<%BANNER%>

Data-Driven Stochastic Optimization

MISSING IMAGE

Material Information

Title:
Data-Driven Stochastic Optimization Integrating Reliability with Cost Effectiveness
Physical Description:
1 online resource (132 p.)
Language:
english
Creator:
Jiang, Ruiwei
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Industrial and Systems Engineering
Committee Chair:
Guan, Yongpei
Committee Members:
Smith, Jonathan Cole
Richard, Jean-Philippe P
Hager, William Ward

Subjects

Subjects / Keywords:
cost -- optimization -- reliability -- stochastic
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre:
Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
Decision making processes in practice often involve trade-offs between reliability and cost effectiveness. When a decision maker is faced with an uncertain environment, more often than not she has an ambiguous knowledge about the uncertainty involved, and only a series of historical data samples are available. In such a circumstance, making strong assumptions (e.g., assuming perfect knowledge of the underlying probability distribution) on the uncertainty can lead to suboptimal decisions, in which case either or both of reliability and cost effectiveness can be sacrificed. This dissertation proposes a decision making framework which integrates reliability with cost effectiveness in a data-driven manner. More specifically, we acknowledge the distributional ambiguity of the uncertainty and attempt to capture that by using a nonparametric statistical estimation. Accordingly, we develop risk-averse models for decision making problems under uncertainty that explicitly incorporate distributional ambiguity, and derive their equivalent reformulations which can be readily solved by available tools. Through the reformulations, we can observe how the reliability and cost effectiveness are integrated in a straightforward manner in many cases. Furthermore, we discuss the relationship between the risk-averse models and the available data samples. By performing convergence analysis, we find that the proposed risk-averse models eventually converges to their perfect-knowledge counterparts as the sample size grows to infinity. This observation indicates that the proposed risk-averse models can be applied in a data-driven decision making scheme.
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.
Statement of Responsibility:
by Ruiwei Jiang.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Guan, Yongpei.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-08-31

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0045885:00001

MISSING IMAGE

Material Information

Title:
Data-Driven Stochastic Optimization Integrating Reliability with Cost Effectiveness
Physical Description:
1 online resource (132 p.)
Language:
english
Creator:
Jiang, Ruiwei
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Industrial and Systems Engineering
Committee Chair:
Guan, Yongpei
Committee Members:
Smith, Jonathan Cole
Richard, Jean-Philippe P
Hager, William Ward

Subjects

Subjects / Keywords:
cost -- optimization -- reliability -- stochastic
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre:
Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
Decision making processes in practice often involve trade-offs between reliability and cost effectiveness. When a decision maker is faced with an uncertain environment, more often than not she has an ambiguous knowledge about the uncertainty involved, and only a series of historical data samples are available. In such a circumstance, making strong assumptions (e.g., assuming perfect knowledge of the underlying probability distribution) on the uncertainty can lead to suboptimal decisions, in which case either or both of reliability and cost effectiveness can be sacrificed. This dissertation proposes a decision making framework which integrates reliability with cost effectiveness in a data-driven manner. More specifically, we acknowledge the distributional ambiguity of the uncertainty and attempt to capture that by using a nonparametric statistical estimation. Accordingly, we develop risk-averse models for decision making problems under uncertainty that explicitly incorporate distributional ambiguity, and derive their equivalent reformulations which can be readily solved by available tools. Through the reformulations, we can observe how the reliability and cost effectiveness are integrated in a straightforward manner in many cases. Furthermore, we discuss the relationship between the risk-averse models and the available data samples. By performing convergence analysis, we find that the proposed risk-averse models eventually converges to their perfect-knowledge counterparts as the sample size grows to infinity. This observation indicates that the proposed risk-averse models can be applied in a data-driven decision making scheme.
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.
Statement of Responsibility:
by Ruiwei Jiang.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Guan, Yongpei.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-08-31

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0045885:00001


This item has the following downloads:


Full Text

PAGE 1

DATA-DRIVENSTOCHASTICOPTIMIZATION:INTEGRATINGRELIABILITYWITHCOSTEFFECTIVENESSByRUIWEIJIANGADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

PAGE 2

c2013RuiweiJiang 2

PAGE 3

Tomyparents QingxinYeandJiguangJiang 3

PAGE 4

ACKNOWLEDGMENTS IamdeeplythankfultomyadvisorYongpeiGuanforhisguidanceandfriendshipinthepastfouryears.IthasbeenthegreatestprivilegeIcaneverimaginetoworkwithYongpeiasagraduatestudentandasafriend.Ihavebenetedintellectuallyfromhistastesforgoodresearchandhishonestadvices.Itishispassion,patience,wisdomandsupportthatmakethisdissertationpossible.Myappreciationalsogoestomydissertationcommitteemembers.IwouldliketothankJ.ColeSmithforhisintelligenthumor,remarkableadvice,anddetailedsuggestionsonChapter2ofthisdissertation,Jean-PhilippeRichardforhisexcellentinstructionandilluminatingremarks,andWilliamHagerforhisvaluablecomments.IamalsoindebtedtothegreatmathinstructorsIwasfortunatetolistentoatUF.IwouldliketothankDr.PaulRobinsonforhissuperbteachingandallowingmetoaudithisanalysisseriesforthreesemesters,andDr.AndrewRosalskyforteachingmeprobabilitytheorythatlaysthefoundationofthisdissertation.TheIndustrialandSystemsEngineeringatUFisagreathome.ManythanksgotomyamazinggraduatestudentcolleaguesZhiliZhou,YiqiangSu,QianfanWang,ZhuofeiLi,HongshengXu,SoheilHemmati,Clay&ReneeKoschnick,MikePrince,AndrewRomich,CinthiaPerez,JingMa,ChaoyueZhao,FangHe,LaiWei,LeiFan,KaiPanandmanyothers,whosefriendshipandallthelaughterswehavebeensharingmakethegraduatestudentlifeoneofthegreatestexperienceforme.Inaddition,IwouldliketothankHaipengBi,LiWang,HaiyueYu,WeijunDingandallofmyfriendswhoarenotassociatedwiththisdissertationfortheirfriendshipandencouragementalongtheway.Finally,Iwouldliketoexpressmydeepestgratitudetomyparentswhoencourageandsupportmetochasemyownlifeanddream,andtomywife,SiqianShen,forherincrediblefriendshipandlove. 4

PAGE 5

TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 11 2DATA-DRIVENCHANCECONSTRAINEDSTOCHASTICPROGRAM ..... 15 2.1ProblemDescriptionandLiteratureReview ................. 15 2.1.1MotivationandLiteratureReview ................... 15 2.1.2ModelSettingsandCondenceSets ................. 18 2.2DCCwithMoment-basedCondenceSet .................. 21 2.2.1ConstructionofCondenceSet .................... 22 2.2.2ReformulationoftheSingleDCCs .................. 23 2.2.3ReformulationoftheJointDCCs ................... 31 2.2.4ReformulationoftheWorst-CaseValue-at-RiskConstraint ..... 34 2.3DCCwithDensity-basedCondenceSet .................. 35 2.3.1ConstructionofCondenceSet .................... 35 2.3.2ReformulationofDCC ......................... 37 2.3.3TheValueofData ........................... 43 2.4NumericalExperiments ............................ 45 2.5ConcludingRemarks .............................. 48 3RISK-AVERSETWO-STAGESTOCHASTICPROGRAMWITHDISTRIBUTIONALAMBIGUITY ..................................... 50 3.1MotivationandLiteratureReview ....................... 50 3.2Risk-AverseTwo-StageStochasticProgram ................. 52 3.2.1CondenceSetConstruction ..................... 53 3.2.2EquivalentReformulation ....................... 55 3.2.3ConvergenceAnalysis ......................... 60 3.3SolutionApproaches .............................. 65 3.4ConcludingRemarks .............................. 67 4RISK-AVERSESTOCHASTICUNITCOMMITMENTWITHINCOMPLETEINFORMATION .................................... 69 4.1ProblemDescriptionandLiteratureReview ................. 69 4.2Risk-AverseStochasticUnitCommitmentModels .............. 72 5

PAGE 6

4.2.1NomenclatureandANominalModel ................. 72 4.2.2AChanceConstrainedUCModel ................... 76 4.2.3ATwo-StageUCModelwithRecourse ................ 77 4.2.4CondenceSetConstruction ..................... 79 4.3ReformulationsandSolutionApproaches .................. 81 4.3.1ReformulationofTheDCCandWorst-CaseExpectations ..... 81 4.3.2SolutionApproaches .......................... 86 4.4ConvergenceAnalysis ............................. 90 4.5CaseStudies .................................. 95 4.6ConcludingRemarks .............................. 97 5CONCLUSIONS ................................... 98 APPENDIX ADETAILEDPROOFSFORCHAPTER2 ...................... 99 A.1S-Lemma .................................... 99 A.2ProofofCorollary 2 .............................. 99 A.3ProofofCorollary 3 .............................. 101 A.4ProofofProposition 1 ............................. 106 A.5ProofofProposition 2 ............................. 107 A.6ProofofProposition 4 ............................. 107 A.7ProofofLemma 2 ............................... 107 A.8ProofofCorollary 5 .............................. 108 A.9ProofofProposition 5 ............................. 109 A.10ProofofProposition 6 ............................. 112 A.11ProofofProposition 7 ............................. 113 A.12ProofofProposition 8 ............................. 114 A.13ProofofProposition 9 ............................. 116 A.14ProofofProposition 10 ............................. 117 BDETAILEDPROOFSFORCHAPTER4 ...................... 119 B.1ProofofProposition 12 ............................. 119 B.2ProofofProposition 13 ............................. 120 B.3ProofofProposition 14 ............................. 120 B.4ProofofProposition 17 ............................. 121 B.5ProofofLemma 7 ............................... 122 REFERENCES ....................................... 126 BIOGRAPHICALSKETCH ................................ 132 6

PAGE 7

LISTOFTABLES Table page 2-1Comparisonofaverageendwealthandriskin100replicationsthroughyears2008-2011 ...................................... 47 4-1Comparisonofthetotalcostandrenewableenergyutilizationbetweenmodel(1-SUCI)and(CCUC) ................................ 96 4-2Comparisonofthetotalcostandtheaveragecostinsimulationbetweenmodel(2-SUCI)and(SUC) ................................. 97 7

PAGE 8

LISTOFFIGURES Figure page 2-1Evolutionofvalues1)]TJ /F4 11.955 Tf 12.31 0 Td[(0andVoDagainstsamplesizeunderrisklevel=0.90andcondencelevels=0.01,0.05,0.10 .................. 46 2-2Comparisonofwealthevolutionin100replicationsthroughyears2008-2011 48 4-1Anexampleofusingahistogramestimationofwindenergycapacityfromawindfarm ....................................... 79 8

PAGE 9

AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyDATA-DRIVENSTOCHASTICOPTIMIZATION:INTEGRATINGRELIABILITYWITHCOSTEFFECTIVENESSByRuiweiJiangAugust2013Chair:YongpeiGuanMajor:IndustrialandSystemsEngineering Decisionmakingprocessesinpracticeofteninvolvetrade-offsbetweenreliabilityandcosteffectiveness.Whenadecisionmakerisfacedwithanuncertainenvironment,moreoftenthannotshehasanambiguousknowledgeabouttheuncertaintyinvolved,andonlyaseriesofhistoricaldatasamplesareavailable.Insuchacircumstance,makingstrongassumptions(e.g.,assumingperfectknowledgeoftheunderlyingprobabilitydistribution)ontheuncertaintycanleadtosuboptimaldecisions,inwhichcaseeitherorbothofreliabilityandcosteffectivenesscanbesacriced.Thisdissertationproposesadecisionmakingframeworkwhichintegratesreliabilitywithcosteffectivenessinadata-drivenmanner.Morespecically,weacknowledgethedistributionalambiguityoftheuncertaintyandattempttocapturethatbyusinganonparametricstatisticalestimation.Accordingly,wedeveloprisk-aversemodelsfordecisionmakingproblemsunderuncertaintythatexplicitlyincorporatedistributionalambiguity,andderivetheirequivalentreformulationswhichcanbereadilysolvedbyavailabletools.Throughthereformulations,wecanobservehowthereliabilityandcosteffectivenessareintegratedinastraightforwardmannerinmanycases.Furthermore,wediscusstherelationshipbetweentherisk-aversemodelsandtheavailabledatasamples.Byperformingconvergenceanalysis,wendthattheproposedrisk-aversemodelseventuallyconvergestotheirperfect-knowledgecounterpartsasthesamplesize 9

PAGE 10

growstoinnity.Thisobservationindicatesthattheproposedrisk-aversemodelscanbeappliedinadata-drivendecisionmakingscheme. 10

PAGE 11

CHAPTER1INTRODUCTION Decisionmakingprocessesinpracticetakereliabilityandcosteffectivenessintoconsideration,andofteninvolveinvolvetrade-offsbetweenthem.Forexample,powersystemoperatorswanttokeepthepowersystemreliableandatmeanwhileascost-effectiveaspossiblewhenfacedwithvarioussupplyandloadscenarios,andinvestmentbankersshallalmostalwayskeepaneyeontherobustnessoftheirportfolioswhentheytrymakingrevenueoutofthem.Whenadecisionmakerisfacedwithanuncertainenvironment,moreoftenthannotshehasanambiguousknowledgeabouttheuncertaintyinvolved,andonlyaseriesofhistoricaldatasamplesareavailable.Insuchacircumstance,makingstrongassumptionsontheuncertaintycanleadtosuboptimaldecisions,inwhichcaseeitherorbothofreliabilityandcosteffectivenesscanbesacriced.Forexample,assumingthatrenewableenergyfollowsacertainstaticprobabilitydistributioncanleadtounreliableoperationaldecisionsinapowersystem,inwhichcasewemayfacetransmissionlineoutagesorhavetocurtailcertainamountofrenewableenergy.Foranotherexample,ifaninvestmentbankerassumesthattherateofreturnforherportfoliocanfollowanyprobabilitydistributions,thenshewillprobablybeover-conservativeininvestingandearnlessmoneythanshould.Toaddressthechallengesfromdistributionalambiguityandthetrade-offindecisionmakingunderuncertainty,thisdissertationproposesaframeworkwhichintegratesreliabilitywithcosteffectivenessinadata-drivenmanner.Morespecically,weacknowledgethedistributionalambiguityoftheuncertaintyandattempttocapturethatbyusinganonparametricstatisticalestimation.Forexample,wenolongassumethatwehaveperfectknowledgeoftheunderlyingprobabilitydistribution,andinsteadweconstructacondencesetforitbyusingtheavailabledatasamplesonhand.Accordingly,wedeveloprisk-aversemodelsfordecisionmakingproblemsunderuncertaintythatexplicitlyincorporatedistributionalambiguity,andderivetheir 11

PAGE 12

equivalentreformulationswhichcanbereadilysolvedbyavailabletools.Throughthereformulations,weobservethatwecanexplainhowthereliabilityandcosteffectivenessareintegratedinastraightforwardmannerinmanycases.Furthermore,wediscusstherelationshipbetweentherisk-aversemodelsandtheavailabledatasamples.Byperformingconvergenceanalysis,wendthattheproposedrisk-aversemodelseventuallyconvergestotheirperfect-knowledgecounterpartsasthesamplesizegrowstoinnity.Thisobservationindicatesthattheproposedrisk-aversemodelscanbeappliedinadata-drivendecisionmakingscheme.Inthisdissertation,weunfoldthediscussionasfollows. InChapter 2 ,weinvestigatethedata-drivenchanceconstrainedstochasticprogram.Chanceconstrained programming isaneffectiveandconvenient approachtocontrolriskin decisionmaking underuncertainty However ,dueto unknown probabilitydistributionsofrandomparameters,thesolutionobtainedfromachanceconstrainedoptimizationproblemcanbe biased Inaddition ,insteadofknowingthetruedistributionsofrandomparameters, inpractice ,onlyaseriesofhistoricaldata, whichcanbeconsideredassamplestakenfromthetrue(whileambiguous)distribution, canbeobservedandstored .InChapter 2 wederivestochasticprogramswithdata-drivenchanceconstraints(DCCs)totackletheseproblemsanddevelopequivalentreformulations Foragiven historicaldata set ,weconstructtwotypesofcondencesetsfortheambiguousdistributionthroughnonparametricstatisticalestimationofitsmomentsanddensityfunctions, dependingontheamountofavailabledata .WethenformulateDCCsfromtheperspectiveofrobustfeasibility,byallowingtheambiguousdistributiontorunadverselywithinitscondenceset. After derivingequivalentreformulations, weprovideexactandapproximatesolutionapproachesforstochasticprogramswithDCCsunderbothmoment-basedanddensity-basedcondencesets. Inaddition,we derive therelation ship betweenthe conservatismof 12

PAGE 13

DCCs andthesamplesizeofhistoricaldata,which showsquantitativelywhatwecallthevalueofdata. InChapter 3 ,weinvestigatetherisk-aversetwo-stagestochasticprogram.Two-stagestochasticprogramming(TSP)providesaviableapproachforarisk-neutraldecisionmakertoachievecosteffectivenessinanuncertainenvironment.However,duetoambiguousprobabilitydistributionsofrandomparameters,thesolutionobtainedfromtheTSPmodelcanunderestimatethe`bad'scenarioswhichincurhigher-than-expectationcosts,andsobecomesub-optimalinarisk-aversecircumstance.Inthispaper,wedeveloparisk-aversetwo-stagestochasticprogram(RTSP)modelwhichexplicitlyincorporatesthedistributionalambiguitytoaddressthischallenge.Startingfromasetofhistoricaldatasamples,weconstructacondencesetfortheambiguousprobabilitydistributionthroughnonparametricstatisticalestimationofitsdensityfunction.WethenformulateRTSPfromtheperspectiveofdistributionalrobustnessbyallowingtheambiguousdistributiontorunadverselywithinitscondenceset.Also,wederiveanequivalentreformulationforRTSPandaccordinglyprovideasolutionalgorithmbasedonthesampleaverageapproximationmethod.Inaddition,weperformconvergenceanalysistoshowthattherisk-aversenessofRTSPvanishesasthesamplesizegrowstoinnity,inthesensethattheoptimalobjectivevalueofRTSPconvergetothatofTSP. InChapter 4 ,weapplytherisk-aversemodelsproposedinthepreviouschapterstoapowersystemoperationalproblem.Renewableenergy(suchaswindandsolar)canhelphumanracesdevelopamoresustainablefuture.Thankstoitslowcarbonemissions,wehavebeenseeingrenewableenergyincreasinglypenetratingintomanypowersystems.However,thepenetrationraisesnewchallengesforthepowersystemoperatorstokeepthesystemsreliableandcost-effectivebecausetherenewableenergyisintermittentanddifculttopredict.Inaddition,theinformationabouttherenewableenergyisusuallyincomplete.Thatis,insteadofknowingthetrueprobabilitydistributionofrenewableenergy,onlyasetofhistoricaldatasamplescanbecollected 13

PAGE 14

fromthetrue(whileambiguous)distribution.InChapter 4 ,westudytworisk-aversestochasticunitcommitmentmodelswithincompleteinformation(SUCI),withthersttobeachance-constrainedUCmodelandthesecondtobeatwo-stageUCmodelwithrecourse.Basedonhistoricaldataofrenewableenergy,weconstructacondencesetfortheprobabilitydistributionoftherenewableenergyandproposedata-drivenSUCImodelstohedgeagainsttheinformationincompleteness.Ourmodelsalsoensuresthat,withahighprobability,alargeportionofrenewableenergyisutilized.Furthermore,wedevelopsolutionapproachesforthemodelsbasedonstrongvalidinequalitiesandBenders'decomposition.Inaddition,weshowthattherisk-aversenessofbothmodelsdecreaseaswecollectmoredatasamplesandeventuallyvanishasthesamplesizegoestoinnity.Ourcasestudiesverifytheeffectivenessofourproposedmodelandsolutionapproach. Finally,inChapter 5 ,weconcludethisdissertationanddiscussthefutureresearchdirections. 14

