ),wehave s1
PAGE 297

g1(x))]TJ /F4 11.955 Tf 12.37 0 Td[(g1)]TJ /F4 11.955 Tf 7.47 -6.67 Td[(S1)]TJ /F3 11.955 Tf 11.95 0 Td[(c0=w(x)+c1x)]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 243.06 -42.88 Td[( 2x)]TJ /F9 11.955 Tf 11.96 13.27 Td[(hw)]TJ /F4 11.955 Tf 7.48 -6.67 Td[(S1+c1S1)]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 388.68 -42.88 Td[( 2S1i)]TJ /F3 11.955 Tf 11.96 0 Td[(c0=w(x)+c1x)]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 243.06 -71.37 Td[( 2x)]TJ /F9 11.955 Tf 11.96 13.27 Td[(hw)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S1+c1S1)]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 388.68 -71.37 Td[( 2S1i)]TJ /F3 11.955 Tf 11.96 0 Td[(c0+(K1)]TJ /F3 11.955 Tf 11.96 0 Td[(K2)e)]TJ ET BT /F5 7.97 Tf 231.33 -99.86 Td[( 2x)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ ET BT /F5 7.97 Tf 272.31 -99.86 Td[( 2S1( 5\00062 )=g1(x))]TJ /F3 11.955 Tf 11.95 0 Td[(g1)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S1)]TJ /F3 11.955 Tf 11.95 0 Td[(c0+(K1)]TJ /F3 11.955 Tf 11.96 0 Td[(K2)e)]TJ ET BT /F5 7.97 Tf 358.79 -128.36 Td[( 2x)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ ET BT /F5 7.97 Tf 399.77 -128.36 Td[( 2S1K1andx
PAGE 298

(c)From( 5 )and( 5 ),wehave g01 (x)= w0 (x)+c1+K2 2e)]TJ ET BT /F5 7.97 Tf 233.51 -70.78 Td[( 2x8>>>><>>>>:>0forx0forx>S1:(D)Nowwestarttoprovethemainclaims.(i)Weprovex2;1>s2.Thisclaimholdsbecausec1>c2and(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)(x2;1)]TJ /F3 11.955 Tf 11.95 0 Td[(s2)( D\000207 )=)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c1S1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2S2+G2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S1)]TJ /F3 11.955 Tf 11.95 0 Td[(G2(S2))]TJ /F3 11.955 Tf 11.96 0 Td[(c1s2+c2s2( D\00036 )=)]TJ /F4 11.955 Tf 11.96 0 Td[((c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c1S1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2S2+G2)]TJ /F4 11.955 Tf 7.48 -6.67 Td[(S1)]TJ /F3 11.955 Tf 11.95 0 Td[(G2(S2))]TJ /F3 11.955 Tf 11.96 0 Td[(c1s2+c2S2+[c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d]+[G2(S2))]TJ /F3 11.955 Tf 11.95 0 Td[(G2(s2)]=c0+c1)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S1)]TJ /F3 11.955 Tf 11.96 0 Td[(s2+G2)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S1)]TJ /F3 11.955 Tf 11.95 0 Td[(G2(s2)( 5\00063 )=c0+c1)]TJ /F4 11.955 Tf 7.47 -6.67 Td[(S1)]TJ /F3 11.955 Tf 11.96 0 Td[(s2+w)]TJ /F4 11.955 Tf 7.47 -6.67 Td[(S1)]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 270.67 -349.88 Td[( 2S1)]TJ /F3 11.955 Tf 11.95 0 Td[(w(s2)+K2e)]TJ ET BT /F5 7.97 Tf 375.33 -349.88 Td[( 2s2=c0+c1S1+w)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S1)]TJ /F3 11.955 Tf 11.96 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 232.86 -376.78 Td[( 2S1)]TJ /F9 11.955 Tf 11.95 9.68 Td[(c1s2+w(s2))]TJ /F3 11.955 Tf 11.95 0 Td[(K1e)]TJ ET BT /F5 7.97 Tf 377.13 -376.78 Td[( 2s2+(K1)]TJ /F3 11.955 Tf 11.96 0 Td[(K2)e)]TJ ET BT /F5 7.97 Tf 186.33 -403.67 Td[( 2S1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ ET BT /F5 7.97 Tf 231.84 -403.67 Td[( 2s2( 5\00062 )=c0+g1)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S1)]TJ /F3 11.955 Tf 11.96 0 Td[(g1(s2)+(K1)]TJ /F3 11.955 Tf 11.95 0 Td[(K2)e)]TJ ET BT /F5 7.97 Tf 292.03 -432.17 Td[( 2S1)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ ET BT /F5 7.97 Tf 337.54 -432.17 Td[( 2s2>c0+g1)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S1)]TJ /F3 11.955 Tf 11.96 0 Td[(g1(s2)( D\000206 )c0+g1(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(g1(s2)( D\000208 )c0+g1(S1))]TJ /F3 11.955 Tf 11.95 0 Td[(g1(s1)( 5\00064 )=0;wheretherstinequalityholdsbecauseofK2>K1undercase1ands2
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(ii)Inordertoproves20forx>S1basedon( D )andS1s2,weprove L V(x)=minf L1 V(x); L2 V(x)g>V(x)byproving(i) L1 V(x)>V(x)and(ii) L2 V(x)>V(x)separately.For(i),notingthats2V(x)forx>s2bydiscussingtwocases:(a)s2w(x))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 225.9 -589.22 Td[( 2x=V(x); 299