PAGE 15

CHAPTER2DATA-DRIVENCHANCECONSTRAINEDSTOCHASTICPROGRAM 2.1ProblemDescriptionandLiteratureReview 2.1.1MotivationandLiteratureReview Toassistdecisionmakinginuncertainenvironments, signicant researchprogresshasbeenmadeinstochasticoptimizationformulationsandtheirsolutionapproaches.Oneeffectiveandconvenientwayofhandlinguncertainty arising inconstraintparametersemployschanceconstraints.Inachanceconstrainedoptimizationproblem,decisionmakersareinterestedinsatisfyingaconstraint,whichissubjecttouncertainty,byatleastapre-speciedprobabilityatthesmallestcost,minx (x)(2{1a) s.t.PfA()xb()g1)]TJ /F4 11.955 Tf 11.96 0 Td[(,(2{1b) x2X,(2{1c) where :Rn!R often representsaconvexcostfunction,Xrepresentsacomputableboundedconvexset(e.g.,apolytope)inRn,2RKrepresentsaK-dimensionalrandomvectordenedonaprobabilityspace(,F, Pr ),andthesetfunctionPfgrepresentstheprobabilitydistributiononRKinducedby,i.e.,PfCg= Pr f)]TJ /F12 7.97 Tf 6.59 0 Td[(1(C)g,8C2B(RK). Inaddition ,thelinearinequalitysystemA()xb()representstheconstraintstobesatised,whereA()2Rmnandb()2Rmdenotethetechnologymatrixandright-handsidesubjecttouncertainty,andx2Rndenotesthedecisionvariable.Constraint 2b iscalledasinglechanceconstraintwhenm=1(i.e.,thematrixA()reducestoarowvector),andotherwiseitiscalledajointchanceconstraint.Inaddition, value representstherisklevel(ortoleranceofconstraintviolation) allowedby thedecisionmakers,andusuallyischosentobesmall,e.g.,0.10or0.05. Chanceconstraintsemergenaturallyasamodelingtoolinvariousdecisionmakingcircumstances.Forexample,decisionmakersin the nanceindustrymayattempt 15

PAGE 16

toensurethatthereturnoftheirportfoliomeetsatargetvaluewithhighprobability.ThestudyofchanceconstrainedoptimizationproblemshasalonghistorydatingbacktoCharnesetal.[ 18 ],MillerandWagner[ 40 ],andPrekopa[ 50 ].Unfortunately,constraint 2b remain s challengingtohandlebecauseof two key difculties :(i)chanceconstraintsarenon-convexingeneral,and(ii)theprobabilityassociatedwiththechanceconstraints canbehardtocompute sinceitrequiresamulti-dimensionalintegral.Toaddresstherst difculty andrecaptureconvexity,previousresearchidenties important casesunderwhichchanceconstraintsarenonlinearbutconvex(see,e.g.,CharnesandCooper[ 17 ],Prekopa[ 51 ],andCalaoreandElGhaoui[ 16 ]),andprovidesconservativeconvexapproximations(see,e.g.,Pinter[ 48 ],NemirovskiandShapiro[ 42 ],RockafellarandUryasev[ 53 ],andChenetal.[ 19 ]).Toaddressthesecond difculty ,previousresearchproposesscenarioapproximationapproaches(see,e.g.,CalaoreandCampi[ 14 15 ],NemirovskiandShapiro[ 41 ],LuedtkeandAhmed[ 38 ],and Pagnoncellietal.[ 44 ] ),which are computationallytractableandcanguaranteeto obtain asolution satisfying achanceconstraintwithhighprobability. Inaddition, i nteger p rogramming(IP)techniquesaresuccessfullyappliedinexactlysolvingchance-constrainedproblems(see,e.g.,Luedtkeetal.[ 39 ],Kucukyavuz[ 34 ],Luedtke[ 37 ], andLejeune[ 35 ] ). Analternativeof the chanceconstraint approach is the r obust o ptimization(RO)approach(see,e.g.,Soyster[ 62 ],Ben-TalandNemirovski[ 7 ],BertsimasandSim[ 10 ],Calaore[ 13 ]andBen-Taletal.[ 5 ]),whichrequirestheconstraintA()xb()tobesatisedforeachinapre-deneduncertaintysetURK,i.e., sup 2UfA()x)]TJ /F9 11.955 Tf 11.96 0 Td[(b()g0,(2) wheretheoperator sup 2Ufgisconsideredconstraint-wisewithoutlossofgenerality.Oneimportantmeritof the ROapproachisthat,byaprioriadjustingtheuncertaintysetU,onecanensurethat theconstraint PfA()xb()g1)]TJ /F4 11.955 Tf 12.19 0 Td[(0forapre-speciedrisk 16

PAGE 17

level0 issatisedunder anyprobabilitydistributionP.Hence, the ROapproachcanbeviewedasaconservativeapproximationofchanceconstraints. Abasic,andperhapsthemostchallenging,questiononchanceconstraintsistheaccessibilityoftheprobabilitydistributionP.Mostliteratureonchanceconstraintslistedaboveassume s thedecisionmakershaveperfectknowledgeofP.Inpractice,however,itmightbe unrealistic tomakesuchanassumption. Normally, decisionmakershaveonlyaseriesof historical datapoints, whichcanbeconsideredassamplestaken fromthetrue(whileambiguous)distribution.Basedonthegivendata set therearetwopotentialissuesfor the classical chanceconstrainedmodel 2 :(i) itmightbechallenging t oassumeaspecicprobabilitydistributionandtogenerate alargenumberofscenarios accordinglyinthescenarioapproximations and(ii) t hesolutionmightbesensitivetotheambiguousprobabilitydistributionand thus questionableinpractice.Toaddressthesedrawbacks, d istributionally r obust(orambiguous) c hance c onstrain ed (DRCC)models areproposed (see,e.g.,ErdoganandIyengar[ 27 ],CalaoreandElGhaoui[ 16 ],NemirovskiandShapiro[ 42 ],Vandenbergheetal.[ 66 ],Zymleretal.[ 74 ],Xuetal.[ 71 ],and ElGhaouietal.[ 26 ] ).InDRCCmodels,Pisassumedtobelongtoapre-denedcondencesetD withgivenrstandsecondmomentvalues (ratherthanbeingknownwithcertainty),andthechanceconstraintsarerequiredtobesatisedundereachprobabilitydistributioninD: infP2DPfA()xb()g1)]TJ /F4 11.955 Tf 11.96 0 Td[(.(2) MostcurrentliteratureproposesapproximationapproachestosolveDRCC,andto thebestof ourknowledge,onlyVandenbergheetal.[ 66 ]andZymleretal.[ 74 ] targetderivingexactequivalentreformulationsfortheproblemwiththeformerprovidinganequivalentsemideniteprogrammingreformulationandthelatterprovidingaconditionalvalue-at-risk(CVaR)approximationwhichistightforthesinglechanceconstraintcase .Inthischapter,we focusondata-drivenapproachestoconstructthecondence 17

PAGE 18

setD.Accordingly,werefer theconstraintsintheformof 2 d ata-driven c hance c onstraints(DCCs). Inourapproach ,weattempttoconstructthecondencesetDbasedonlyonthehistoricaldatasampledfromthetrueprobabilitydistributionandthestatisticalinferencesobtainedfromthedata.Intuitively,sincerealdata are involvedinitsestimation,D can gettighteraroundthetrueprobabilitydistributionPwithmoredatasamples,and accordingly theDCC can becomelessconservative.Inthischapter, we proposeexactapproachestohandle bothsingleand jointDCCsunderdifferenttypesofcondencesetsbyderiving their equivalentreformulations. Furthermore,weshowtherelationshipbetweentheconservatismofDCCsandthesamplesizeofhistoricaldata,whichdepictsquantitativelythevalueofdata. 2.1.2ModelSettingsandCondenceSets Inuncertainenvironments,peopleutilizehistoricaldatainvariouswaystohelpdescriberandomparametersthroughstatisticalinference.Forexample,decisionmakersin the nanceindustry commonly describetheuncertaintyin r ateof r eturn(RoR)oftheinvestmentsby theirmeanandcovariancematrix statisticallyinferredbythehistoricaldata.Hence,acondencesetDcanbenaturallydenedasallprobabilitydistributionswhoserstandsecondmomentsagreewiththeinference.Forthis case,wecandenotethecondencesetDasD1shownasfollows(cf.DelageandYe[ 23 ]) : D1=fP2M+:(E[])]TJ /F4 11.955 Tf 11.95 0 Td[()>)]TJ /F12 7.97 Tf 6.59 0 Td[(1(E[])]TJ /F4 11.955 Tf 11.95 0 Td[()1,E[()]TJ /F4 11.955 Tf 11.96 0 Td[()()]TJ /F4 11.955 Tf 11.96 0 Td[()>]2g,(2) whereM+representsthesetofallprobabilitydistributions,and0representtheinferredmeanandcovariancematrixrespectively,and1>0and2>1aretwoparametersobtainedfromtheprocessofinference.One advantage ofD1isthatwecan construct itwitharelativelysmaller amount ofdata,sinceusuallytherstandsecondmomentsofcanbeeffectivelyestimatedbythesamplemeanandcovariancematrix. Inaddition ,D1considersmomentambiguity(i.e.,momentestimationerrorsareallowed)anddevelopsnonparametricboundsonthemeanandcovariancematrix. 18

PAGE 19

OnespecialcaseofD1canbeconstructedwithoutconsideringmomentambiguity,e.g.,D01=fP2M+:E[]=,E[>]=>+g.Forthisspecialcondencesetsetting,readersarereferredtoVandenbergheetal.[ 66 ]andZymleretal.[ 74 ]forthedetaileddiscussions. Besides themoments,decisionmakers can also resorttothedensityfunctionoftherandomvector.Forexample,powersystemoperatorsoftendescriberandomwindpoweravailableinatimeunitatawindfarmbyestimatingitsdensityfunction.Anaturalextensionofsuch a pointestimat ion isa condenceset estimation builtaroundthe point estimate,i.e.,the decisionmakers mightbelievethatalthoughtheirestimatecouldsufferfromsomeerrors,thetruedensity function isnottoofarawayfromit.Oneconvenientandcommonlyusedwayofmodelingthedistancebetweendensityfunctionsisby-divergence,whichisdenedas D(fjjf0)=ZRKf() f0()f0()d, wherefandf0denotethetruedensityfunctionanditsestimaterespectively,and:R!RisaconvexfunctiononR+suchthat (C1) (1)=0, (C2) 0(x=0):=8><>:xlimp!+1(p)=pifx>0,0ifx=0, (C3) (x)=+1forx<0. Threeexamplesof-divergenceareasfollows: Kullback-Leibler(KL)divergence:(x)=xlogx)]TJ /F9 11.955 Tf 11.96 0 Td[(x+1forx0, divergenceoforder2:(x)=(x)]TJ /F3 11.955 Tf 11.96 0 Td[(1)2forx0, Variationdistance:(x)=jx)]TJ /F3 11.955 Tf 11.95 0 Td[(1jforx0. Forgeneral-divergenceandotherexamples,interestedreadersarereferredtoPardo[ 46 ]andBen-Taletal.[ 5 ].Basedon-divergence,the decisionmakers canbuild 19

PAGE 20

acondencesetas follows: D2=fP2M+:D(fjjf0)d,f=dP=dg,(2) wherethedivergencetolerancedcanbechosenbythe decisionmakers torepresenttheirriskaversion level,orcanbeobtainedfromstatisticalinference. Itcanbeobservedthatusing D2asacondencesetforDCCscanperformbetterthanusingD1, because wecandepicttheproleoftheambiguousdistributionPmoreaccuratelybyitsdensityfunctionthanrsttwomomentsalone. Therefore, DCCsbasedonD2canbelessconservativethan those basedonD1. Inthischapter,wedevelopmodelingandsolutionapproachesforDCCsunderbothmoment-based(e.g.,D1)anddensity-basedcondencesets(e.g.,D2).Wedescribetheconstructionofcondencesets,showhowtoequivalentlyreformulateDCCs,anddiscussexactandapproximatesolutionapproachesfordata-drivenchanceconstrainedprograms(DCCPs)basedontheirequivalentreformulations.Tothebestofourknowledge,thischapterprovides the rststudyofDCCsunderD1andD2. Theremainderofthischapterisorganizedasfollows.Attheendofthissection,weintroducenotationanduncertaintysettingstobeusedthroughoutthischapter.Wediscuss the moment-basedcondencesets(D1)inSection 2.2 InSection 2.3 ,wediscuss the density-basedcondenceset(D2).Inaddition, inthissection ,we discovertherelationship betweenthe conservatism ofDCCsandthesamplesizeofhistoricaldata,which showsquantitativelythevalueofdata InSection 2.4 ,weexecuteanumericalstudytocomparetheperformancesofD1,D2andmyopicapproachesandverifytheeffectivenessofourproposedapproaches. Finally,wesummarizethischapterinSection 2.5 NotationandUncertaintySettings. Foragivenvectorx2RK,weletjjxjjrepresenttheL2-normofx,i.e.,jjxjj=p x>x,andjjxjjrepresentthenormofxinducedbyasymmetricandpositivesemidenitematrix2SKK+,i.e.,jjxjj=p x>x. Wespecify 20

PAGE 21

thetechnologymatrixA()andright-handsideb()byassumingthatA()andb()areafnelydependenton (aK-dimensionalvectorintheform=(1,...,K)>) ,i.e., A()=A0+KXk=1Akk,b()=b0+KXk=1bkk,(2) whereA0andb0representthedeterministicpartofA()andb(),and eachelementinA()andb()isanafnefunctionof .Thisuncertaintysettinghasbeenadoptedintheliteratureofstochasticprogrammingandrobustoptimization (cf.Chenetal.[ 19 ]andChenandZhang[ 20 ]) .Underthisuncertaintysetting,wecanreformulateconstraintA()xb()asfollows: A()xb(),A0x+KXk=1(Akx)kb0+KXk=1bkk,A(x)b(x), wherevectorb(x)=b0)]TJ /F9 11.955 Tf 11.95 0 Td[(A0x,andA(x)is an mKmatrixdenedas A(x)=A1x)]TJ /F9 11.955 Tf 11.95 0 Td[(b1,A2x)]TJ /F9 11.955 Tf 11.96 0 Td[(b2,...,AKx)]TJ /F9 11.955 Tf 11.95 0 Td[(bK. Accordingly ,DCC 2 canbegenerallyreformulatedas infP2DPf2Cg1)]TJ /F4 11.955 Tf 11.95 0 Td[(,(2) whereC=f2RK:AbgisapolyhedronwhoseparametersAandbdependuponx.Inthe remainderofthischapter ,weuseDCC 2 anditsgeneralreformulation 2 interchangeablyfornotationbrevity. 2.2DCCwithMoment-basedCondenceSet Inthissection,weconsiderconstraint 2 withD=D1.InSection 2.2.1 ,wediscusstheconstructionofD1byusingaseriesofindependent historicaldata figNi=1obeyingthetruedistributionP.InSection 2.2.2 weshowageneralequivalencerelationshipbetweenagroupofinnitenonconvexconstraintsandagroupoflinearmatrixinequalities(LMIs).WeapplythisequivalencerelationshiptoreformulatethesingleDCCunderD1,i.e.,infP2D1Pf2Cg1)]TJ /F4 11.955 Tf 11.63 0 Td[(,asLMIswithpolyhedronCreplaced 21

PAGE 22

byitsinterior, int (C)=f2RK:A, whereandaremaximumlikelihoodestimatorsofE[]andcov(),respectively.Second,we followtheproceduredescribed inDelageandYe[ 23 ]toconstructanonparametriccondencesetforthemeanandcovariancematrixofasfollows: D1=nP2M+:(E[])]TJ /F4 11.955 Tf 11.96 0 Td[()>)]TJ /F12 7.97 Tf 6.59 0 Td[(1(E[])]TJ /F4 11.955 Tf 11.95 0 Td[()1,E[()]TJ /F4 11.955 Tf 11.95 0 Td[()()]TJ /F4 11.955 Tf 11.96 0 Td[()>]2o, wheretheparameters10and2>1canbeobtainedfromtheprocessofinference(seeDelageandYe[ 23 ]fordetails). 22

PAGE 23

2.2.2ReformulationoftheSingleDCCs Inthissubsection,weinvestigatethereformulationofthe singleDCCunderD1 infP2D1Pfa>bg) inlaterProposition 1 ThereformulationreliesonageneralequivalencerelationshipbetweenagroupofinnitenonconvexconstraintsandagroupofLMIs.Moreover,wendthatthisequivalencerelationshipcan beextended toothersingleand even jointDCCs. Werst showtheequivalencerelationshipinLemma 1 Lemma1. Foragivengroupof symmetricandpositivesemidenitematricesMi2RKK(i.e.,Mi2SKK+),vectorsai2RK,andscalarsbi2R,i=1,...,m, thefollowingtwoclaimsareequivalentcorrespondingto asymmetricmatrixH2SKK,avectorp2RK,andascalarq: (i) >H+p>+qIC(),82RK,whereIC()=1if2CandIC()=0otherwise, (ii) Thereexistfyigmi=10,suchthat8>>>>>>>><>>>>>>>>:264)]TJ /F9 11.955 Tf 9.3 0 Td[(H)]TJ /F12 7.97 Tf 10.49 4.71 Td[(1 2p)]TJ /F12 7.97 Tf 10.5 4.71 Td[(1 2p>1)]TJ /F9 11.955 Tf 11.96 0 Td[(q3750,264)]TJ /F9 11.955 Tf 9.3 0 Td[(H)]TJ /F9 11.955 Tf 11.95 0 Td[(yiMi)]TJ /F12 7.97 Tf 10.5 4.7 Td[(1 2(p+yiai))]TJ /F12 7.97 Tf 10.5 4.71 Td[(1 2(p+yiai)>yibi)]TJ /F9 11.955 Tf 11.96 0 Td[(q3750,8i=1,...,m, whereC:=\mi=1CiwithCi:=f2RK:>Mi+a>iMi+a>i
PAGE 24

andhence statement(i)isequivalentto>H+p>+qI[>Mi+a>iH+p>+q1,82RK, (2a) >Mi+a>ibi)>H+p>+q0,82RK,8i=1,...,m. (2b) Nextwediscussthereformulationofconstraint 2a andimplication 2b .First,weobservethatconstraint 2a equivalentlyrequiresthatthequadraticfunction>H+p>+q)]TJ /F3 11.955 Tf 11.96 0 Td[(1isnonpositiveeverywhereinRK,andhence isequivalentto 264)]TJ /F9 11.955 Tf 9.3 0 Td[(H)]TJ /F12 7.97 Tf 10.49 4.71 Td[(1 2p)]TJ /F12 7.97 Tf 10.5 4.71 Td[(1 2p>1)]TJ /F9 11.955 Tf 11.96 0 Td[(q3750.(2) Second,theimplication 2b isequivalenttothestatementthat, foreachi=1,...,m ,the following quadraticsystem 8>><>>:>H+p>+q>0>Mi+a>i)]TJ /F3 11.955 Tf 12.04 2.66 Td[(bi0(2) hasnosolutioninRK.Weclaimthatthisisequivalenttothefollowing(innite)quadraticsystemhavinganonnegativesolutionyi: )]TJ /F3 11.955 Tf 11.95 0 Td[((>H+p>+q))]TJ /F9 11.955 Tf 11.95 0 Td[(yi(>Mi+a>i)]TJ /F3 11.955 Tf 12.05 2.66 Td[(bi)0,82RK.(2) Tosee this ,wediscussthefollowingcases: Case1 IfmatrixMihasastrictlypositiveeigenvalue,thenthereexistsa2RKsuchthat>Mi+a>i)]TJ /F3 11.955 Tf 10.74 2.66 Td[(bi>0sincewecanchoosetomake>Mi+a>i)]TJ /F3 11.955 Tf 10.74 2.66 Td[(biarbitrarilylarge. Thus ,inthiscase, theconditionoftheS-Lemma (cf.Yakubovich[ 72 ],PolikandTerlaky[ 49 ], andAppendix A.1 ) issatisedandaccordingly theequivalenceisguaranteed following theS-Lemma. 24