PAGE 300

wherethesecondandthelastequalitiesholdbecauseof( 5 )andyx>s2under(a),andthethirdequalityholdsbecause)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(w(y))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 286.69 -31.52 Td[( 2y+c0+c1y0= w0 (y)+c1+K2 2e)]TJ ET BT /F5 7.97 Tf 38.26 -55.43 Td[( 2y0forallyunderCase2.1asdescribedin( 5 ).For(a)withs2w(x))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 156.41 -228.58 Td[( 2x=V(x);wherethesecondandthelastequalitiesholdbecauseof( 5 )andyx>s2,thethirdequalityholdsbecause)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(w(y))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 242.17 -315.84 Td[( 2y+c0+c1y0= w0 (y)+c1+K2 2e)]TJ ET BT /F5 7.97 Tf 38.26 -359.88 Td[( 2y8>>>><>>>>:>0fory0fory>S1:basedon( D ),andtheinequalityholdsbecausew)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S1)]TJ /F3 11.955 Tf 10.4 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 71.78 -410.54 Td[( 2S1+c0+c1)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)]TJ /F9 11.955 Tf 10.4 9.68 Td[()]TJ /F3 11.955 Tf 5.48 -9.68 Td[(w(x))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 261.73 -410.54 Td[( 2x( 5\000128 )=g1)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S1)]TJ /F4 11.955 Tf 10.82 0 Td[(g1(x)+c0( 5\000135 )>0fors2w(x))]TJ /F3 11.955 Tf 11.95 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 219.8 -607.26 Td[( 2x=V(x); 300

PAGE 301

wherethesecondandthelastequalitiesholdbecauseof( 5 )andxS1( D\000205 )>s2,andthethirdequalityholdsbecause)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(w(y))]TJ /F3 11.955 Tf 11.95 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 265.44 -36.36 Td[( 2y+c0+c1y0= w0 (y)+c1+K2 2e)]TJ ET BT /F5 7.97 Tf 38.26 -60.27 Td[( 2y0foryxS1basedon( D ).Thus,wehaveproved L1 V(x)>V(x)forx>s2.For(ii),notingthatS2>s2basedon( 5 ),weprove L2 V(x)>V(x)forx>s2bydiscussingtwocases:(c)s2w(x))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 188.18 -319.57 Td[( 2x=V(x);wherethesecondandthelastequalitiesholdbecauseof( 5 )andyx>s2,thefourthequalityholdsbecauseof( 5 )andyx>s2,theinequalityholdsbecauseg2(s2))]TJ /F3 11.955 Tf 11.32 0 Td[(c2x)]TJ /F9 11.955 Tf 11.33 9.68 Td[()]TJ /F3 11.955 Tf 5.48 -9.68 Td[(w(x))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 144.06 -420.15 Td[( 2x( 5\00062 )=g2(s2))]TJ /F3 11.955 Tf 11.33 0 Td[(g2(x)>0fors2w(x))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 188.18 -605.24 Td[( 2x=V(x); 301