PAGE 25

Case2 IfalltheeigenvaluesofmatrixMi are nonpositive,thenMi=0becauseMi0byassumption.Wefurtherdiscussthefollowingcases: Case2.1 Ifai6=0,weobservethatthereexistsa2RKsuchthata>i)]TJ /F3 11.955 Tf 12.61 2.66 Td[(bi>0sincewecanchoosetomakea>iarbitrarilylarge.Inthiscase,theequivalenceisagainguaranteedbytheS-Lemma. Case2.2 Ifai=0andbi0:First,supposethat 2 hasnosolution.Sincea>i)]TJ /F3 11.955 Tf 12.61 2.65 Td[(bi=)]TJ /F3 11.955 Tf 9.4 2.65 Td[(bi0issatised,itfollowsthat>H+p>+q>0hasnosolution,i.e.,>H+p>+q0forall2RK,whichimpliesthat 2 hasasolutionyi=0.Second,supposethat 2 hasasolutionyi0.SinceMi=0,ai=0,andbi0,itfollowsthat)]TJ /F3 11.955 Tf 9.3 0 Td[((>H+p>+q))]TJ /F9 11.955 Tf 21.92 0 Td[(yibi0,whichimpliesthat 2 hasnosolution.Hence,theequivalenceisguaranteedinthiscase. Case2.3 Ifai=0andbi>0:Sincea>i)]TJ /F3 11.955 Tf 12.48 2.66 Td[(bi=)]TJ /F3 11.955 Tf 9.39 2.66 Td[(bi<0, 2 hasnosolution,andweonlyneedtoshowthat 2 hasanonnegativesolution.Butinviewof 2a ,weknowthatyi=1bi>0isasolutionfor 2 Wehaveprovedtheequivalencebetween 2 and 2 .Anotherwayofstating 2 isthatthereexistssomeyi0,suchthatthequadraticfunction)]TJ /F4 11.955 Tf 9.3 0 Td[(>(H+yiMi))]TJ /F3 11.955 Tf 12.12 0 Td[((p+yiai)>+yibi)]TJ /F9 11.955 Tf 11.96 0 Td[(qisnonnegativeeverywhereinRK,whichisequivalentto 264)]TJ /F9 11.955 Tf 9.29 0 Td[(H)]TJ /F9 11.955 Tf 11.96 0 Td[(yiMi)]TJ /F12 7.97 Tf 10.49 4.7 Td[(1 2(p+yiai))]TJ /F12 7.97 Tf 10.49 4.71 Td[(1 2(p+yiai)>yibi)]TJ /F9 11.955 Tf 11.95 0 Td[(q3750.(2) Therefore,wehaveequivalentlyreformulatedstatement(i)asLMIs 2 and 2 fori=1,...,m ,whichcompletestheproof. Second,weapplyLemma 1 toreformulatethesingleDCC 2 asLMIs. 25

PAGE 26

Theorem1. Given a vectoraand a scalarb,theDCCinfP2D1Pfa>1)]TJ /F9 11.955 Tf 11.96 0 Td[(r375+264001375264Hpp>q375y,(2{14a) 264G)]TJ /F9 11.955 Tf 9.3 0 Td[(p)]TJ /F9 11.955 Tf 9.29 0 Td[(p>1)]TJ /F9 11.955 Tf 11.96 0 Td[(r37526401 2a1 2a>y+a>)]TJ /F3 11.955 Tf 12.05 2.66 Td[(b375,(2{14b) 264G)]TJ /F9 11.955 Tf 9.3 0 Td[(p)]TJ /F9 11.955 Tf 9.29 0 Td[(p>1)]TJ /F9 11.955 Tf 11.96 0 Td[(r3752S(K+1)(K+1)+,264Hpp>q3752S(K+1)(K+1)+,y0,(2{14c) whereoperatorinconstraint 2a representstheFrobeniusinnerproduct. Proof. First,weletzD1representtheoptimalobjectivevalueoftheoptimizationproblemontheleft-handsideoftheDCC,i.e.,zD1=infP2D1Pfa>1375dP0,(2{15b) ZRK()]TJ /F4 11.955 Tf 11.96 0 Td[()()]TJ /F4 11.955 Tf 11.96 0 Td[()>dP2,(2{15c) ZRKdP=1,(2{15d) whereconstraints 2b describesthecondencesetofE[] basedonSchurcomplement 2c describesthecondencesetofE[()]TJ /F4 11.955 Tf 12.32 0 Td[()()]TJ /F4 11.955 Tf 12.32 0 Td[()>],andconstraint 2d guarantees that weareconsideringprobabilitydistributionsonRK.Weapplythe 26

PAGE 27

dualitytheoryforconiclinearprogrammingproblemsanddualizeproblem 2 as [Dual] zD1=maxG,H,p,q,r)]TJ /F4 11.955 Tf 9.3 0 Td[(2G+r)]TJ /F3 11.955 Tf 11.96 0 Td[(H)]TJ /F4 11.955 Tf 11.96 0 Td[(1q(2{16a) s.t.()]TJ /F4 11.955 Tf 11.96 0 Td[()>()]TJ /F9 11.955 Tf 9.3 0 Td[(G)()]TJ /F4 11.955 Tf 11.96 0 Td[()+2p>()]TJ /F4 11.955 Tf 11.95 0 Td[()+rI[a>q3752S(K+1)(K+1)+,(2{16c) wherematrix264Hpp>q375,matrixG,andscalarrrepresentthedualvariablesforconstraints 2b 2c ,and 2d ,respectively.Noteherethatstrongdualityholdsforproblems[Primal]and[Dual]basedonestablishedconiclinearprogrammingtheory( cf. Isii[ 31 ],Smith[ 61 ], and Shapiro[ 58 ]).Second,wereformulateconstraint 2b .Byreplacingwith+,wehaveconstraint 2b equivalentto>()]TJ /F9 11.955 Tf 9.3 0 Td[(G)+2p>+rI[a>](),82RK, which,inviewofLemma 1 ,isfurtherequivalentto264G)]TJ /F9 11.955 Tf 9.3 0 Td[(p)]TJ /F9 11.955 Tf 9.3 0 Td[(p>1)]TJ /F9 11.955 Tf 11.95 0 Td[(r37526401 2ya1 2ya>1+y(a>)]TJ /F3 11.955 Tf 12.05 2.66 Td[(b)375,264G)]TJ /F9 11.955 Tf 9.3 0 Td[(p)]TJ /F9 11.955 Tf 9.3 0 Td[(p>1)]TJ /F9 11.955 Tf 11.95 0 Td[(r3750, forsomey0.Hence,infP2D1Pfa>1)]TJ /F9 11.955 Tf 11.95 0 Td[(r375+264001375264Hpp>q375,(2{17a) 264G)]TJ /F9 11.955 Tf 9.3 0 Td[(p)]TJ /F9 11.955 Tf 9.3 0 Td[(p>1)]TJ /F9 11.955 Tf 11.95 0 Td[(r37526401 2ya1 2ya>1+y(a>)]TJ /F3 11.955 Tf 12.05 2.66 Td[(b)375,(2{17b) 264G)]TJ /F9 11.955 Tf 9.3 0 Td[(p)]TJ /F9 11.955 Tf 9.3 0 Td[(p>1)]TJ /F9 11.955 Tf 11.95 0 Td[(r3752S(K+1)(K+1)+,264Hpp>q3752S(K+1)(K+1)+,y0.(2{17c) 27

PAGE 28

Nowweobservethaty>0,becauseotherwise(i.e.,y=0)byconstraint 2b wehave264G)]TJ /F9 11.955 Tf 9.29 0 Td[(p)]TJ /F9 11.955 Tf 9.3 0 Td[(p>1)]TJ /F9 11.955 Tf 11.96 0 Td[(r3752640001375, andhence2642001375264G)]TJ /F9 11.955 Tf 9.3 0 Td[(p)]TJ /F9 11.955 Tf 9.29 0 Td[(p>1)]TJ /F9 11.955 Tf 11.96 0 Td[(r375+264001375264Hpp>q37526420013752640001375=1, whichviolatesconstraint 2a because2(0,1).Hence,welety=1=y0,replacematrices264G)]TJ /F9 11.955 Tf 9.3 0 Td[(p)]TJ /F9 11.955 Tf 9.3 0 Td[(p>1)]TJ /F9 11.955 Tf 11.96 0 Td[(r375and264Hpp>q375byy264G)]TJ /F9 11.955 Tf 9.3 0 Td[(p)]TJ /F9 11.955 Tf 9.3 0 Td[(p>1)]TJ /F9 11.955 Tf 11.96 0 Td[(r375andy264Hpp>q375, respectivelyinLMIs 2a 2c ,andobtainthereformulation 2a 2c Third,weextendtheapplicationofLemma 1 toothersingleDCCsundervariouscondencesets.Wepresenttheextensionsinthefollowingcorollaries,whosedetailedproofsareprovidedinAppendices A.2 A.3 forbrevity. (Nonlinearinequality) Theabove linearinequalitya>M+a>M+a>1)]TJ /F9 11.955 Tf 11.95 0 Td[(r375+264001375264Hpp>q375y,(2{18a) 264G)]TJ /F9 11.955 Tf 9.3 0 Td[(p)]TJ /F9 11.955 Tf 9.3 0 Td[(p>1)]TJ /F9 11.955 Tf 11.95 0 Td[(r375264M1 2(a+2M)1 2(a+2M)>y+>M+a>)]TJ /F3 11.955 Tf 12.05 2.66 Td[(b375,(2{18b) 264G)]TJ /F9 11.955 Tf 9.3 0 Td[(p)]TJ /F9 11.955 Tf 9.3 0 Td[(p>1)]TJ /F9 11.955 Tf 11.95 0 Td[(r3752S(K+1)(K+1)+,264Hpp>q3752S(K+1)(K+1)+,y0.(2{18c) 28

PAGE 29

(Polytopicambiguity)D1incorporatesthemomentambiguitybyconsideringnonlinearconvexsets(e.g.,consideringthemeanwithinanellipsoid). Analternative istoconsider polytopicambiguity .Thatis,decisionmakerscannameasetofpossiblemeansandsecondmomentsP:=f(i,i)2RKSKK+:ii>i,i=1,...,Ig, wheretheconditionii>iisusedtomakePwelldened ,and constructtheprobabilitydistributioncondencesetasfollows: Dpol1=nP2M+:(E[],E[>])2conv(P)o, whereconvrepresentstheconvexhull,i.e.,thereexists2RI+,suchthatE[]=PIi=1ii,E[>]=PIi=1ii,andPIi=1i=1.TheDCCunderDpol1canbereformulatedasLMIs shown inthefollowingcorollary,whosedetailedproofisprovidedinAppendix A.2 Corollary2. TheDCCinfP2Dpol1Pfa>ip)]TJ /F9 11.955 Tf 11.95 0 Td[(r0,8i=1,...,I,(2{19b) 264)]TJ /F9 11.955 Tf 9.3 0 Td[(H)]TJ /F12 7.97 Tf 10.5 4.71 Td[(1 2p)]TJ /F12 7.97 Tf 10.49 4.7 Td[(1 2p>1)]TJ /F9 11.955 Tf 11.95 0 Td[(q37526401 2a1 2a>y)]TJ /F3 11.955 Tf 12.05 2.65 Td[(b375,(2{19c) 264)]TJ /F9 11.955 Tf 9.3 0 Td[(H)]TJ /F12 7.97 Tf 10.5 4.7 Td[(1 2p)]TJ /F12 7.97 Tf 10.49 4.71 Td[(1 2p>1)]TJ /F9 11.955 Tf 11.95 0 Td[(q3752S(K+1)(K+1)+,y0.(2{19d) Remark1. Onespecialcaseofcorollary 2 iswhenI=1, inwhich Dpol1reducestobefP2M+:E[]=1,E[>]=1g,whichissimplerthanD1anddoesnotallowmomentambiguity.ThereformulationoftheDCCunder thisspecialcase isdiscussedinVandenbergheetal.[ 66 ]andZymleretal.[ 74 ]. (Marginalmoments)AnothervariantofD1istoestimatethersttwomomentsofeachcomponentkseparately.Thatis,ifweomitthecorrelationofeachpairkand`and 29

PAGE 30

buildcondenceintervalsforE[k]andE[2k]fork=1,...,K,itresultsin Dmgn1=nP2M+:LkE[k]Uk,E[2k]k,8k=1,...,Ko, where[Lk,Uk]isacondenceintervalofE[k]andkisanupperboundofE[2k],respectively. Also,weassumethatdiag(1,...,K)>foreachLU tomake Dmgn1welldened. Noteherethat wecanalwaysincreasekvaluestomakeithappen. ItiseasiertoconstructDmgn1thanD1 byusingasmallersizedataset ,sinceweonlyconsiderthemarginaldistributionoftherandomvector. Westatethecorollaryasfollowsandthedetailedproofisprovided inAppendix A.3 Corollary3. TheDCCinfP2Dmgn1Pfa>
PAGE 31

Proposition1. Denotez(C)astheoptimalobjectivevalueofanoptimizationproblemwithDCC,i.e.,z(C)=minx2X (x)s.t.infP2D1Pf2Cg1)]TJ /F4 11.955 Tf 11.95 0 Td[(, whereXisaconvexset, ()isaconvexfunction,C=f2RK:a>bgwithaandbbeingafnefunctionsofx,andtheSlaterconditionholdsfortheoptimizationproblem.Thenz(C)=z(int(C)). Remark2. Proposition 1 istrivialiftheprobabilitydistributionPisabsolutelycontinuouswithrespecttoLebesguemeasureonRKbecause inthiscasePfa>=bg=0 .However,thispropositionholdsforgeneraldistributions. Remark3. SimilarcontinuityresultsasProposition 1 holdforothersingleDCCsdiscussedincorollaries 1 3 .Thatis,Proposition 1 holdswithpolyhedronCreplacedbyf2RK:>M+a>bg,withD1replacedbyDpol1,andwithD1replacedbyDmgn1,respectively. 2.2.3ReformulationoftheJointDCCs Inthissubsection,we study thejointDCCs.We observe thatthejointDCCsunderD1cannolongerbereformulatedasLMIsalthoughitsleft-handside(i.e.,infP2D1PfA
PAGE 32

Corollary4. GivenamatrixAandavectorb,thejointDCCinfP2D1PfA1)]TJ /F9 11.955 Tf 11.96 0 Td[(r375)]TJ /F15 11.955 Tf 11.95 27.62 Td[(264001375264Hpp>q3751)]TJ /F4 11.955 Tf 11.96 0 Td[(,(2{21a) 264G)]TJ /F9 11.955 Tf 9.3 0 Td[(p)]TJ /F9 11.955 Tf 9.3 0 Td[(p>1)]TJ /F9 11.955 Tf 11.95 0 Td[(r37526401 2yiai1 2yia>i1+yi(a>i)]TJ /F3 11.955 Tf 12.05 2.66 Td[(bi)375,8i=1,...,m,(2{21b) 264G)]TJ /F9 11.955 Tf 9.3 0 Td[(p)]TJ /F9 11.955 Tf 9.3 0 Td[(p>1)]TJ /F9 11.955 Tf 11.95 0 Td[(r3752S(K+1)(K+1)+,264Hpp>q3752S(K+1)(K+1)+,y0.(2{21c) wherea>1,...,a>mrepresentthemrowvectorsconsistingofmatrixA. Notethatthereformulatingconstraints 2a 2c arenolongerLMIsbecauseofthebilineartermsyiaiinconstraints 2b ,whereyirepresentsanonnegativemultiplierandaiisanafnefunctionofdecisionvariablex.However,withthemultipliersyxed,constraints 2a 2c recoverasLMIs.Inthiscase,theworst-caseprobabilityboundinfP2D1PfAi)]TJ /F3 11.955 Tf 12.05 2.66 Td[(big01)]TJ /F4 11.955 Tf 11.96 0 Td[(, wherethefunctionmaxi=1,...,mfa>i)]TJ /F3 11.955 Tf 12.71 2.66 Td[(bigisconvexandpiece-wiselinearover.Thisobservationmotivatesustoapproximatethisfunctionbyusingaquadratic function intheform>M+c>+d,inviewthatwecanreformulatetheresultingapproximationasLMIsbyusingcorollary 1 .Tomakethisapproximationconservative,weensurethat 32

PAGE 33

>M+c>+da>i)]TJ /F3 11.955 Tf 12.05 2.66 Td[(biforall2RKandalli=1,...,m,whichimpliesthatinfP2D1Pmaxi=1,...,mfa>i)]TJ /F3 11.955 Tf 12.05 2.66 Td[(big0infP2D1P>M+c>+d01)]TJ /F4 11.955 Tf 11.96 0 Td[(. Inaddition,tomakethisapproximationastightaspossible,wecantreattheapproximationcoefcientsM,c,anddasdecisionvariables,andsowecanndanoptimalapproximationonthewayofoptimizingoverthejointDCC.Wesummarizetheapproximationresultbythefollowingproposition,whosedetailedproofisprovidedinAppendix A.5 Proposition2. ThejointDCCinfP2D1PfAbg1)]TJ /F4 11.955 Tf 12.87 0 Td[(canbeconservativelyapproximatedbythefollowingLMIs:2642001375264G)]TJ /F9 11.955 Tf 9.3 0 Td[(p)]TJ /F9 11.955 Tf 9.3 0 Td[(p>1)]TJ /F9 11.955 Tf 11.95 0 Td[(r375+264001375264Hpp>q375y,(2{22a) 264G)]TJ /F9 11.955 Tf 9.3 0 Td[(p)]TJ /F9 11.955 Tf 9.3 0 Td[(p>1)]TJ /F9 11.955 Tf 11.95 0 Td[(r375264M1 2(c+2M)1 2(c+2M)>y+>M+c>+d375,(2{22b) 264M1 2(c)]TJ /F3 11.955 Tf 11.87 0 Td[(ai)1 2(c)]TJ /F3 11.955 Tf 11.87 0 Td[(ai)>d+bi3750,8i=1,...,m,(2{22c) 264G)]TJ /F9 11.955 Tf 9.3 0 Td[(p)]TJ /F9 11.955 Tf 9.3 0 Td[(p>1)]TJ /F9 11.955 Tf 11.95 0 Td[(r3752S(K+1)(K+1)+,264Hpp>q3752S(K+1)(K+1)+,y0.(2{22d) ArelaxationofthejointDCCcanbeobtainedbynotingthatinfP2D1Pa>ibiinfP2D1Pmaxi=1,...,mfa>i)]TJ /F3 11.955 Tf 12.05 2.66 Td[(big01)]TJ /F4 11.955 Tf 11.96 0 Td[(,8i=1,...,m, wherethejointDCCisrelaxedtobeasingle(andtractable)DCC.WecanalsotightenthisrelaxationbyaddinginsingleDCCrelaxationforeachi=1,...,m,whichispresentedbythefollowingproposition. 33

PAGE 34

Proposition3. ThejointDCCinfP2D1PfAbg1)]TJ /F4 11.955 Tf 12.59 0 Td[(canberelaxedtobethefollowingLMIs:2642001375264Gi)]TJ /F9 11.955 Tf 9.3 0 Td[(pi)]TJ /F9 11.955 Tf 9.3 0 Td[(p>i1)]TJ /F9 11.955 Tf 11.96 0 Td[(ri375+264001375264Hipip>iqi375yi,8i=1,...,m,264Gi)]TJ /F9 11.955 Tf 9.3 0 Td[(pi)]TJ /F9 11.955 Tf 9.3 0 Td[(p>i1)]TJ /F9 11.955 Tf 11.96 0 Td[(ri37526401 2ai1 2a>iyi+a>i)]TJ /F3 11.955 Tf 12.04 2.66 Td[(bi375,8i=1,...,m,264Gi)]TJ /F9 11.955 Tf 9.3 0 Td[(pi)]TJ /F9 11.955 Tf 9.3 0 Td[(p>i1)]TJ /F9 11.955 Tf 11.96 0 Td[(ri3752S(K+1)(K+1)+,264Hipip>iqi3752S(K+1)(K+1)+,yi0,8i=1,...,m. 2.2.4ReformulationoftheWorst-CaseValue-at-RiskConstraint Inthissubsection,we extendourstudy ontheDCCstotheworst-casevalue-at-riskconstraint.First,value-at-risk(VaR)isariskmeasurefrequentlyusedinportfoliooptimizationproblems(see,e.g.,RockafellarandUryasev[ 53 ],andElGhaouietal.[ 26 ]).GivenarandomvectorwithprobabilitydistributionP,aconstraintontheVaRofrandomvariablea>withrisklevel(denotedasVaR(a>))canbedenedasVaR(a>):=inf`2R:Pfa>`g1)]TJ /F4 11.955 Tf 11.96 0 Td[(b, wherebisagivenupperboundforVaR. Inpractice,Pisnotalwaysperfectlyknownoraccuratelyestimated,andadecisionmakercanthusconsideranalternativeworst-casevalue-at-risk(WVaR)constraintdenedas VaR(a>)b,8P2D1,(2) whereD1canbeestimatedbasedonhistoricaldatasamplesof,andconstraint 2 isrobustwithregardtotheunderlyingprobabilitydistributionof.Inthefollowingproposition,weshowtheequivalencebetweenconstraint 2 andthesingleDCCinfP2D1Pfa>bg1)]TJ /F4 11.955 Tf 11.96 0 Td[(,andweprovidethedetailedproofinAppendix A.6 34

PAGE 35

Proposition4. TheWVaRconstraint 2 isequivalenttotheLMIs 2a 2c denedinTheorem 1 Remark4. FromProposition 4 ,itisveryinterestingtoobservethatthedata-drivenchanceconstraintand the worst-casevalue-at-riskconstraintleadtothesamereformu-lation.Inotherwords,withtheconsiderationofthedata-drivenrobustnessinthechanceconstraint,thereisnoneedtoconsidertheworst-casevalue-at-riskmeasuretoimprovetherobustness. 2.3DCCwithDensity-basedCondenceSet Inthissection,weconsiderDCC 2 whendensityinformationistakenintoaccount,i.e.,withD=D2.WerstdiscusstheconstructionofD2inSection 2.3.1 .Then,inSection 2.3.2 ,weshowthatthejointDCCsunderD2isequivalenttoa classical chanceconstraint 2b withadeterministicprobabilitydistribution,whichcanbefurthersolvedbyscenarioapproximationapproaches.Inaddition, by deriv ing howthesamplesizeof the datareectsthelevelof conservatism ofDCCs,wedepictquantitativelythevalueofdatainSection 2.3.3 2.3.1ConstructionofCondenceSet Inthissubsection,wediscusstheconstructionofcondencesetD2,whichisbasedonageneral-divergencedenedasD(fjjf0)=ZRKf() f0()f0()d, wherefandf0denotethedensityfunction(resp.probabilitymassfunctionfordiscretedistribution)ofPanditsestimaterespectively,andtheintegraliswithrespecttoLebesguemeasureonRK(resp.withrespecttothecountingmeasurefordiscretedistribution).Fordiscretedistributions,histogramsareusefulinpracticefordepictingdensityfunctionproles.SupposethatwehaveadatasetfigNi=1.Todrawahistogram,werstconstructa nonempty partitionfBj:j=1,..., B gofthesamplespace,where=S B j=1BjandeachBj iscalled abin.Second,wecountthefrequency 35

PAGE 36

Nj=PNi=1I[Bj](i)foreachbinBj,whereI[Bj](i)equal s oneifi2Bj,andzerootherwise.Finally,wecanestimatetheprobabilityoflandingineachbin,i.e.,PfBjg,byitsempiricalrelativefrequencyNj=N.Forcontinuousdistributions,wecannotdirectlyusethehistogramestimateforfsinceitisnotabsolutelycontinuouswithregardtotheLebesguemeasure. Instead ,wereplacethehistogramestimatebyitscounterpartinestimatingcontinuousdensityfunction s ,called the kerneldensityestimator(KDE), whichisdenedasfollows (seeRosenblatt[ 54 ]andParzen[ 47 ]): fN()=1 NhKNNXi=1H)]TJ /F4 11.955 Tf 11.96 0 Td[(i hN, wherehNisapositiveconstant, Kisthedimensionof ,andH()isasmoothfunctionsatisfyingH()0,RH()d=1,RH()d=0,andR2H()d>0.OneexampleforH()isthestandardnormaldensityfunction.It is showninDevroyeandGyor[ 25 ]thatKDEconverges tothetruedensity inL1norm ,i.e.,withprobabilityone, ZRKjfN())]TJ /F9 11.955 Tf 11.95 0 Td[(f()jd!0asN!1. TherearevariouswaystodecidethedivergencetolerancedinD2.First,decisionmakerscandecidethevalueofdtoreecttheirrisk-aversionpreference,andadjustdtoperformpost-optimizationsensitivityanalysis.Forexample,decisionmakersmaydecided=J=ln(N),whereJisaconstantandNisthetotalnumberofdatasamples,toreectanintuitionthatD2becomestighterasNincreases.Second,fordiscretedistributions,wecanestimatethevalueofdbyusingthehistogramestimate.Pardo[ 46 ](see ,e.g., Theorem3.1 in[ 46 ] )showsthat2N 00(1)D(fjjf0)convergesindistributiontoaChi-squaredistributedrandomvariablewithB)]TJ /F3 11.955 Tf 12.75 0 Td[(1degreesoffreedomasNgoestoinnity,providedthat00(1)existsandisnonzero.Thisobservationmotivatesustoapproximatethedivergencetolerancedbysettingd=00(1)2B)]TJ /F12 7.97 Tf 6.59 0 Td[(1,1)]TJ /F6 7.97 Tf 6.58 0 Td[(=(2N)withlargeN,where2B)]TJ /F12 7.97 Tf 6.59 0 Td[(1,1)]TJ /F6 7.97 Tf 6.58 0 Td[(representsthe100(1)]TJ /F4 11.955 Tf 12.91 0 Td[()%(e.g.,=0.05)percentileofthe2B)]TJ /F12 7.97 Tf 6.58 0 Td[(1distribution.Forcontinuousdistributions,however,therearefewliterature 36

PAGE 37

discussingtheestimationofdforageneral-divergence. Onepossibleapproachistoestimatedby 00(1)2B)]TJ /F12 7.97 Tf 6.59 0 Td[(1,1)]TJ /F6 7.97 Tf 6.59 0 Td[(=(2N)forlargeN, asforthediscretecase,motivatedby theobservationthatboththehistogramandtheKDEconvergetothetruedistributionasNgoestoinnity. Ingeneral ,inobservationofmultiplepossiblewaysofestimatingthe-divergencetoleranced, inthischapter ,weassumethatdisingeneralafunctionofdatasamplesizeN,i.e.,d:=d(N),disnonincreasingasNincreases,anddtendstozeroasNtendstoinnity, inourtheoreticalanalysis 2.3.2ReformulationofDCC Inthissubsection,weaddressthereformulationoftheDCCsunderD2 with ageneral-divergence measure considered.Beforegivingthemainresult,wereviewthedenitionofconjugateduality.Givenafunctiong:R!R,theconjugateg:R!R[f+1gisdenedas g(t)=supx2Rftx)]TJ /F9 11.955 Tf 11.96 0 Td[(g(x)g. Also,wepresentsomepropertiesoftheconjugateoffunctionwhichisusedtodene-divergencemeasures.WeprovidethedetailedproofofthefollowingLemmainAppendix A.7 forcompleteness. Lemma2. Let:R!Rbeaconvexfunctionsuchthat(1)=0and(x)=+1forx<0.Thensatisesthefollowingproperties: (i) isconvex; (ii) isnondecreasing; (iii) (x)xforallx2R; (iv) Ifisaniteconstantonaninterval[a,b]fora,b2Randa
PAGE 38

Denition1. Let:R!Rbeaconvexfunctionsuchthat(1)=0and(x)=+1forx<0. Dene m ():=supfm2R:isaniteconstanton(,m]gand m():=inffm2R:(m)=+1g. Theorem2. Givendensityestimatef0,letP0representtheprobabilitydistributiondenedbyf0.ThentheDCCinfD(fjjf0)dPfAbg1)]TJ /F4 11.955 Tf 11.95 0 Td[(canbereformulatedas P0fAbg1)]TJ /F4 11.955 Tf 11.96 0 Td[(0+,(2)where0=1)]TJ /F3 11.955 Tf 44.44 0 Td[(infz>0,m ()z0+z m()n(z0+z))]TJ /F9 11.955 Tf 11.95 0 Td[(z0)]TJ /F4 11.955 Tf 11.96 0 Td[(z+d (z0+z))]TJ /F4 11.955 Tf 11.95 0 Td[((z0)o, andx+=maxfx,0gforx2R,ifoneofthefollowingtwoconditionsissatised: (i) limx!+1(x)=x=+1, (ii) f0()>0almostsurelyonRK. Proof. DenotingsetC=f2RK:Abg,werewrite the left-handsideofthechanceconstraintaszD2=minfZRKIC()f()d(2{26a) s.t.ZRKf0()f() f0()dd,(2{26b) ZRKf()d=1,(2{26c) f()0,82RK,(2{26d) whereconstraint 2b boundsthe-divergenceD(fjjf0)fromabovebyd,andconstraints 2c and 2d guaranteethatfisadensityfunction.Sinceproblem 2 is,onceagain,asemi-inniteproblem,weresorttoduality.TheLagrangiandualofproblem 2 canbewrittenasL=supz0,z02Rinff()0(ZRKIC()f())]TJ /F9 11.955 Tf 11.96 0 Td[(z0f()+zf0()f() f0()d+z0)]TJ /F9 11.955 Tf 11.95 0 Td[(zd)=supz0,z02R(z0)]TJ /F9 11.955 Tf 11.95 0 Td[(zd+inff()0nZRKh(IC())]TJ /F9 11.955 Tf 11.96 0 Td[(z0)f()+zf0()f() f0()ido), (2) 38

PAGE 39

wherezandz0representthedualvariablesofconstraints 2b and 2c ,respectively. SinceP02D2,D2isregularinthesensethatD(f0jjf0)=00. ThenwehaveL=supz>0,z02R(z0)]TJ /F9 11.955 Tf 11.96 0 Td[(zd)]TJ /F9 11.955 Tf 11.96 0 Td[(zsupf()0nZRKhz0)]TJ /F9 11.955 Tf 11.96 0 Td[(IC() zf())]TJ /F9 11.955 Tf 11.95 0 Td[(f0()f() f0()ido)=supz>0,z02R(z0)]TJ /F9 11.955 Tf 11.96 0 Td[(zd)]TJ /F9 11.955 Tf 11.96 0 Td[(zsupf()0nZ[f0()>0][[f0()=0]hz0)]TJ /F9 11.955 Tf 11.95 0 Td[(IC() zf())]TJ /F9 11.955 Tf 11.95 0 Td[(f0()f() f0()ido)=supz>0,z02R(z0)]TJ /F9 11.955 Tf 11.96 0 Td[(zd)]TJ /F9 11.955 Tf 11.96 0 Td[(zsupf()0nZ[f0()>0]hz0)]TJ /F9 11.955 Tf 11.95 0 Td[(IC() zf() f0())]TJ /F4 11.955 Tf 11.96 0 Td[(f() f0()if0()do) (2)=supz>0,z02R(z0)]TJ /F9 11.955 Tf 11.96 0 Td[(zd)]TJ /F9 11.955 Tf 11.96 0 Td[(zZ[f0()>0]supf()=f0()0nz0)]TJ /F9 11.955 Tf 11.96 0 Td[(IC() zf() f0())]TJ /F4 11.955 Tf 11.96 0 Td[(f() f0()of0()d)=supz>0,z02R(z0)]TJ /F9 11.955 Tf 11.96 0 Td[(zd)]TJ /F9 11.955 Tf 11.96 0 Td[(zZ[f0()>0]z0)]TJ /F9 11.955 Tf 11.96 0 Td[(IC() zf0()d) (2)=supz>0,z02Rnz0)]TJ /F9 11.955 Tf 11.96 0 Td[(zd)]TJ /F9 11.955 Tf 11.96 0 Td[(zP0fCgz0)]TJ /F3 11.955 Tf 11.96 0 Td[(1 z)]TJ /F9 11.955 Tf 11.95 0 Td[(z)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F11 11.955 Tf 11.96 0 Td[(P0fCgz0 zo, (2) whereequality 2 followsfromthetwoassumptionsbecause(i)iflimx!+1(x)=x=+1,thenwheneverf0()=0,f()hastobezerotoachieveoptimality (seeproperty(C2)offunctioninSection 2.1.2 ) ,and(ii)iff0()>0almostsurelyonRK,thentheequalityisclear.Equality 2 followsfromthedenitionofconjugate,andequality 2 followsfromconditionalprobability,conditioningonthevalueofIC(). Furthermore,tomaketheDCCsatised,weobservethatweshouldletthedecisionvariablesz0andzbesuchthatz0=z2[m (), m()].Toseethat,wediscussthefollowingcases: 39

PAGE 40

Case1. Supposethatz0=z> m().Thenwehave(z0=z)=+1andsoz0)]TJ /F9 11.955 Tf 11.95 0 Td[(zd)]TJ /F9 11.955 Tf 11.95 0 Td[(zP0fCgz0)]TJ /F3 11.955 Tf 11.95 0 Td[(1 z)]TJ /F9 11.955 Tf 11.96 0 Td[(z)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(1)]TJ /F11 11.955 Tf 11.95 0 Td[(P0fCgz0 z=, inwhichcasetheDCCisviolated. Case2. Supposethatz0=z0andz0suchthatz0=z2[m (), m()]and P0fCg1)]TJ /F9 11.955 Tf 11.96 0 Td[(z0)]TJ /F4 11.955 Tf 11.95 0 Td[(+z)]TJ /F8 7.97 Tf 6.67 -4.8 Td[(z0 z+zd z)]TJ /F8 7.97 Tf 6.67 -4.8 Td[(z0 z)]TJ /F9 11.955 Tf 11.96 0 Td[(z)]TJ /F8 7.97 Tf 6.67 -4.8 Td[(z0)]TJ /F12 7.97 Tf 6.59 0 Td[(1 z.(2) Therefore,thereformulation 2 isobtainedby simultaneously replacingzandz0by1=zand(z0+z)=z,respectively,ininequality 2 Second,supposethatz=0.ThenwehaveL=supz02Rnz0+inff()0ZRK(IC())]TJ /F9 11.955 Tf -427.52 -26.6 Td[(z0)f()do.Itfollowsthat Case1. IfC6=RK,thenatoptimalityz0=0withL=0 becauseIC())]TJ /F9 11.955 Tf 12.06 0 Td[(z00foreach2RK ItfollowsthattheDCCzD21)]TJ /F4 11.955 Tf 11.95 0 Td[(isviolatedbecause2(0,1). 40

PAGE 41

Case2. IfC=RK,thenatoptimalityz0=1withL=1 becauseIC())]TJ /F9 11.955 Tf 12.03 0 Td[(z0=1)]TJ /F9 11.955 Tf 12.03 0 Td[(z00foreach2RK ItfollowsthattheDCCzD2=11)]TJ /F4 11.955 Tf 11.97 0 Td[(issatisedbecause>0.Forthiscase,thereformulation 2 isvalidbecause1)]TJ /F4 11.955 Tf 11.96 0 Td[(0+1. Remark5. ThetwoconditionsinTheorem 2 aremildbecause(i)manytypesof-divergencemeasuresatisfylimx!+1(x)=x=+1,e.g.,KLdivergence(with(x):=xlogx)]TJ /F9 11.955 Tf 12.29 0 Td[(x+1),divergenceoforder(with(x):=jx)]TJ /F3 11.955 Tf 12.29 0 Td[(1jand>1),andCressieandReaddivergence(with(x):=1)]TJ /F6 7.97 Tf 6.58 0 Td[(+x)]TJ /F8 7.97 Tf 6.58 0 Td[(x (1)]TJ /F6 7.97 Tf 6.58 0 Td[()and>1),and(ii)anydensityestimatef0canbeslightlymodiedtosatisfyf0>0,andanimportantexampleisaKDEf0bychoosingH()asthestandardnormaldensity. Corollary5. TheDCCreformulationP0fAbg1)]TJ /F4 11.955 Tf 11.99 0 Td[(0+isaconservativeapproxima-tionofthenominalchanceconstraintP0fAbg1)]TJ /F4 11.955 Tf 11.96 0 Td[(,i.e.,1)]TJ /F4 11.955 Tf 11.95 0 Td[(0=infz>0,m ()z0+z m()n(z0+z))]TJ /F9 11.955 Tf 11.95 0 Td[(z0)]TJ /F4 11.955 Tf 11.96 0 Td[(z+d (z0+z))]TJ /F4 11.955 Tf 11.95 0 Td[((z0)o1)]TJ /F4 11.955 Tf 11.96 0 Td[(. Proof. SeeAppendix A.8 Henceforth,wecallthechanceconstraint 2 areformulatedDCC.Theorem 2 andcorollary 5 showthatareformulatedDCCisa classical chanceconstraintwiththeambiguousprobabilitydistributionPreplacedbyitsestimateP0,andtheriskleveldecreasedto0.Ascomparedtothe classical chanceconstraints,reformulatedDCCsprovideusthefollowingtheoreticalmerits: 1. Intermsofmodeling,unlikerelyingonanambiguousprobabilitydistributionP,wewaivetheperfectinformationassumptionandresorttoP0whichcanbeestimatedfromthehistoricaldata.Meanwhile,wecanmakeamoreaccurateP0estimatewithmoredataonhand. 2. Intermsofalgorithmdevelopment,theestimateP0ismoreaccessiblethantheambiguousP.First,thesamplestakenfromP0,whichisdeterministic,ismoretrustablethanfromaguessoftheambiguousP.Second,wecanfacilitatethesamplingprocedurebychoosingdensityfunctionsthatareeasiertosamplefrom.Forexample,wecandevelopP0asaKDEandchoosefunctionH()asastandardnormaldistribution. ThisobservationmotivatesustosolveoptimizationproblemswithreformulatedDCCsbyusingthescenarioapproximationapproach. 41

PAGE 42

3. Intermsofconservatism,itisintuitivethatwhenfacedwithprobabilitydistributionambiguity,onecanreducetomakea classical chanceconstraintmoreconservative.Theorem 2 canhelptoquantifyhowmuchneedstobereduced,andhenceaccuratelydepictsthe relationship betweentherisklevelandtheconservatism. ForD2undergeneral-divergencemeasures,theperturbedrisklevel0inthereformulatedDCCcanbeobtainedbysolvingatwo-dimensionalnonlinearoptimizationproblem.Manyoff-the-shelfoptimizationsoftware(e.g.,GAMSandAIMMS)andgeneral-purposeoptimizationpackages(e.g.,MINOSandSNOPT)canbeused.Inthischapter,wetakethreetypesof-divergencemeasure,i.e.,divergenceoforder2,KLdivergence,andvariationdistance,asexamplestoshowhowtheperturbedrisklevelcanbeobtained.Wendthattheperturbedrisklevelforbothdivergenceoforder2andvariationdistancecanbeobtainedinaclosedform.FortheKLdivergence,itseemshardtoobtainaclosed-formperturbedrisklevel. However ,wecanefcientlycomputeitbyusingbisectionlinesearch, becausethecomputation effort canbeshowntobeequivalenttominimizingaunivariateconvexfunction .Wesummarizeourresultsinthefollowingthreepropositions,whoseproofsareprovidedinAppendices A.9 A.11 forbrevity. Proposition5. SupposethatwedevelopD2byusingthedivergenceoforder2with(x):=(x)]TJ /F3 11.955 Tf 11.95 0 Td[(1)2and<1=2.Thentheperturbedrisklevelis0=)]TJ /F15 11.955 Tf 13.15 18.53 Td[(p d2+4d()]TJ /F4 11.955 Tf 11.96 0 Td[(2))]TJ /F3 11.955 Tf 11.95 0 Td[((1)]TJ /F3 11.955 Tf 11.95 0 Td[(2)d 2d+2. Proposition6. SupposethatwedevelopD2byusingthevariationdistancewith(x):=jx)]TJ /F3 11.955 Tf 11.95 0 Td[(1j.Thentheperturbedrisklevelis0=)]TJ /F9 11.955 Tf 13.15 8.09 Td[(d 2. 42