PAGE 302

wherethesecondandthelastequalitiesholdbecauseof( 5 )andxS2( 5\00053 )>s2,thefourthequalityholdsbecauseg02(y)0foryS2basedon( 5 ).Thus,wehaveproved L2 V(x)>V(x)forx>s2andsowehave L V(x)=minf L1 V(x); L2 V(x)g>V(x): D.24ProofofProposition 5.17 Proof. Basedon( 5 ),wehave L V(x)=minf L1 V(x); L2 V(x)g,whereL1V(x)isdescribedin( 5 )andL2V(x)isdescribedin( 5 ).Forxs2,weprove L V(x)=V(x)=V(S2)+(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(x)byproving(i) L1 V(x)>V(x)and(ii) L2 V(x)=V(x)=V(S2)+(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(x)forxs2.Notethatforxs2,from( 5 ),wecanwrite L1 V(x)as L1 V(x)=inff L1l V(x); L1r V(x)g;(D)where L1l V(x)=infx
PAGE 303

and L2r V(x)=infys2fV(y)+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x)g:(D)Inrstpartoftheremainingproof,weprove(i) L1 V(x)>V(x)byproving(a) L1l V(x)>V(x)and(b) L1r V(x)>V(x)forxs2.For(a)withxs2,wehave L1l V(x)( D\000211 )=minxys2fV(y)+c0+c1(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x)g=minxys2w(S2))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 200.64 -153.86 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d+c2(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(y)+c0+c1(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x)=w(S2))]TJ /F3 11.955 Tf 11.95 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 141.19 -177.77 Td[( 2S2+c2(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(x)+2c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d=V(x)+c0>V(x);whereandthesecondandthelastequalitiesholdbecauseof( 5 )andxys2.For(b)withxs2,werstproveaninequalityasfollowsw(s2))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 186.9 -337.52 Td[( 2s2+c0+c1(s2)]TJ /F3 11.955 Tf 11.96 0 Td[(x))]TJ /F3 11.955 Tf 11.95 0 Td[(U2(x) (D)( 5\000130 )=w(s2))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 186.9 -364.42 Td[( 2s2+c0+c1(s2)]TJ /F3 11.955 Tf 11.96 0 Td[(x))]TJ /F9 11.955 Tf 11.95 9.68 Td[(w(S2))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 208.18 -391.31 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2S2)]TJ /F3 11.955 Tf 11.96 0 Td[(c2x=w(s2))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 186.9 -418.21 Td[( 2s2+c0+c1s2)]TJ /F4 11.955 Tf 11.95 0 Td[((c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)x)]TJ /F9 11.955 Tf 11.95 9.68 Td[(w(S2))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 208.18 -445.11 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2S2w(s2))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 186.9 -472 Td[( 2s2+c0+c2s2)]TJ /F9 11.955 Tf 11.95 9.69 Td[(w(S2))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 208.18 -498.9 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2S2( 5\00062 )=g2(s2))]TJ /F3 11.955 Tf 11.96 0 Td[(g2(S2))]TJ /F4 11.955 Tf 11.95 0 Td[((c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d( 5\00065 )=c0>0;wheretheinequalityholdsbecausexs2andc1>c2basedonAssumption 5.2 .For(b)withxs2andunderCase2.1,wehave 303

PAGE 304

L1r V(x)( D\000212 )=minys2fV(y)+c0+c1(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x)g( 5\00010 )=minys2w(y))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 264.7 -49.26 Td[( 2y+c0+c1(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x)=w(s2))]TJ /F3 11.955 Tf 11.95 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 199.35 -73.16 Td[( 2s2+c0+c1(s2)]TJ /F3 11.955 Tf 11.95 0 Td[(x)( D\000216 )>U2(x)( 5\000130 )=w(S2))]TJ /F3 11.955 Tf 11.95 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 228.62 -121.03 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2S2)]TJ /F3 11.955 Tf 11.95 0 Td[(c2x( 5\00010 )=V(x);wherethethirdequalityholdsbecause)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(w(y))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 278.05 -172.73 Td[( 2y+c0+c1y0= w0 (y)+c1+K2 2e)]TJ ET BT /F5 7.97 Tf 38.26 -196.64 Td[( 2y0forallyunderCase2.1describedin( 5 ),andthelastinequalityholdsbecauseof( 5 )andxs2.For(b)withxs2andunderCase2.2,wehave L1r V(x)( D\000212 )=minys2fV(y)+c0+c1(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x)g( 5\00010 )=minys2w(y))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 206.29 -321.95 Td[( 2y+c0+c1(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x)=minw(s2))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 169.41 -345.86 Td[( 2s2+c0+c1(s2)]TJ /F3 11.955 Tf 11.96 0 Td[(x);w)]TJ /F4 11.955 Tf 7.48 -6.66 Td[(S1)]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 363.5 -345.86 Td[( 2S1+c0+c1)]TJ /F4 11.955 Tf 7.47 -6.66 Td[(S1)]TJ /F3 11.955 Tf 11.95 0 Td[(x( 5\000129 )=minw(s2))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 197 -369.77 Td[( 2s2+c0+c1(s2)]TJ /F3 11.955 Tf 11.96 0 Td[(x);U1(x)>U2(x)( 5\000130 )=w(S2))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 170.21 -417.58 Td[( 2S2+c2S2+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d)]TJ /F3 11.955 Tf 11.95 0 Td[(c2x( 5\00010 )=V(x);wherethethirdequalityholdsbecause)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(w(y))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 98.01 -545.37 Td[( 2y+c0+c1y0= w0 (y)+c1+K2 2e)]TJ ET BT /F5 7.97 Tf 294.59 -545.37 Td[( 2y8>>>><>>>>:>0fory0fory>S1: 304