PAGE 43

Proposition7. SupposethatwedevelopD2byusingtheKLdivergencewith(x):=xlogx)]TJ /F9 11.955 Tf 11.96 0 Td[(x+1.Thentheperturbedrisklevelis0=1)]TJ /F3 11.955 Tf 19.44 0 Td[(infx2(0,1)ne)]TJ /F8 7.97 Tf 6.58 0 Td[(dx1)]TJ /F6 7.97 Tf 6.59 0 Td[()]TJ /F3 11.955 Tf 11.96 0 Td[(1 x)]TJ /F3 11.955 Tf 11.95 0 Td[(1o, andcanbecomputedbyusingbisectionlinesearchafterlog2(1 )stepstoachieveaccuracy. 2.3.3TheValueofData Intuitively,asthesamplesizeNincreaseswecandepicttheproleoftheambiguousprobabilitydistributionPmoreaccuratelywithasmaller-divergencetoleranced.BasedonTheorem 2 ,onemaybeinterestedintwoquestions: Willtheperturbedrisklevel0increasetoasdtendstozero(resp.asNtendstoinnity)? Howfastwill0convergeasddecreases(resp.asNincreases)? Thesetwoquestionsareimportantinpractice.First,anafrmativeanswertotherstquestionindicatesthattheconservatismoftheDCCswillvanishasNtendstoinnity.Second,sincecollectingdatafrequentlyincurscost,theconvergencerateof0helpstoestimatehowanewsetofdatadecreasestheconservatismofDCC,i.e.,thevalueofdata.Morespecically,fornominalrisklevelgiven,wecandenethevalueofdataasVoD=d0 dN, whichrepresentstheincreaseof0valueifwemarginallyenlargethedataset.Inthissubsection,weanswertherstquestionintheafrmativeforD2underageneral-divergence.Moreover,wefocusontwotypesof-divergence,i.e.,thedivergenceoforder2,andtheKLdivergence,anddiscusstheconvergencebehaviorof0andtheircorrespondingVoD.Fordiscussionconsistency,weassumethatdischosentobeadifferentialfunctionofN,i.e.,d=d(N),disnonincreasingasNincreases,anddtendstozeroasNtendstoinnity.Forexample,wecanfollowPardo[ 46 ]andsetthe 43

PAGE 44

divergencetoleranced=00(1)2B)]TJ /F12 7.97 Tf 6.59 0 Td[(1,1)]TJ /F6 7.97 Tf 6.58 0 Td[(=(2N).Werstshowthefollowingpropositionforthegeneral-divergence,whosedetailedproofisprovidedinAppendix A.12 Proposition8. Forageneral-divergencewithx=1asitsuniqueminimizer,theperturbedrisklevel0denedas1)]TJ /F3 11.955 Tf 44.43 0 Td[(infz>0,m ()z0+z m()n(z0+z))]TJ /F9 11.955 Tf 11.96 0 Td[(z0)]TJ /F4 11.955 Tf 11.95 0 Td[(z+d (z0+z))]TJ /F4 11.955 Tf 11.96 0 Td[((z0)o increasestoasddecreasestozero(resp.asNincreasestoinnity). Nextweshowtheconvergenceresultof0andthecorrespondingVoDforthedivergenceoforder2andtheKLdivergence,respectively.Thedetailedproofs are providedinAppendices A.13 A.14 Proposition9. SupposethatwedevelopD2byusingthedivergenceoforder2.Then0increasestoasddecreasestozero.Furthermore,thevalueofdatasatisesVoD=1 2(d(N)+1)2"(22)]TJ /F3 11.955 Tf 11.95 0 Td[(2+1)d(N)+2()]TJ /F4 11.955 Tf 11.96 0 Td[(2) p d(N)2+4d(N)()]TJ /F4 11.955 Tf 11.95 0 Td[(2))]TJ /F3 11.955 Tf 11.96 0 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[(2)#d0(N), whered0(N)representsthederivativeofd(N)overN. Proposition10. SupposethatwedevelopD2byusingtheKLdivergence.Then0increasestoasddecreasestozero.Furthermore, wecanobtain d=log 0+(1)]TJ /F4 11.955 Tf 11.96 0 Td[()log1)]TJ /F4 11.955 Tf 11.96 0 Td[( 1)]TJ /F4 11.955 Tf 11.96 0 Td[(0, andthevalueofdatasatisesVoD=0(1)]TJ /F4 11.955 Tf 11.96 0 Td[(0) 0)]TJ /F4 11.955 Tf 11.96 0 Td[(d0(N), whered0(N)representsthederivativeofd(N)overN. WeremarkthatthevalueofVoDdependsonthechoiceofd(N),i.e.,therelationshipbetweenthe-divergencetoleranced and thedatasamplesizeN.Fordiscretedistributions,thechoiced(N):=00(1)2B)]TJ /F12 7.97 Tf 6.59 0 Td[(1,1)]TJ /F6 7.97 Tf 6.59 0 Td[(=(2N)andaccordinglyVoD 44

PAGE 45

becomesaccurateasNincreasestoinnity.Forcontinuousdistributions,d(N)isusedtoapproximateD2,andaccordinglyVoDcannotguaranteetobeaccurateingeneral.Inpractice,wecanuseVoDtohelpdecisionmakersapproximatelyestimatethevalueofanewsetofdata,andaccurateestimatorsworthfuturestudies.Forillustration,wedepictthe relationships between1)]TJ /F4 11.955 Tf 12.41 0 Td[(0andNandbetweenVoDandNinFigure 2-1 byusingKLdivergenceundercondencelevelswith=0.90andbinsizeB=30 (e.g.,fordiscretedistributions) W eobservethatwithsamplesizeNincreas ing ,both1)]TJ /F4 11.955 Tf 12.18 0 Td[(0andVoDdecayquickly.However,weneedalargesamplesize(e.g.,N>2000)toguarantee0convergingtoandVoDconvergingtozero.Thatis,wehavetodraw a large set of historical datatoguaranteeanalmostexactdescriptionoftheunknownprobabilitydistributionP,andaccordinglytherisklevelof the reformulatedDCC,0,canbechosentobenearitsdeterministiccounterpart.Thisobservationmakessense because weneedalargesamplesizetoconstructthehistogramintherstplace, especiallywhenthedimensionofrandomvectorsbecomesverylarge .Inpractice,wesuggest using density-basedcondencesetsin an industrythathasrichaccesstodataandreliesheavilyondatatomakedecisions.Ascomparedtodensity-basedcondencesets,moment-basedcondencesetsaremoreconservative and onlyneedsmalldataset s forconstruction,andhence are suitabletobeusedin an industrythathaslimitedaccesstodata. 2.4NumericalExperiments Inthissection,weconductasimplenumericalexperimenttoillustratetheapplicationofDCCPs.WemodelDCCswithbothmoment-basedanddensity-basedcondencesetsinaportfoliooptimizationproblem.Inthisexperiment,agenericDCCP 45

PAGE 46

A1)]TJ /F25 9.963 Tf 9.96 0 Td[(0v.s.N BVoDv.s.N Figure2-1. Evolutionofvalues1)]TJ /F4 11.955 Tf 11.96 0 Td[(0andVoDagainstsamplesizeunderrisklevel=0.90andcondencelevels=0.01,0.05,0.10 for the portfoliooptimizationproblemcanbeformulatedas[DCPO]maxx0nXi=1E[i]xis.t.infP2DP8>><>>:Pni=1ixiT0(Pni=1xi),Pi2NjixiTj(Pi2Njxi),8j=1,...,J9>>=>>;1)]TJ /F4 11.955 Tf 11.95 0 Td[(,nXi=1xi=1, wherenrepresentsthetotalnumberofinvestments,irepresentstherateofreturn(RoR)ofinvestmenti,xirepresentstheshareofinvestmenti,N1,...,NJrepresentdifferentportfoliosegments withSJj=1Nj=f1,...,ng (e.g.,N1consistsofstocks,N2consistsofbonds,andsoon),andT0,...,TJrepresentthe investment targetsofdifferentportfoliosegments.Tospecifywhichcondencesetweuse,wedenote[DCPO-M]whenusing a moment-basedcondenceset( e.g. ,D=D1),anddenote[DCPO-D]whenusing a density-basedcondenceset( e.g. ,D=D2undertheKLdivergence).InspiredbyDelageandYe[ 23 ],weevaluate[DCPO]inthisexperimentbyusingahistoricaldatasetof30assetsfromyears2008to2011,obtainedfromthe 46

PAGE 47

Yahoo!Financewebsite1.Ineachexperiment,werandomlychoose4assets, randomlyassignthemintoJ=2portfoliosegments, andbuildadynamicportfoliowiththeseassets. Theassetsintheportfolioareupdatedeverythirtytransactiondays ,throughyears2008to2011,byadoptingoptimalinvestmentdecisionsobtainedfrom[DCPO]. During anydayof the experiment,wecollectthemostrecent2000daysofRoRdatatoconstructbothcondencesetsD1andD2. Inthisexperiment,weestimateeachmeanRoR,i.e.,E[i]foreachi=1,...,n,bythesamplemeanRoRbasedonthemostrecent2000daysdata.WeemploytheconservativeapproximationdiscussedinSection 2.2.3 tosolve[DCPO-M],andthescenarioapproximationapproachtosolve[DCPO-D]. Inthisexperiment,weevaluatetheperformanceoftheinvestmentdecisionsobtainedfrom[DCPO] during eachtradingday against therealdatainthefollowingthirtydays.Thatis,aftermakingtheinvestmentdecision during eachtradingday,wewillholdtheassetsforthirtydaysandseehowitperformsintherealmarket.Tosetabenchmark,wecompare[DCPO-M]and[DCPO-D]toamyopicmodel,whichmaximizesanaveragereturnoverthelast2000days.Inthisexperiment,werun100replicationsintotalbychoosingfourassets (amongthe30assets) ineachreplication,andsummarizetheresultsobtainedfromallreplicationsinTable 2-1 andFigure 2-2 Table2-1. Comparisonofaverageendwealthandriskin100replicationsthroughyears2008-2011 Avg.St.dev10thPerc.90thPerc. DCPO-D1.1180.1940.9691.415DCPO-M1.0100.1660.8231.245Myopic0.9910.5070.4991.394 1The30assetsareAARCorp.,AT&T,AveryDenisonCorp.,BoeingCorp.,Bristol-Myers-Squibb,CiscoSystems,DellComputerCorp.,DowChemical,DukeEnergyCompany,DuPont,EliLillyandCo.,ExelonCorp.,FMCCorp.,GeneralElectric,HewlettPackard,Hitachi,Honeywell,IBMCorp.,IngersollRand,IntelCorp.,LockheedMartin,MerckandCo.,Microsoft,Motorola,NorthernTelecom,Oracle,PinnacleWest,TexasInstruments,UnitedTechnologiesanda0%-interest-ratedeposit. 47

PAGE 48

FromTable 2-1 andFigure 2-2 ,weobservethatboth[DCPO-D]and[DCPO-M]outperformtheMyopicapproachinbothendwealthandriskcontrol forthe years2008-2011.Inparticular,[DCPO-D]largelyoutperformstheothertwoapproacheswithatleast9.6%moreinendwealthandamuchsmallerstandarddeviationascomparedtotheMyopicapproach.Itindicatesthatthedensity-basedDCCPscanmakerobustandprotableportfolioselection.[DCPO-M]slightlyoutperformstheMyopicapproachintermsofendwealth,buthasthesmalleststandarddeviation.FromFigure 2-2 ,wecanalsoobservethat[DCPO-M]clearlyoutperformstheothertwoapproachesduringyears2008-2009whenthemarketisdepressed.Thisobservationmakessensebecause[DCPO-M]considersmomentambiguityandweemployaconservativeapproximationtosolve[DCPO-M]. Figure2-2. Comparisonofwealthevolutionin100replicationsthroughyears2008-2011 2.5ConcludingRemarks Inthischapter,wedevelop ed exactandapproximateapproachesforDCCPs.Startingfromthehistoricaldata,wedescribedhowtoconstructmoment-basedanddensity-basedcondencesetsfortheambiguousprobabilitydistributions,howtoequivalentlyreformulateDCCs,andhowtoeffectivelysolveDCCPs.Ingeneral,inthis 48

PAGE 49

study,weproposedaframeworktoproviderobustdecisionsbasedontheavailabledatasetinformation.Besidesguaranteeingtherobustness,ourframeworkensuresthattheproposedapproachislessconservativewhenmoredatainformationisonhand.PossiblefutureresearchdirectionsincludethestudyofDCCsunderdifferentcondencesetsandtheirsolutionapproaches.Itisalsointerestingtostudytheaccurateestimatorsofthevalueofdataforgeneral-divergence. 49

PAGE 50

CHAPTER3RISK-AVERSETWO-STAGESTOCHASTICPROGRAMWITHDISTRIBUTIONALAMBIGUITY 3.1MotivationandLiteratureReview Stochasticprogrammingprovidesaneffectiveandconvenientapproachtoassistdecisionmakingunderuncertainty.Bycapturingtheuncertainparametersbyitsprobabilitydistribution,adecisionmakeraimstosolveanoptimizationproblem minx2XE[ (x,)],(3) wherexrepresentsthedecisionstomake,Xdescribestheconstraintsorrequirementsonx,and (,)representsageneralcostfunctionunderuncertaintydescribedbyaK-dimensionalrandomvectoronaprobabilityspace(,F,P),whereRKrepresentsthesamplespacefor,Frepresentsa-algebraon,andPrepresentstheunderlyingprobabilitydistributionof.Byconsideringitsexpectationasaproxyoftheuncertaincostfunction,problem 3 makesdecisioninarisk-neutralmanner.Followingtheseminalworks(see,e.g.,[ 4 ]and[ 22 ])datedbacktothe1950s,signicantresearchhasbeenmadeonaspecicclassofSPmodelscalledtwo-stagestochasticprogramming(TSP)intheform (TSP)minx2Xc>x+E[Q(x,)],(3)Q(x,)representsthevaluefunctionofanoptimizationproblemforgivenxandrealizedrandomvector,i.e., Q(x,)=minyq>y (3a)s.t.Wy+TxS, (3b) (TSP)dividesthedecisionmakingprocessintotwostages,wherexrepresentsasetofhere-and-nowdecisionswhichhavetobemadebeforetherandomvectorrealizesandyrepresentsasetofwait-and-seedecisionswhichcanbemadeafter 50

PAGE 51

therealizationofasarecourse.Todescribethecostsincurredandconstraintstorespect,weletc2Rn1representthecostvectorfortherst-stageproblem,X:=fx2R(n1)]TJ /F8 7.97 Tf 6.58 0 Td[(p)Bp:Axbgrepresentthefeasibleregionfortherst-stagedecisionxwithA2Rm1n1,b2Rm1,andp2[0,n1]\(e.g.,p>0impliesintegralconstraintsforX),q2Rn2representthecostvectorforthesecond-stageproblem,W2Rm2n2representthetechnologymatrix,T2Rm2n1representtherecoursematrix,andS2Rm2Krepresenttheinvolvementofrandomvectoroneachsecond-stageconstraint.Fromthestandpointofadecisionmaker,(TSP)providesaviableandexibleapproachtoassistdecisionmakingunderanuncertainenvironmentwherewehavetomakesomecrucialdecisionswhileatleastpartoftheproblemparametersarenotknownorcannotbeaccuratelyestimated.Sinceproposed,(TSP)hasbeenappliedinavarietyofpracticalproblemsincludingpowersystemoperation(see,e.g,[ 64 ]),telecommunicationnetworkplanning(see,e.g.[ 57 ]),supplychainmanagement(see,e.g.,[ 56 ]),andmanyothers(see[ 33 ],[ 11 ],[ 59 ],andreferencestherein). Although(TSP)emergesnaturallyasamodelingtoolandiswidelyappliedinpractice,therearestillchallenges.Forexample,(TSP)assumesthatthedecisionmakerisrisk-neutralandevaluatesthecostbyitsexpectation.Thisassumptiondoesnotapplytotherisk-aversedecisionmakersthattakeadditionalcareofthose`risky'scenarioswherehigher-than-expectationcostsareincurred,andsothesolutionto(TSP)canbecomesuboptimalinarisk-aversecircumstance.Moreimportantly,(TSP)assumesthatwecanaccuratelyestimatetheunderlyingprobabilitydistributionPwhenxisdecided.ThisassumptionisnotnecessarilypracticalbecauseourknowledgeofPisusuallyambiguous,andwhatwehaveisonlyaseriesofdatasamplesfngNn=1whichcanbecollectedfromP.Basedonthesesamples,apointestimatorP0ofPcanbebiasedandaccordinglythesolutionto(TSP)canbecomesuboptimalwithoutconsideringthedistributionalambiguity.Inthischapter,weproposearisk-averseTSP(RTSP)modeltoaddressthesechallengesbyexplicitlyincorporatingthedistributional 51

PAGE 52

ambiguityintoaclassicalTSPmodel.Startingfromasetofhistoricaldatasamples,weconstructacondencesetfortheambiguousprobabilitydistributionthroughnonparametricstatisticalestimationofitsdensityfunction.WethenformulateRTSPfromtheperspectiveofdistributionalrobustnessbyallowingtheambiguousdistributiontorunadverselywithinitscondenceset.InSection 3.2 ,westatetheRTSPmodelandderiveanequivalentreformulationwhichreectsitsrisk-aversenature.Furthermore,weperformconvergenceanalysisfortheoptimalobjectivevalueoftheRTSPmodel,andshowthatitconvergestothatofTSPasthesamplesizegrowstoinnity.ThisobservationindicatesthattheRTSPmodelcanbeappliedinadata-drivendecisionmakingschemewhereadecisionmakercanadjustherrisk-aversenessaccordingtothedatasamplesshehasonhand.InSection 3.3 ,weinvestigateasolutionalgorithmforRTSPbasedonthesampleaverageapproximation(SAA)method.WealsoshowthattheoptimalobjectivevalueoftheSAAproblemconvergestothatoftheRTSPmodelasthesamplesizeoftheSAAmethodgrowstoinnity.Finally,wesummarizethechapterinSection 3.4 3.2Risk-AverseTwo-StageStochasticProgram Inthissection,weinvestigatearisk-aversetwo-stagestochasticprogram(RTSP)asfollows: (RTSP)minx2Xc>x+supP2DE[Q(x,)],(3) whereDisacondencesetfortheambiguousprobabilitydistributionP.Ascomparedto(TSP)presentedin 3 3 ,(RTSP)allowstheambiguousdistributiontorunadverselywithinitscondencesetD.Weobservethat(RTSP)possessesthefollowingfeatures: 1. Intermsofmodeling,(RTSP)reectsbothcosteffectivenessandreliabilityinanexplicitway.Oneonehand,itattemptstoachievecosteffectivenessbyminimizinganexpectedtotalcost;ontheotherhand,ittakescareofreliabilitybyconsideringaworst-caseprobabilitydistribution.Inthissense,(RTSP)liesinamiddlestripbetweenthestochasticoptimizationandtherobustoptimizationmodels. 52