PAGE 305

basedon( D ),andtheinequalityholdsbecausew(s2))]TJ /F3 11.955 Tf 11.26 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 349.72 -7.62 Td[( 2s2+c0+c1(s2)]TJ /F3 11.955 Tf 11.95 0 Td[(x)>U2(x)forx0forxs2( 5\000137 )V(x)and(d) L2r V(x)=V(x)=V(S1)+(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(x)forxs2.For(c)withxs2,wehave L2l V(x)( D\000214 )=minxys2fV(y)+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x)g=minxys2w(S2))]TJ /F3 11.955 Tf 11.95 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 242.83 -297.31 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(y)+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d+c2(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x)g=w(S2))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 183.38 -345.12 Td[( 2S2+c2(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(x)+2c0+2(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d=V(x)+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d>V(x);whereandthesecondandthelastequalitiesholdbecauseof( 5 )andxys2.For(d)withxs2,wehave L2r V(x)( D\000215 )=minys2fV(y)+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x)g( 5\00010 )=minys2w(y))]TJ /F3 11.955 Tf 11.95 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 226.23 -523.2 Td[( 2y+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d+c2(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x)( 5\00062 )=minys2fg2(y)+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d)]TJ /F3 11.955 Tf 11.95 0 Td[(c2xg=g2(S2)+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d)]TJ /F3 11.955 Tf 11.96 0 Td[(c2x( 5\00062 )=w(S2))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 185.92 -594.93 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(x)=V(x);(D) 305

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wherethefourthequalityholdsbecauseminxs2g2(x)=g2(S2)basedon( 5 ),andthelastequalityholdsbecauseof( 5 )andxs2.Inaddition,wehave L2r V(x)=w(S2))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 208.22 -114.61 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(x);=V(S2)+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(x)=V(S1)+(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(x) (D)wheretherstequalityholdsbecauseofthelastsecondequationin( D ),thesecondequalityholdsbecauseof( 5 ),S2>s2basedon( 5 ),andthelastequalityfollowsfromthedenitionof(x)in( 5 ),S2)]TJ /F3 11.955 Tf 12.72 0 Td[(xS2)]TJ /F3 11.955 Tf 12.71 0 Td[(s2basedonxs2,andS2)]TJ /F3 11.955 Tf 11.95 0 Td[(s2>dbasedon( 5 ).Thus,wehavethatforxs2, L2 V(x)=minf L2l V(x); L2r V(x)g=V(x)=V(S1)+(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(x):(D)From( 5 ),( D ),and( D ),wehave L V(x)=minf L1 V(x); L2 V(x)g=V(x)forxs2. D.25ProofofProposition 5.19 Proof. Intherstpartofthisproof,weshowAV(x)+h(x)=0forx>s2.From( 5 ),wehaveV(x)=w(x))]TJ /F3 11.955 Tf 11.95 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 153.88 -510.31 Td[( 2xforx>s2.Thus,AV(x)+h(x)= Aw(x))-221(A)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(K2e)]TJ ET BT /F5 7.97 Tf 298.8 -551.95 Td[( 2x+h(x) = Aw(x)+h(x) =0; 306