PAGE 53

2. Intermsofrisk-averseness,theobjectivefunctionof(RTSP)isequivalenttoacoherentriskmeasure.Indeed,aspointedoutinseveralpioneerworks(see,e.g.,[ 1 ],[ 24 ],and[ 29 ]),itcanbeshownthattheobjectivefunction 3 isequivalenttominx2Xc>x+[Q(x,)] foracoherentriskmeasureonRundersomeregularityconditions1.InSection 3.2.2 ,wespecifyasacoherentriskmeasurebyderivinganequivalentreformulationof 3 3. Intermsofapplicability,(RTSP)canbeappliedinadata-drivencircumstancewheredecisionmakerscanconstructDbyusingdatasamplesofcollectedinpractice.Furthermore,wecanexpectthattherisk-aversenessof(RTSP)decreasesasthesamplesizegrows.InSection 3.2.1 ,wediscusshowDcanbeconstructedinadata-drivenmanner,andweestablishconvergenceanalysisfortherisk-aversenessof(RTSP)asthesamplesizegrowsinSection 3.2.3 Inthischapter,weassumethat(RTSP)satisesthecompleterecourseproperty,i.e.,foreachx2Rn1and2,thereexistsysuchthatconstraints 3b aresatised.Thisassumptioncanbemadewithoutlossofgeneralitybecausewecanaddslackvariablesforconstraints 3b andcorrespondingpenaltycostsintheobjectivefunction 3a .WealsoassumethatvaluefunctionQ(x,)isboundedfrombelow,i.e.,Q(x,)>foranyx2Xand2.Thisassumptionismildbecauseotherwisethereexistssomerst-stagedecisionx0andascenario0suchthatwehaveancost(orequivalently+1revenue),whichdoesnotmakesenseinpractice. 3.2.1CondenceSetConstruction Inthissection,wediscusshowtoobtainapointestimatorP0forPandthecondencesetDaroundP0.SupposethatwecollectasetofdatasamplesfngNn=1fromtheambiguousdistributionP.Inpractice,histogramsareoftenusedtoestimateortoidentifytheproleofPifitisdiscrete.IfPiscontinuous,however,histogramsare 1Werefertheinterestedreaderstotheaforementionedpapersandthereferencesthereinforthedenitionofcoherentriskmeasuresandtheregularityconditions. 53

PAGE 54

notusuallyaccuratebecausetheyarenotabsolutelycontinuouswithregardtotheLeabesguemeasure.Inthischapter,wefocusonthecasewhenPiscontinuous,i.e.,thereexistsadensityfunctionf:RK!R,suchthatf=dP=d.WeproposetoestimatePbyusingthekerneldensityestimator(KDE)denedasf0()=1 NjHj1=2NXn=1P)]TJ /F9 11.955 Tf 5.48 -9.68 Td[(H)]TJ /F12 7.97 Tf 6.58 0 Td[(1=2()]TJ /F4 11.955 Tf 11.96 0 Td[(n), wheref0:RK!R+representsthedensityfunctionofP0onRK(i.e.,f0=dP0=d),H2RKKrepresentsasymmetricandpositivedenitebandwidthmatrix,andP:Rk!R+representsasymmetrickernelfunctionsatisfyingP()0,RP()d=1,andR2P()d>0.OneexampleforPisthestandardmultivariatenormaldensity,i.e.,P()=(2))]TJ /F8 7.97 Tf 6.59 0 Td[(K=2expf)]TJ /F3 11.955 Tf 15.27 0 Td[((1=2)>g.Itisshownin[ 25 ]thatf0convergestofinL1-normwithprobabilityone,i.e.,ZRKjf())]TJ /F9 11.955 Tf 11.95 0 Td[(f0()jd!0asN!1. ThisobservationmotivatesustoconstructacondencesetDaroundP0basedontheL1-normasfollows: D=P2M+:ZRKjf())]TJ /F9 11.955 Tf 11.96 0 Td[(f0()jdd,f=dP=d,(3) whereM+representsthesetofallprobabilitydistributions,andDincorporatesalltheprobabilitydistributionswhoseL1-metricdistanceawayfromthepointestimatorP0isboundedbyadistancetoleranced.Inpractice,thevalueofdcanbechosenbydecisionmakerstoreecttheirrisk-aversenessorperformpost-optimizationsensitivityanalysis.Intuitively,inviewoftheL1convergence,thevalueofddecreasesassamplesizeNgrowsandconvergestozeroasNgoestoinnity. 54

PAGE 55

3.2.2EquivalentReformulation Inthissection,wederiveanequivalentreformulationfor(RTSP).Wederiveareformulationfortheworst-caseexpectationsupP2DE[g(x,)] forageneralcostfunctiong(x,),andthenweapplythisgeneralreformulationresultsto(RTSP).Werststatethegeneralreformulationresultasfollows. Theorem3. Foranyxedx2Rn1andageneralfunctiong(x,),wehavesupP2DE[g(x,)]=(1)]TJ /F9 11.955 Tf 11.96 0 Td[(d=2)CVaR(d=2)[g(x,)]+(d=2)max2g(x,) ifd2[0,2),andsupP2DE[g(x,)]=max2g(x,) ifd2. Proof. WebeginbystatingsupP2DE[g(x,)]asthefollowingoptimizationproblem: maxf()0Zg(x,)f()d (3a)s.t.Zjf())]TJ /F9 11.955 Tf 11.95 0 Td[(f0()jdd (3b)Zf()d=1, (3c) whereconstraints 3b 3c describeP2D,frepresentstheprobabilitydensityfunctionofPifhasacontinuousdistribution(i.e.,f=dP=d),ortheprobabilitymassfunctionifhasadiscretedistribution.TheLagrangiandualofproblem 3 canbewrittenasinfz0,z0supf()0Zg(x,)f()d+z01)]TJ /F15 11.955 Tf 11.96 16.27 Td[(Zf()d)]TJ /F9 11.955 Tf 11.95 0 Td[(zZjf())]TJ /F9 11.955 Tf 11.96 0 Td[(f0()j)]TJ /F9 11.955 Tf 17.93 0 Td[(d 55

PAGE 56

wherezandz0representdualvariablesforconstraints 3b and 3c respectively.Wethenproceedtothereformulationasfollows: infz0,z0supf()0Zg(x,)f()d+z01)]TJ /F15 11.955 Tf 11.95 16.28 Td[(Zf()d)]TJ /F9 11.955 Tf 11.95 0 Td[(zZjf())]TJ /F9 11.955 Tf 11.95 0 Td[(f0()j)]TJ /F9 11.955 Tf 17.94 0 Td[(d=infz0,z0z0+zd+supf()0Zh(g(x,))]TJ /F9 11.955 Tf 11.95 0 Td[(z0)f())]TJ /F9 11.955 Tf 11.95 0 Td[(zjf())]TJ /F9 11.955 Tf 11.95 0 Td[(f0()jid,=infz0,z0z0+zd+Zsupf()0h(g(x,))]TJ /F9 11.955 Tf 11.96 0 Td[(z0)f())]TJ /F9 11.955 Tf 11.96 0 Td[(zjf())]TJ /F9 11.955 Tf 11.96 0 Td[(f0()jid,=infz0,z0z0+zd+Z\[g(x,))]TJ /F8 7.97 Tf 6.59 0 Td[(z0<)]TJ /F8 7.97 Tf 6.59 0 Td[(z][)]TJ /F9 11.955 Tf 9.3 0 Td[(zf0()]d+Z\[)]TJ /F8 7.97 Tf 6.58 0 Td[(zg(x,))]TJ /F8 7.97 Tf 6.58 0 Td[(z0z](g(x,))]TJ /F9 11.955 Tf 11.96 0 Td[(z0)f0()d (3a)s.t.max2g(x,)z0+z (3b)=infz0,z0z0+zd+Z[g(x,))]TJ /F8 7.97 Tf 6.59 0 Td[(z0<)]TJ /F8 7.97 Tf 6.59 0 Td[(z][)]TJ /F9 11.955 Tf 9.3 0 Td[(zf0()]d+Z[)]TJ /F8 7.97 Tf 6.59 0 Td[(zg(x,))]TJ /F8 7.97 Tf 6.58 0 Td[(z0](g(x,))]TJ /F9 11.955 Tf 11.96 0 Td[(z0)f0()d (3c)s.t.max2g(x,)z0+z=infz0,z0z0+zd+Z[g(x,))]TJ /F8 7.97 Tf 6.59 0 Td[(z0<)]TJ /F8 7.97 Tf 6.59 0 Td[(z][)]TJ /F9 11.955 Tf 9.3 0 Td[(zf0()]d+ZRK(g(x,))]TJ /F9 11.955 Tf 11.96 0 Td[(z0)f0()d)]TJ /F15 11.955 Tf 11.95 16.28 Td[(Z[g(x,))]TJ /F8 7.97 Tf 6.59 0 Td[(z0<)]TJ /F8 7.97 Tf 6.59 0 Td[(z](g(x,))]TJ /F9 11.955 Tf 11.95 0 Td[(z0)f0()d (3d)s.t.max2g(x,)z0+z=infz0,z0zd+EP0[g(x,)]+Z[z0)]TJ /F8 7.97 Tf 6.58 0 Td[(z)]TJ /F8 7.97 Tf 6.58 0 Td[(g(x,)>0](z0)]TJ /F9 11.955 Tf 11.96 0 Td[(z)]TJ /F9 11.955 Tf 11.96 0 Td[(g(x,))f0()d (3e)s.t.max2g(x,)z0+z=infz0zd+EP0[g(x,)]+Z[m)]TJ /F12 7.97 Tf 6.58 0 Td[(2z)]TJ /F8 7.97 Tf 6.59 0 Td[(g(x,)0](m)]TJ /F3 11.955 Tf 11.96 0 Td[(2z)]TJ /F9 11.955 Tf 11.96 0 Td[(g(x,))f0()d, (3f) whereequality 3a followsbysolvingtheone-dimensionalproblemV(z0,z):=supf()0h(g(x,))]TJ /F9 11.955 Tf 11.95 0 Td[(z0)f())]TJ /F9 11.955 Tf 11.95 0 Td[(zjf())]TJ /F9 11.955 Tf 11.96 0 Td[(f0()ji 56

PAGE 57

foreach2,whichhastheoptimalobjectivevalueV(z0,z)=8>>>>><>>>>>:)]TJ /F9 11.955 Tf 9.29 0 Td[(zf0(),ifg(x,))]TJ /F9 11.955 Tf 11.95 0 Td[(z0z, andhenceweshouldaddconstraints 3b intothedualproblem.Equality 3c isduetoconstraints 3b andthefactsthatf0()=0foreach=2.Equality 3d followsfromthepartitionRK=[)]TJ /F9 11.955 Tf 9.3 0 Td[(zg(x,))]TJ /F9 11.955 Tf 11.96 0 Td[(z0][[g(x,))]TJ /F9 11.955 Tf 11.96 0 Td[(z0<)]TJ /F9 11.955 Tf 9.3 0 Td[(z].Equality 3e isduetoZRK(g(x,))]TJ /F9 11.955 Tf 11.96 0 Td[(z0)f0()d=ZRKg(x,)f0()d)]TJ /F15 11.955 Tf 11.96 16.27 Td[(ZRKz0f0()d=EP0[g(x,)])]TJ /F9 11.955 Tf 11.95 0 Td[(z0, whereP0representstheprobabilitymeasureinducedbyf0()(i.e.,P0(A)=RAf0()dforeachARK).Equality 3f isbecausez0=m)]TJ /F9 11.955 Tf 12.5 0 Td[(zisanoptimalsolutiontotheoptimizationproblem 3e overvariablez0foraxedz,wherem:=max2g(x,).Toseethis,weconsidertwofeasiblesolutionz20>z10m)]TJ /F9 11.955 Tf 11.96 0 Td[(z.ThenwehaveZ[z20)]TJ /F8 7.97 Tf 6.59 0 Td[(z)]TJ /F8 7.97 Tf 6.59 0 Td[(g(x,)>0](z20)]TJ /F9 11.955 Tf 11.96 0 Td[(z)]TJ /F9 11.955 Tf 11.95 0 Td[(g(x,))f0()d)]TJ /F15 11.955 Tf 11.95 16.27 Td[(Z[z10)]TJ /F8 7.97 Tf 6.58 0 Td[(z)]TJ /F8 7.97 Tf 6.58 0 Td[(g(x,)>0](z10)]TJ /F9 11.955 Tf 11.96 0 Td[(z)]TJ /F9 11.955 Tf 11.96 0 Td[(g(x,))f0()d=Z[z10)]TJ /F8 7.97 Tf 6.59 0 Td[(z)]TJ /F8 7.97 Tf 6.59 0 Td[(g(x,)>0](z20)]TJ /F9 11.955 Tf 11.96 0 Td[(z)]TJ /F9 11.955 Tf 11.95 0 Td[(g(x,))f0()d+Z[z10)]TJ /F8 7.97 Tf 6.58 0 Td[(zg(x,)0](z10)]TJ /F9 11.955 Tf 11.96 0 Td[(z)]TJ /F9 11.955 Tf 11.96 0 Td[(g(x,))f0()d (3)=Z[z10)]TJ /F8 7.97 Tf 6.59 0 Td[(z)]TJ /F8 7.97 Tf 6.59 0 Td[(g(x,)>0](z20)]TJ /F9 11.955 Tf 11.96 0 Td[(z10)f0()d+Z[z10)]TJ /F8 7.97 Tf 6.58 0 Td[(zg(x,)0]=[z10)]TJ /F9 11.955 Tf 12.41 0 Td[(z)]TJ /F9 11.955 Tf 12.41 0 Td[(g(x,)>0][[z10)]TJ /F9 11.955 Tf 12.63 0 Td[(zg(x,)0.Hence,z0=m)]TJ /F9 11.955 Tf 11.95 0 Td[(zisoptimaltoproblem 3e 57

PAGE 58

Itremainstosolvetheone-dimensionaloptimizationproblem 3f .Leth(z)representitsobjectivefunctionandwehaveh0(z)=d)]TJ /F3 11.955 Tf 11.95 0 Td[(2Z[m)]TJ /F12 7.97 Tf 6.58 0 Td[(2z)]TJ /F8 7.97 Tf 6.59 0 Td[(g(x,)]f0()d,h00(z)=4f0(m)]TJ /F3 11.955 Tf 11.96 0 Td[(2z)0. Itfollowsthath(z)isconvexoverz,andhencewehavethefollowingtwocases (i) Ifd2,wehaved=21andhenceh0(z)=2[d=2)]TJ /F11 11.955 Tf 11.96 0 Td[(P0fg(x,)m)]TJ /F3 11.955 Tf 11.95 0 Td[(2zg]0.Therefore,z=0isoptimaltoproblem 3f whoseoptimalobjectivevalueish(0)=EP0[g(x,)]+Z[m)]TJ /F8 7.97 Tf 6.58 0 Td[(g(x,)0](m)]TJ /F9 11.955 Tf 11.95 0 Td[(g(x,))f0()d=EP0[g(x,)]+ZRK(m)]TJ /F9 11.955 Tf 11.95 0 Td[(g(x,))f0()d (3)=EP0[g(x,)]+m)]TJ /F11 11.955 Tf 11.95 0 Td[(EP0[g(x,)]=m, whereequality 3 isduetog(x,)mforeach2andf0()=0foreach=2.ThisprovesthatsupP2DE[g(x,)]=max2g(x,)whend2. (ii) Ifd2[0,2),xingh0(z)=0yieldsP0fg(x,)m)]TJ /F3 11.955 Tf 11.96 0 Td[(2zg=d 2, whichisequivalentto m)]TJ /F3 11.955 Tf 11.96 0 Td[(2z=inff:P0fg(x,)gd=2g=VaR(d=2)[g(x,)].(3) 58

PAGE 59

Itfollowsthat h(z)=zd+EP0[g(x,)]+Z[g(x,)VaR(d=2)[g(x,)]])]TJ /F1 11.955 Tf 5.48 -9.68 Td[(VaR(d=2)[g(x,)])]TJ /F9 11.955 Tf 11.96 0 Td[(g(x,)f0()d=(d=2)m)]TJ /F3 11.955 Tf 11.95 -.17 Td[((d=2)VaR(d=2)[g(x,)]+ZRKg(x,)f0()d+Z[g(x,)VaR(d=2)[g(x,)]]VaR(d=2)[g(x,)]f0()d)]TJ /F15 11.955 Tf 11.95 16.27 Td[(Z[g(x,)VaR(d=2)[g(x,)]]g(x,)f0()d (3a)=(d=2)m+VaR(d=2)[g(x,)])]TJ /F11 11.955 Tf 5.48 -9.69 Td[(P0fg(x,)VaR(d=2)[g(x,)]g)]TJ /F9 11.955 Tf 20.59 0 Td[(d=2+Z[g(x,)>VaR(d=2)[g(x,)]]g(x,)f0()d (3b)=(d=2)m+(1)]TJ /F9 11.955 Tf 11.95 0 Td[(d=2)P0fg(x,)VaR(d=2)[g(x,)]g)]TJ /F9 11.955 Tf 20.59 0 Td[(d=2 1)]TJ /F9 11.955 Tf 11.96 0 Td[(d=2VaR(d=2)[g(x,)]+P0fg(x,)>VaR(d=2)[g(x,)]g 1)]TJ /F9 11.955 Tf 11.96 0 Td[(d=2CVaR+(d=2)[g(x,)] (3c)=(1)]TJ /F9 11.955 Tf 11.96 0 Td[(d=2)CVaR(d=2)[g(x,)]+(d=2)m, (3d) whereequality 3a followsfromequation 3 andthedenitionofEP0[g(x,)],equality 3b combinestermsandusesthefactthatZ[g(x,)VaR(d=2)[g(x,)]]f0()d=P0fg(x,)VaR(d=2)[g(x,)]g, equalities 3c and 3d areduetothedenitionofCVaR+(d=2)[g(x,)]andCVaR(d=2)[g(x,)](see[ 52 ]forreference),respectively.Thiscompletestheproofforthecased2[0,2). Remark6. Theorem 3 speciesthecoherentriskmeasure(g(x,))asaconvexcombinationoftheconditionalvalue-at-riskandtheworst-casecost.TheweightofCVaRis1)]TJ /F3 11.955 Tf 11.95 0 Td[(minfd=2,1gandthatoftheworst-casecostisminfd=2,1g. 59

PAGE 60

Remark7. Theorem 3 developsafullspectrumofriskmeasuresfromtheworst-casecostatthemostconservativeextremetoexpectationattheleastconservativeextreme.Specically,itindicatesthat(i)whenthesamplesizeislimited(andaccordinglydissufcientlylarge),wehavetoprotecttheworst-casescenariosof,(ii)whenthesamplesizeincreasesanddbecomessmaller,theweightofCVaRgrowsandthatoftheworst-casecostreduces,and(iii)asthesamplesizegoestoinnityanddgoestozero,theriskmeasureconvergestotheexpectation. NextweapplyTheorem 3 to(RTSP)whichleadstothefollowingproposition. Proposition11. Ifd2[0,2),then(RTSP)isequivalenttothefollowingoptimizationproblemminx2X,2Rc>x+(1)]TJ /F9 11.955 Tf 11.96 0 Td[(d=2)+ZRK(Q(x,))]TJ /F4 11.955 Tf 11.95 0 Td[()+f0()d+(d=2)max2Q(x,). Otherwise,ifd2,then(RTSP)isequivalenttominx2Xc>x+max2Q(x,). Proof. Ifd2,theconclusionfollowsimmediatelyfromTheorem 3 Ifd2[0,2),byTheorem 3 ,(RTSP)isequivalenttotheoptimizationproblemminx2Xc>x+(1)]TJ /F9 11.955 Tf 11.96 0 Td[(d=2)CVaR(d=2)[g(x,)]+(d=2)max2Q(x,). TheconclusionfollowsfromthefactthatCVaR(d=2)[g(x,)]=min2R+1 1)]TJ /F9 11.955 Tf 11.95 0 Td[(d=2ZRK(Q(x,))]TJ /F4 11.955 Tf 11.95 0 Td[()+f0()d, whichisduetoTheorem10in[ 52 ].Thiscompletestheproof. 3.2.3ConvergenceAnalysis Inthissection,weanalyzetheconvergencepropertyof(RTSP)asthesamplesizegrowstoinnity(andaccordinglyddecreasestozero).Wendthatasddecreasestozero,theoptimalobjectivevalueof(RTSP)convergestothatof(TSP),with 60

PAGE 61

theambiguousdistributionPreplacedwithitspointestimatorP0,undersomemildconditions.Thisresultconrmsanintuitionthattherisk-aversenessof(RTSP)vanishesasthesamplesizegrowstoinnity.Wesummarizeourmainresultinthefollowingtheorem. Theorem4. Letz(d)representtheoptimalobjectivevalueof(RTSP)withregardtoadistancetoleranced.Thenz(d)convergestotheoptimalobjectivevalueof(TSP)withregardtoP0,i.e.,limd!0z(d)=z(0) ifXiscompact. Remark8. TheconditionXbeingcompactissatisedwhen(i)rst-stagedecisionvariablesxarepure-integral,and(ii)rst-stagedecisionvariablesxarepure-continuousormixed-integral,andisbounded(e.g.,inapolytope).Thisconditionismildbecauseitissatisedinmostofpracticalproblems. ToproveTheorem 4 ,westateandprovethefollowinglemmaswhichareusedlater. Lemma3. ExpectedvaluefunctionEP0[Q(x,))]TJ /F4 11.955 Tf 12.15 0 Td[(]+iscontinuousovervariables(x,)onRn1+1. Proof. Consideranypoint(x,)2Rn1+1andanysequencef(xi,i)gi2NinRn1+1convergingto(x,).First,foreach2,since(i)Q(x,)isconvexoverxonRn1,(ii)Q(x,)<+1foreachx2Rn1bycompleterecourse,and(iii)functiong(x):=(x)+isconvexandnondecreasing,(Q(x,))]TJ /F4 11.955 Tf 11.25 0 Td[()+isconvexandcontinuousover(x,)onRn1+1,i.e., limi!1(Q(xi,))]TJ /F4 11.955 Tf 11.96 0 Td[(i)+=(Q(x,))]TJ /F3 11.955 Tf 12.8 2.65 Td[()+,82.(3) 61

PAGE 62

Second,since(Q(x,))]TJ /F4 11.955 Tf 11.96 0 Td[()+<+1foreach(x,)2Rn1+1andeach2,wehave limi!1EP0[Q(xi,))]TJ /F4 11.955 Tf 11.96 0 Td[(i]+=EP0hlimi!1(Q(xi,))]TJ /F4 11.955 Tf 11.95 0 Td[(i)+i (3a)=EP0[Q(x,))]TJ /F3 11.955 Tf 12.8 2.66 Td[(]+, (3b) whereequality 3a followsfromthedominatedconvergencetheorem,andequality 3b followsfromequation 3 .Therefore,theproofiscomplete. Lemma4. Worst-casevaluefunctionmax2Q(x,)isconvexandcontinuousovervariablesxonRn1. Proof. First,sinceQ(x,)isconvexoverxonRn1foreach2,worst-casevaluefunctionmax2Q(x,)isalsoconvexoverxonRn1becausethemaximizationoperatorpreservesconvexity. Second,sincemax2Q(x,)isconvexoverxonRn1andmax2Q(x,)<+1foreachx2Rn1bycompleterecourse,continuityofmax2Q(x,)followsfromconvexity. Lemma5. (TSP)withregardtoP0isequivalenttothefollowingoptimizationproblem:minx2X,2Rc>x++ZRK(Q(x,))]TJ /F4 11.955 Tf 11.95 0 Td[()+f0()d. Proof. First,let=minx2Xmin2Q(x,).SinceQ(x,)>foranyx2Xand2byassumption,>.Itfollowsthat minx2X,2Rc>x++ZRK(Q(x,))]TJ /F4 11.955 Tf 11.95 0 Td[()+f0()dminx2Xc>x++ZRK(Q(x,))]TJ /F4 11.955 Tf 11.96 0 Td[()+f0()d (3a)=minx2Xc>x++ZRK(Q(x,))]TJ /F4 11.955 Tf 11.96 0 Td[()f0()d (3b)=minx2Xc>x+ZRKQ(x,)f0()d, 62

PAGE 63

whereinequality 3a isbecausewex=andsoobtainanupperbound,andequality 3b isbecauseQ(x,)foreachx2Xand2. Second,inviewthat(Q(x,))]TJ /F4 11.955 Tf 11.95 0 Td[()+Q(x,))]TJ /F4 11.955 Tf 11.96 0 Td[(,wehaveminx2X,2Rc>x++ZRK(Q(x,))]TJ /F4 11.955 Tf 11.95 0 Td[()+f0()dminx2Xc>x++ZRK(Q(x,))]TJ /F4 11.955 Tf 11.95 0 Td[()f0()d=minx2Xc>x+ZRKQ(x,)f0()d, whichcompletestheproof. NowwearereadytoproveTheorem 4 asfollows. (ProofofTheorem 4 )First,wecanassumethatd2withoutlossofgeneralitybecauseweanalyzethecasewhendgoestozero.Weintroducesomeadditionalnotationforpresentationbrevity.Welet^h(x,,d)representtheobjectivefunctionof(RTSP)reformulationwithregardtodistancetoleranced,i.e.,^h(x,,d):=c>x+(1)]TJ /F9 11.955 Tf 11.96 0 Td[(d=2)+ZRK(Q(x,))]TJ /F4 11.955 Tf 11.95 0 Td[()+f0()d+(d=2)max2Q(x,), andh(x,)representtheobjectivefunctionof(TSP)byLemma 5 ,i.e.,h(x,):=c>x++ZRK(Q(x,))]TJ /F4 11.955 Tf 11.95 0 Td[()+f0()d Second,sinceEP0[Q(x,))]TJ /F4 11.955 Tf 12.65 0 Td[(]+iscontinuousover(x,)onRn1+1byLemma 3 ,h(x,)iscontinuousover(x,)onXR.Itfollowsthatforany>0,thereexistsaneighborhoodU1(x,)foreach(x,)2XR,suchthatU1(x,)XRand sup(x,)2U1(x,)jh(x,))]TJ /F9 11.955 Tf 11.95 0 Td[(h(x,)j=3.(3) Third,sincemax2Q(x,)iscontinuousoverxonXbyLemma 4 ,^h(x,,2)iscontinuousover(x,)onXRwhendisxedat2.Itfollowsthatforany>0,there 63

PAGE 64

existsaneighborhoodU2(x,)foreach(x,)2XR,suchthatU2(x,)XRand sup(x,)2U2(x,)j^h(x,,2))]TJ /F3 11.955 Tf 12.05 2.66 Td[(^h(x,,2)j=3.(3) Furthermore,wedene U(x,)=U1(x,)\U2(x,)(3) foreach(x,)2XR.Since^h(x,,d)=(1)]TJ /F9 11.955 Tf 12.3 0 Td[(d=2)h(x,)+(d=2)^h(x,,2),wehave^h(x,,d)maxfh(x,),^h(x,,2)gforeachd2[0,2]andsosup(x,)2U(x,)j^h(x,,d))]TJ /F3 11.955 Tf 12.06 2.65 Td[(^h(x,,d)jmax(sup(x,)2U(x,)j^h(x,,2))]TJ /F3 11.955 Tf 12.05 2.66 Td[(^h(x,,2)j,sup(x,)2U(x,)jh(x,))]TJ /F9 11.955 Tf 11.95 0 Td[(h(x,)j)maxf=3,=3g==3, (3) whereinequality 3 followsfrominequalities 3 3 andtheinclusion 3 Fourth,sinceXiscompact, B:=maxx2Xmax2Q(x,)<+1andB :=minx2Xmin2Q(x,)>byassumption.Hence,wecanrestrictthat2[B B]forboth(TSP)and(RTSP)withoutlossofgenerality.Itfollowsthatforeachx2Xand2[B B],limd!0^h(x,,d)=h(x,)bythedenitionof^handh.Hence,thereexistsd(x,)>0suchthat j^h(x,,d))]TJ /F9 11.955 Tf 11.95 0 Td[(h(x,)j=3,8d2[0,d(x,)].(3) Furthermore,sinceX[B B]iscompact,thereexistsanitesetofpointsf(xm,m)gMm=1suchthat X[B B]M[m=1U(xm,m),(3) i.e.,foreach(x,)2X[B B],thereexistsanm02f1,...,Mgsuchthat(x,)2U(xm0,m0).Noteherethatm0dependson(x,)andwemarkthedependencybyaprimebynotationbrevity. 64

PAGE 65

Therefore,bychoosingd=minm=1,...,Md(xm,m)>0,wehave j^h(x,,d))]TJ /F9 11.955 Tf 11.96 0 Td[(h(x,)jj^h(x,,d))]TJ /F3 11.955 Tf 12.06 2.65 Td[(^h(xm0,m0,d)j+j^h(xm0,m0,d))]TJ /F9 11.955 Tf 11.96 0 Td[(h(xm0,m0)j+jh(xm0,m0))]TJ /F9 11.955 Tf 11.95 0 Td[(h(x,)j (3a)=3+=3+=3= (3b) foreach(x,)2X[B B]andeachd2[0,d],whereinequality 3a followsfromthetriangleinequality(usedtwice)andnitecoverage 3 ,andinequality 3b followsfrominequalities 3 3 3 ,inclusion 3 ,andthedenitionofd.Itfollowsthat^h(x,,d)uniformlyconvergestoh(x,)onX[B B]asdgoestozero. Finally,welet(x(d),(d))representanoptimalsolutionto(RTSP),and(x(0),(0))representanoptimalsolutionto(TSP).Itfollowsthatz(d))]TJ /F9 11.955 Tf 11.96 0 Td[(z(0)^h(x(0),(0),d))]TJ /F9 11.955 Tf 11.96 0 Td[(h(x(0),(0)) andz(0))]TJ /F9 11.955 Tf 11.96 0 Td[(z(d)h(x(d),(d)))]TJ /F3 11.955 Tf 12.06 2.66 Td[(^h(x(d),(d),d), foreachd2[0,d],whichfollowsfromtheuniformconvergencepropertyestablishedabove,andthesub-optimalityof(x(0),(0))for(RTSP)and(x(d),(d))for(TSP),respectively.Hence,jz(d))]TJ /F9 11.955 Tf 9.3 0 Td[(z(0)jmaxn^h(x(0),(0),d))]TJ /F9 11.955 Tf 11.96 0 Td[(h(x(0),(0)),h(x(d),(d)))]TJ /F3 11.955 Tf 12.06 2.65 Td[(^h(x(d),(d),d)o foreachd2[0,d].Thiscompletestheproof. 3.3SolutionApproaches Inthissection,weinvestigateasampleaverageapproximation(SAA)algorithmtosolve(RTSP).Specically,weresamplefromP0toobtainaseriesofsamplesfngNn=1, 65

PAGE 66

andthenformulateaSAAproblem ^zN(d)=minx2X,2Rc>x+(1)]TJ /F9 11.955 Tf 11.95 0 Td[(d=2)+1 NNXn=1(Q(x,n))]TJ /F4 11.955 Tf 11.96 0 Td[()++(d=2)maxn=1,...,NQ(x,n)(3) ifd2[0,2),whereweapproximatetheexpectationEP0[Q(x,))]TJ /F4 11.955 Tf 12.65 0 Td[(]+bythesampleaverage(1=N)PNn=1(Q(x,n))]TJ /F4 11.955 Tf 11.95 0 Td[()+andtheworst-casecostmax2Q(x,)bythesamplemaximummaxn=1,...,NQ(x,n).Ifd2,theSAAproblemreducestoascenario-basedapproximationforthetwo-stageadjustablerobustoptimizationproblem(see,e.g.,[ 6 ]and[ 8 ]) minx2Xc>x+maxn=1,...,NQ(x,n).(3) Inthissection,weanalyzelimitingbehaviorofSAAproblem 3 inapproximating(RTSP)asthesamplesizeNincreases.WeshowthatboththeoptimalobjectivevalueandthesetofoptimalsolutionsofSAAproblem 3 convergestothoseof(RTSP).Wesummarizethemainresultbythefollowingtheorem. Theorem5. LetSand^SNrepresentthesetofoptimalsolutionsof(RTSP)andSAAproblem 3 respectively.Supposethat(i)Xiscompact,and(ii)P0satisesthat:=P0Q(x,)>max2Q(x,))]TJ /F4 11.955 Tf 11.96 0 Td[(>0 forany>0.ThentheoptimalobjectivevalueofSAAproblem 3 convergestothatof(RTSP)asNincreasestoinnitywithprobability1,i.e.,limN!1^zN(d)=z(d).Furthermore,^SNconvergestoSasNincreasestoinnitywithprobability1inthesensethatlimN!1supx2^SNdist(x,S)=0. Proof. Let^h(x,,d)representstheobjectivefunctionof(RTSP),and^hN(x,,d).Weprovethatthepointwiseconvergenceholdbetween^hN(x,,d)and^h(x,,d),i.e.,foreach(x,)2XR,limN!1^hN(x,,d)=^h(x,,d) withprobability1.TheconclusionthenfollowsfromTheorem5.3in[ 59 ]. 66

PAGE 67

Foreach(x,)2XR,itisclearthatlimN!11 NNXn=1(Q(x,n))]TJ /F4 11.955 Tf 11.96 0 Td[()+=ZRK(Q(x,))]TJ /F4 11.955 Tf 11.95 0 Td[()+f0()d bythestronglawoflargenumbers.Itremainstoshowthatthesamplemaximumconvergestotheworst-casecost,i.e., limN!1maxn=1,...,NQ(x,n)=max2Q(x,).(3) Indeed,bycondition(ii)wehavethatforany>0,Pmaxn=1,...,NQ(x,n)>max2Q(x,))]TJ /F4 11.955 Tf 11.95 0 Td[(=1)]TJ /F9 11.955 Tf 11.95 0 Td[(Pmaxn=1,...,NQ(x,n)max2Q(x,))]TJ /F4 11.955 Tf 11.95 0 Td[(=1)]TJ /F15 11.955 Tf 21.56 11.35 Td[(Yn=1,...,NPQ(x,n)max2Q(x,))]TJ /F4 11.955 Tf 11.95 0 Td[(=1)]TJ /F3 11.955 Tf 11.95 0 Td[((1)]TJ /F4 11.955 Tf 11.95 0 Td[()N. Since>0,itfollowsthat1XN=1Pmaxn=1,...,NQ(x,n))]TJ /F3 11.955 Tf 11.96 0 Td[(max2Q(x,)>=1XN=1(1)]TJ /F4 11.955 Tf 11.95 0 Td[()N<1. Therefore,theconvergence 3 followsfromtheBorel-Cantellitheorem,whichcompletestheproof. Remark9. Condition(ii)ofTheorem 5 issatisedwhen(i)P0iscontinuouson(e.g.,P0isakerneldensityestimatoron),or(ii)P0isdiscreteandhasapositivemassoneachextremepointsof. 3.4ConcludingRemarks Inthischapter,weproposearisk-aversecounterpartfortheclassicaltwo-stagestochasticprogrambyexplicitlyincorporatingthedistributionalambiguity.Throughderivinganequivalentreformulationoftheproposedmodel,webuildtherelationship 67

PAGE 68

betweentheobjectivefunctionofthemodelandacoherentriskmeasure.Also,weproposeaSAA-basedsolutionalgorithmfortheproposedmodel.Furthermore,weanalyzetheconvergenceanalysisoftheproposedrisk-aversemodelandshowthatithasthepotentialtobeappliedinadata-drivendecisionmakingscheme. 68

PAGE 69

CHAPTER4RISK-AVERSESTOCHASTICUNITCOMMITMENTWITHINCOMPLETEINFORMATION 4.1ProblemDescriptionandLiteratureReview Renewableenergy(e.g.,windandsolar)isbittersweetintoday'spowersystemoperations.Ononehand,itisenvironmentallyfriendlyandhaslowergenerationcostsascomparedtotheconventionalenergyresourcessuchascoalandgas.Ontheotherhand,itisnaturallyintermittentandvolatile,andhencedifculttopredictorharvest.Thankstothepositivecharacteristicsofrenewableenergy,lastdecadehasbeenseeingitssteadypenetrationintopowersystemsallovertheworld.Forexample,U.S.departmentofenergyanalyzesascenarioinwhichwindpowercontributes20%tothetotalenergyutilization(see[ 36 ]).Also,somepowersystemoperators(e.g.,inGermany[ 28 ])arerequiredtoutilizerenewableenergyasapriorityoverotherconventionalones.However,Theincreasingpenetrationofrenewableenergyraisesnewchallengesforthepowersystemoperatorstokeepthesystemsreliableandcosteffective.Duetotheinherentintermittencyandvolatility,itisverydifculttoperfectlyaccommodatetheuncertainrenewableenergywhenfacedwithtransmissioncongestionandthephysicalrestrictionsofothergenerationresources(e.g.,minimumon/offtime,ramp-up/-downratelimits).Inordertokeepthepowersystemsreliable,largeutilizationofrenewableenergyoftenresultsin(i)largesystemreserveswhichcanincreasetheoperationalcost,and(ii)renewableenergycurtailmentwhichcanincreasesystemcarbonemissionscontributedbyalternativepowergenerationresourcessuchascoalandgas. Onepossiblesolutiontobetteraccommodatingtheincreasingpenetrationofrenewableenergyisbyimprovedunitcommitment(UC),whichaimstobetterscheduleavailablegenerationunitsinthepowersystemtowithstandtherenewableenergyuctuationsintheeconomicdispatch.Forexample,severalindependentsystemoperators(ISOs)forderegulatedelectricitymarkets(e.g.,ERCOT)conductreliabilityunit 69

PAGE 70

commitmentstudiesaftertheclosureofday-aheadmarkets(see,e.g.,[ 70 ]).Reliabilityunitcommitmentstudyensuressufcientgenerationpotentialsinproperlocationstoreliablyservetheforecastdemandundertransmissioncongestionbyleveragingoff-lineresourcesifnecessary.SeveralmethodshavebeenproposedandstudiedintheliteraturetoimproveUCdecisions.OneeffectivemethodiscalledthestochasticUCapproach(see,e.g.,[ 3 ],[ 12 ],[ 65 ],[ 55 ],and[ 45 ]),whichcapturestherenewableenergyuncertaintybyanumberofpossiblescenariosandtheircorrespondingprobability.ThebasicideaofusingthestochasticUCapproachforanISOistoformulatearst-stageproblemfortheday-aheadUCdecisionsandasecond-stageproblemforthereal-timeeconomicdispatchproblem,withtheobjectiveofminimizingtheexpectedtotalcosts.Analternativemethodiscalledthesecurity-constrainedUC(SCUC)approach(see,e.g.,[ 67 ]),whichincorporatesalltherenewableenergyscenariosasconstraintsinthemodel,andhencetheobtainedUCdecisionscanaccommodateallthepossiblescenarios.BoththestochasticUCandSCUCapproachescanbesolvedbytheBenders'decompositionalgorithmwhichcanreducethecomputationalrequirement.Recently,thechance-constrainedapproachhasbeenstudiedandappliedtosolveUCproblems(see,e.g.,[ 43 ],[ 73 ],and[ 68 ]).In[ 68 ],chanceconstraintsareformulatedinastochasticUCproblemtoensurethatalargeportionofavailablewindpowerisutilizedwithahighprobability.Althougheffectiveandwidelyapplied,however,therearestillchallengestothestochasticUCapproaches.Forexample,moststochasticUCapproachesneedacertainprobabilitydistributionoftherenewableenergytogeneratescenariosaccordingly.Inthatcase,itcanbedifculttoaccuratelyestimatethedesiredprobabilitydistributionandassignprobabilitymasstoeachscenario.Furthermore,duetotheambiguousprobabilitydistribution,theobtainedUCdecisioncanbebiasedorsensitivetothedistributionselectedforsolvingthemodel. AnalternativetothestochasticUCapproachesiscalledtherobustUCapproach(see,e.g.,[ 63 ],[ 32 ],and[ 9 ]),whichcapturestherenewableenergyuncertaintybyan 70