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wherethesecondequalityholdsbecauseA)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(e)]TJ ET BT /F5 7.97 Tf 251.64 -7.62 Td[( 2x=0basedonLemma D.2 inAppendix D.15 ,andthelastequalityholdsbecauseAw(x)+h(x)=0basedonLemma D.3 inAppendix D.15 .Inthesecondpartofthisproof,weshowAV(x)+h(x)0forxs2.Toobtainourconclusion,werstprovesomepreliminaryresultsthatwewilluselater: w(S2))]TJ /F3 11.955 Tf 9.34 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 73.78 -174.38 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(x))]TJ /F3 11.955 Tf 9.35 0 Td[(w(x)[ w0 (x)+c2]= 2forxs2;(D)whichisequivalenttog2(S2)+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d)]TJ /F3 11.955 Tf 11.96 0 Td[(f2(x)0forxs2;basedon( 5 )and( 5 ).Toreachthis,notingthatg2(S2)+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d)]TJ /F3 11.955 Tf 12.63 0 Td[(f2(s2)( 5\00068 )=g2(S2)+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d)]TJ /F3 11.955 Tf 12.58 0 Td[(g2(s2)( 5\00065 )=0and(g2(S2)+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d)]TJ /F3 11.955 Tf 11.96 0 Td[(f2(x))0=)]TJ ET BT /F3 11.955 Tf 378.15 -341.19 Td[(f02 (x)>0forxs2basedon( 5 ),wehaveg2(S2)+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d)]TJ /F3 11.955 Tf 12.72 0 Td[(f2(x)g2(S2)+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d)]TJ /F3 11.955 Tf 11.95 0 Td[(f2(s2)=0forxs2.Nowweareabletoproveourmainclaim.Forxs2,wehave 307

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AV(x)+h(x)( 5\00010 )=A)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(w(S2))]TJ /F3 11.955 Tf 11.95 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 216.63 -37.3 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2)d+c2(S2)]TJ /F3 11.955 Tf 11.95 0 Td[(x)+h(x)=A)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(w(S2))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 193.27 -61.21 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2(S2)]TJ /F3 11.955 Tf 11.96 0 Td[(x))]TJ /F3 11.955 Tf 11.95 0 Td[(A0w(x)=A)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(w(S2))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 193.27 -85.12 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2S2)]TJ /F3 11.955 Tf 11.96 0 Td[(A0(w(x)+c2x)( 5\00013 )=)]TJ /F3 11.955 Tf 9.29 0 Td[(r)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(w(S2))]TJ /F3 11.955 Tf 11.96 0 Td[(K2e)]TJ ET BT /F5 7.97 Tf 221.98 -109.03 Td[( 2S2+c0+(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)d+c2S2+r(w(x)+c2x))]TJ /F3 11.955 Tf 9.3 0 Td[(( w0 (x)+c2))]TJ /F4 11.955 Tf 11.96 0 Td[((2=2) w00 (x);( D\000221 ))]TJ /F3 11.955 Tf 37.09 0 Td[(r[ w0 (x)+c2]= 2)]TJ /F3 11.955 Tf 11.95 0 Td[(( w0 (x)+c2))]TJ /F4 11.955 Tf 11.95 0 Td[((2=2) w00 (x)=[ w0 (x)+c2]()]TJ /F3 11.955 Tf 9.29 0 Td[(r= 2)]TJ /F3 11.955 Tf 11.95 0 Td[())]TJ /F4 11.955 Tf 11.96 0 Td[((2=2) w00 (x)=)]TJ /F4 11.955 Tf 11.29 0 Td[((2=2)[ 2( w0 (x)+c2)+ w00 (x)]( 5\00050 )=)]TJ /F4 11.955 Tf 11.29 0 Td[((2=2) 2 f02 (x)0;wherethesecondequalityholdsbecauseAw(x)+h(x)=0basedonLemma D.3 ,thesixthequalityholdsbecauseof)]TJ /F3 11.955 Tf 9.3 0 Td[(r)]TJ /F3 11.955 Tf 12.1 0 Td[( 2+2 22=2=0inLemma D.1 ,thelastinequalityholdsbecauseof f02 (x)<0forxs2basedon( 5 ).Intheremainingpartoftheproof,weshowlimx!V(x)=h(x)<1.From(d)ofAssumption 5.2 and( 5 ),wehavelimx!V(x)=h(x)=limx!()]TJ /F3 11.955 Tf 9.3 0 Td[(c2)= h0 (x)2[0;1=r].From(e)ofAssumption 5.2 ,( 5 ),and( 5 ),wehavelimx!+1V(x)=h(x)=limx!+1w(x)=h(x)<+1. 308

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BIOGRAPHICALSKETCH LaiWeireceivedhisB.S.instatisticsfromPekingUniversity,Beijingin2007.HereceivedhisM.S.inindustrialandoperationsengineeringfromtheUniversityofMichigan,AnnArbor,MIin2009andreceivedhisPh.D.fromindustrialandsystemsengineeringfromtheUniversityofFlorida,Gainesville,FLinthesummerof2015. 315