PAGE 71

uncertaintysetcontainingtheworst-casescenarios.OneadvantageoftherobustUCapproachisthattheobtainedUCdecisionscanprotectagainsttheworst-casescenariosandsothesystemreliabilityisguaranteed.However,therobustUCapproachisoftenconsideredtobeover-conservativeandthecorrespondingtotalcostfortheobtainedUCdecisionsisoftenveryhigh. Inthischapter,weproposeanapproachcalledrisk-aversestochasticUCwithincompleteinformation(SUCI).Inthisapproach,webeginwithasetofhistoricaldatasamplesthatarecollectedfromthetrue(whileambiguous)distributioninpractice,andconstructacondencesetfortheprobabilitydistributionbasedonthedatasamples.Thenweformulaterisk-aversestochasticUCproblemsbyallowingtheambiguousprobabilitydistributiontogoadverselywithinitscondenceset.WeproposetwoSUCImodels,withthersttobeachance-constrainedUCmodeltoguaranteerenewableenergyutilizationandthesecondtobeatwo-stageUCmodelwitheconomicdispatchasrecourse.Also,weprovidetractablereformulationandsolutionapproachforbothmodels.Furthermore,weshowmonotonicityandconvergencepropertiesofbothmodelsasthedatasamplesizegrows.Wesummarizeourmaincontributionsasfollows. 1. Wedonotassumethattheprobabilitydistributionoftherenewableenergycanbeperfectlyknownoraccuratelyestimated.TheproposedapproachbeginswithdatasamplesandweelaboratehowacondencesetcanbeconstructedinSection 4.2.4 .Inthissense,ourapproachisdata-driven.Tothebestofourknowledge,thischapterpresentstherstdata-drivenmodelstoUCproblems. 2. Wereformulatebothmodelsasmixed-integersecond-orderconeprograms(MISOCPs),whichcanbesolvedbytheoff-the-shelfcommercialsoftwaressuchasCPLEX(seePropositions 15 16 ).Furthermore,weprovidesolutionapproachestobothreformulationstoacceleratesolvingthemodels.Morespecically,wedevelopstrongvalidinequalitiesandapolynomial-timeseparationalgorithmforthechance-constrainedUCmodel(seePropositions 17 18 ),andaBenders'decompositionalgorithmforthetwo-stageUCmodel. 3. Weshowthattherisk-aversenessofbothmodelsdecreaseaswecollectmoredatasamplesandeventuallyvanishasthesamplesizegoestoinnity(seeTheorems 71

PAGE 72

7 8 ).Thisresultdepictstherelationshipbetweenconservativenessandthedatasamplesize,andconrmsthatourmodelsbecomelessconservativeasmoredatasamplesarecollected. Theremainderofthischapterisorganizedasfollows.InSection 4.2 ,weintroducetwoSUCImodelsandtheconstructionofthecondenceset.InSection 4.3 ,weprovidereformulationandsolutionapproachestobothSUCImodels.InSection 4.4 ,weshowthemonotonicityandconvergencepropertiesofbothmodels.Finally,summarizethechapterinSection 4.6 4.2Risk-AverseStochasticUnitCommitmentModels Inthissection,wedenenotationandintroducetwoSUCImodels.WebeginwithanomenclatureandanominalmodelinSection 4.2.1 ,andthenweintroduceachance-constrainedUCmodelandatwo-stagestochasticUCmodelwithrecourseinSections 4.2.2 and 4.2.3 respectively.WeclosethissectionbydiscussingconstructionofthecondencesetoftheprobabilitydistributioninSection 4.2.4 4.2.1NomenclatureandANominalModel Weintroducenotationweusethroughoutthischapter.Wesummarizethenotationbysets,parameters,randomvariables,anddecisionvariableslistedinthefollowingnomenclature. Nomenclature Sets T:=f1,...,TgSetofoperationaltimeintervals.B:=f1,...,BgSetofbuses. GbSetofgeneratorsatbusb.EBBSetoftransmissionlines.f1,...,NgSetofscenarios. Parameters 72

PAGE 73

SbiStart-upcostforgeneratoriatbusb.S biShut-downcostforgeneratoriatbusb.FbiFixedcostforgeneratoriatbusbifitison. fbit()Fuelcostfunctionofthegenerationamountforgeneratoriatbusbintimeperiodt. MbiMinimum-uptimeofgeneratoriatbusb.cLSUnitloadsheddingcost. M biMinimum-downtimeofgeneratoriatbusb. QbiMaximumgenerationlimitofgeneratoriatbusb.Q biMinimumgenerationlimitofgeneratoriatbusb. DbtLoadatbusbintimeperiodt.VuiRamp-upratelimitofgeneratori. VuiStart-upramp-ratelimitofgeneratori. VdiRamp-downratelimitofgeneratori. VdiShut-downramp-ratelimitofgeneratori.KbijFlowdistributionfactorofbusbontransmissionline(i,j). CijFlowcapacitylimitoftransmissionline(i,j).Res%Systemspinningreservepencentage.Requiredpercentageofutilizationoftherenewableenergy. Risklevelofthechanceconstraintfortherenewableenergyutilization.Unitpenaltycostofinsufcientutilizationoftherenewableenergy.Unitpenaltycostoftheenergyshortage. Randomvariables 73

PAGE 74

~RbtRenewableenergyoutputcapacityatbusbintimeperiodt.~RRandomvectorcontaining~Rbt,i.e.,~R:=[~R11,~R21,...,~RB1,...,~R1T,~R2T,...,~RBT].~RbntRenewableenergyoutputcapacityinscenarionatbusbintimeperiodt. ~RnRandomvectorcontaining~Rbnt,i.e.,~Rn:=[~R1n1,~R2n1,...,~RBn1,...,~R1nT,~R2nT,...,~RBnT]. Decisionvariables ybitOn/offstatusofgeneratoriatbusbintimeperiodt. ybitStart-upoperationofgeneratoriatbusbintimeperiodt.y bitShut-downoperationofgeneratoriatbusbintimeperiodt. xbitGenerationamountofgeneratoriatbusbintimeperiodt.LSbtLoadsheddingamountatbusbintimeperiodt.rbtCommitedutilizationamountofrenewableenergyoutputatbusbintimeperiodt. 74

PAGE 75

Withthenotationdenedabove,westateanominalmodelofthesecurityconstrainedunitcommitmentmodelasfollows: miny,x,rTXt=1BXb=1Xi2Gb Sbi ybit+S biy bit+Fbiybit+fbit(xbit)+cLSTXt=1BXb=1LSbt (4a)s.t.)]TJ /F9 11.955 Tf 11.95 0 Td[(ybi(t)]TJ /F12 7.97 Tf 6.59 0 Td[(1)+ybit)]TJ /F9 11.955 Tf 11.95 0 Td[(ybik0,1k)]TJ /F3 11.955 Tf 11.95 0 Td[((t)]TJ /F3 11.955 Tf 11.95 0 Td[(1) Mbi,8b2B,8i2Gb,8t2T (4b)ybi(t)]TJ /F12 7.97 Tf 6.59 0 Td[(1))]TJ /F9 11.955 Tf 11.96 0 Td[(ybit+ybik1,1k)]TJ /F3 11.955 Tf 11.95 0 Td[((t)]TJ /F3 11.955 Tf 11.96 0 Td[(1)M bi,8b2B,8i2Gb,8t2T (4c)BXb=1Xi2Gb Qbiybit(1+Res%)BXb=1Dbt,8t2T, (4d)(UC))]TJ /F9 11.955 Tf 11.95 0 Td[(ybi(t)]TJ /F12 7.97 Tf 6.59 0 Td[(1)+ybit)]TJ ET q .478 w 125.02 -198.98 m 131.88 -198.98 l S Q BT /F9 11.955 Tf 125.02 -206.3 Td[(ybit0,8b2B,8i2Gb,8t2T (4e)ybi(t)]TJ /F12 7.97 Tf 6.59 0 Td[(1))]TJ /F9 11.955 Tf 11.96 0 Td[(ybit)]TJ /F9 11.955 Tf 11.95 0 Td[(y bit0,8b2B,8i2Gb,8t2T (4f)Q biybitxbit Qbiybit,8b2B,8i2Gb,8t2T (4g)xbit)]TJ /F9 11.955 Tf 11.96 0 Td[(xbi(t)]TJ /F12 7.97 Tf 6.58 0 Td[(1)(2)]TJ /F9 11.955 Tf 11.95 0 Td[(ybi(t)]TJ /F12 7.97 Tf 6.59 0 Td[(1))]TJ /F9 11.955 Tf 11.96 0 Td[(ybit) Vui+(1+ybi(t)]TJ /F12 7.97 Tf 6.58 0 Td[(1))]TJ /F9 11.955 Tf 11.95 0 Td[(ybit)Vui,8b2B,8i2Gb,8t2T (4h)xbi(t)]TJ /F12 7.97 Tf 6.59 0 Td[(1))]TJ /F9 11.955 Tf 11.96 0 Td[(xbit(2)]TJ /F9 11.955 Tf 11.95 0 Td[(ybi(t)]TJ /F12 7.97 Tf 6.59 0 Td[(1))]TJ /F9 11.955 Tf 11.96 0 Td[(ybit) Vdi+(1)]TJ /F9 11.955 Tf 11.96 0 Td[(ybi(t)]TJ /F12 7.97 Tf 6.58 0 Td[(1)+ybit)Vdi,8b2B,8i2Gb,8t2T (4i)BXb=1 Xi2Gbxbit+rbt+LSbt!=BXb=1Dbt,8t2T (4j))]TJ /F9 11.955 Tf 11.95 0 Td[(CijBXb=1Kbij Xm2Gbxbmt+rbt+LSbt)]TJ /F9 11.955 Tf 11.95 0 Td[(Dbt!Cij,8(i,j)2E,8t2T (4k)ybit, ybit,y bit2f0,1g,xbit,rbt,LSbt0,8b2B,8i2Gb,8t2T, (4l) whereconstraints 4b 4c describetheminimum-upand-downtimeofeachgeneratorrespectively,constraints 4d describethesystemspanningreserverequirement,constraints 4e 4f describethestart-upandshut-downoperationsofeachgeneratorrespectively,constraints 4g describetheminimalandmaximalgenerationlimitsofeachgeneratorrespectively,constraints 4h 4i describetheramping-up(includingstart-upramping)andramping-down(includingshut-down 75

PAGE 76

ramping)ratelimitsofeachgeneratorrespectively,constraints 4j describethegeneration-loadbalanceofthepowersystem,constraints 4k describethetransmissionlinecapacitylimits,andconstraints 4l describetheintegralityandnonnegativityrestrictions.Thisnominalmodelhasanobjectivefunction 4a aimingtominimizeatotaloperationalcostincludingstart-up,shut-down,xedandfuelcostofeachgenerator,plusacostfortheloadshedding.Inthischapter,weapproximatethefuelcostfunctionofeachgeneratorfbit()byapiecewiselinearfunction(see,e.g.,[ 32 ]).Also,noteherethattheproposedrampingconstraints 4h 4i incorporatetherampingraterestrictionwhenageneratorisonandtherst-hourandlast-hourgenerationamountrestriction(see,e.g.,[ 30 ]).Inaddition,weuseaDCapproximationtoformulatethetransmissionconstraints(see,e.g.,[ 69 ]). 4.2.2AChanceConstrainedUCModel Inthismodel,toensurehighutilizationoftherenewableenergyoutput~Rbt,weproposethefollowingdata-drivenchanceconstraint(DCC) infP2DP(TXt=1BXb=1rbtTXt=1BXb=1~Rbt)1)]TJ /F4 11.955 Tf 11.95 0 Td[(,(4) whereDrepresentsacondencesetfortheprobabilitydistributionof~Rbt.TheDCC 4 ensuresthatthroughouttheoperationaltimeintervals,withprobabilityatleast1)]TJ /F4 11.955 Tf 12.58 0 Td[(,atleastaportionofrenewableenergyoutput(e.g.,=80%)isutilizedbythepowersystemforanypossibleprobabilitydistributionwithinthecondencesetD. Sincetheutilizationamountrbtiscommittedbeforetherealizationoftherenewableenergyoutput~Rbt,weshouldtakecareofthefollowingtwocases:(1)thescheduleamountunderestimatestheoutput,i.e.,rbt<~Rbt,and(2)thescheduleamountoverestimatestheoutput,i.e.,rbt>~Rbt.Intherstcase,theactualrenewableoutputislessthanthecommittedutilizationamount,andtheexcessrenewableoutputcanbecurtailed.Whentheutilizationamountofrenewableenergyisinsufcient,i.e.,whenPTt=1PBb=1rbt
PAGE 77

thispenaltycostbasedonalinearfunctionoftheenergyshortage TXt=1BXb=1~Rbt)]TJ /F8 7.97 Tf 17 14.94 Td[(TXt=1BXb=1rbt!+, whererepresentstheunitpenaltycostand(x)+=maxfx,0gforx2R.Inthesecondcase,thecommittedutilizationamountexceedstheactualrenewableoutputandtheenergyshortageincursanadditionalpenaltycost.Inthischapter,wemeasurethispenaltycostbasedonalinearfunctionoftheenergyshortageTXt=1BXb=1rbt)]TJ /F3 11.955 Tf 13.31 2.66 Td[(~Rbt+, whererepresentstheunitpenaltycost.WithDCC 4 andtheadditionalpenaltycostfortheenergyshortage,westatethestochasticsecurityconstrainedunitcommitmentwithincompleteinformation(SUCI)asfollows:z1(d):=miny,x,rTXt=1BXb=1Xi2Gb Sbi ybit+S biy bit+Fbiybit+fbit(xbit)+supP2DEP24 TXt=1BXb=1~Rbt)]TJ /F8 7.97 Tf 17 14.95 Td[(TXt=1BXb=1rbt!++TXt=1BXb=1rbt)]TJ /F3 11.955 Tf 13.31 2.66 Td[(~Rbt+35 (4)(1-SUCI)s.t. 4)-222()]TJ /F3 11.955 Tf 21.25 0 Td[(1b )]TJ ET 0 0 1 RG 0 0 1 rg BT /F3 11.955 Tf 152.8 -409.65 Td[(4)-221()]TJ /F3 11.955 Tf 21.25 0 Td[(1l 4)-222()]TJ /F3 11.955 Tf 21.25 0 Td[(2 whereinobjectivefunction 4 weminimizethetotalcostsplustheworst-caseexpectationofthepenaltycostsforallprobabilitydistributionswithinD. 4.2.3ATwo-StageUCModelwithRecourse Inthismodel,toensurehighutilizationoftherenewableenergyoutput~Rbt,wedecomposethedecisionmakingintotwostages,thatis,wemakeUCdecisionsintherststageanddecideeconomicdispatchinthesecondstage.Westatethemodelas 77

PAGE 78

follows:z2(d):=minyTXt=1BXb=1Xi2Gb Sbi ybit+S biy bit+Fbiybit+supP2DEPhQ(y,~R)i(2-SUCI)s.t. 4)-222()]TJ /F3 11.955 Tf 21.26 0 Td[(1b )]TJ ET 0 0 1 RG 0 0 1 rg BT /F3 11.955 Tf 191.82 -85.24 Td[(4)-222()]TJ /F3 11.955 Tf 21.25 0 Td[(1f ,ybit, ybit,y bit2f0,1g,8b2B,8i2Gb,8t2T, where Q(y,~R)=minx,rTXt=1BXb=1Xi2Gbfbit(xbit) (4a)s.t. 4)-222()]TJ /F3 11.955 Tf 21.25 0 Td[(1g )]TJ ET 0 0 1 RG 0 0 1 rg BT /F3 11.955 Tf 246.23 -221.29 Td[(4)-222()]TJ /F3 11.955 Tf 21.26 0 Td[(1k (4b)BXb=1rbtBXb=1~Rbt,8t2T, (4c)rbt~Rbt,8b2B,8t2T, (4d)xbit,rbt0,8b2B,8i2Gb,8t2T. (4e) Inmodel(2-SUCI),constraints 4c requiresufcientutilizationofrenewableoutput,andconstraints 4d requirethattherenewableusageamountcannotexceeditscapacity.NoteherethatQ(y,~R)representsthevaluefunctionofthesecond-stageeconomicdispatchproblemwithgivenrst-stagedecisionyandtherealizedrenewableoutput~R,andmodel(2-SUCI)strivestominimizetheexpectedtotalcostundertheworst-caseprobabilitydistributionP.Fornotationalbrevity,werewritesecond-stageproblem 4a 4e inanabstractformasfollows: Q(y,~R)=minx0q>x (4a)s.t.Ly+Wx~g, (4b) wherexsubsumesthedecisionvariablesxbitandrbt,constraints 4b subsumesconstraints 4b 4d ,and~grepresentstheright-handsidessubjecttouncertain 78

PAGE 79

windpowercapacity~R.Noteherethat~Ronlyresidesintheright-handssidesofsecond-stageproblem 4a 4e 4.2.4CondenceSetConstruction Inpractice,powersystemoperatorscanadoptvariousmethodstoestimatetheprobabilitydistributionPoftherenewablecapacity~Randitscondenceset.Oneusefulandfrequentlyadoptedmethodishistogram.SupposethatwehavecollectadatasetfmgMm=1ofthewindenergycapacityfromawindfarm.Todrawahistogram,werstconstructanonemptypartitionfBn:n=1,...,Ngofthesamplespace,where=SBn=1BnandeachBniscalledabin.Second,wecountthefrequencyKn=PMm=1InmforeachbinBn,whereInmequalsoneifm2Bn,andzerootherwise.Finally,wecanestimatetheprobabilityoflandingineachbin,i.e.,PfBng,byitsempiricalrelativefrequencyKn=M(seeFigure 4-1 foranexample).Thehistogramestimation Figure4-1. Anexampleofusingahistogramestimationofwindenergycapacityfromawindfarm motivatesustouseadiscretedistributionP0toestimatetheprobabilitydistributionPoftherenewablecapacity~R.Thatis,weapproximatePbyanitesetf~R1,~R2,...,~RNgconsistingatotalofNscenariosobtainedfromthebinsusedinthehistogram,andestimatethecorrespondingprobabilitymassfunctionbyfp01,p02,...,p0Ngobtainedfromthedatasamples,wherep0n>0foreachn=1,...,N.Todevelopacondencesetfortherealprobabilitydistributionof~RaroundP0,weproposetodeneDbasedonthe 79

PAGE 80

L2-normofprobabilitydistributionsasfollows: D:=(P:NXn=1)]TJ /F9 11.955 Tf 5.48 -9.68 Td[(pn)]TJ /F9 11.955 Tf 11.96 0 Td[(p0n2d),(4) wheredrepresentsadeviationtolerance.Themotivationofusingsuchacondencesetdenitionisbasedon(i)bothmodels(1-SUCI)and(2-SUCI)resultintractablereformulationandhavedesiredmonotoneandconvergencepropertiesundercondencesetDbasedontheL2-normofprobabilitydistributions,whicharediscussedindetailsinSections 4.3.1 and 4.4 ,andmoreimportantly(ii)theL2-normofprobabilitydistributionsiscloselyrelatedtothecentrallimittheorem(CLT)andhencecanhelptodevelopDdirectlyfromthedatasamplesandestimatethevalueofd,whichisdiscussedinthefollowing.Supposethatweestimatethedistributionof~Roverthescenariosf~R1,...,~RNgbyusingahistogramcontainingatotalofMdatasamples.Thenforeachn=1,...,N,thepointestimateforpnisp0n=PMm=1Inm=M,andbytheCLTwehavethatasMgoestoinnityp M(p0n)]TJ /F9 11.955 Tf 12.92 0 Td[(pn)convergesindistributiontoN(0,2n),i.e.,anormalrandomvariablewithmeanzeroandvariance2n=Var(Inm)=pn(1)]TJ /F9 11.955 Tf 12.93 0 Td[(pn).Hence,wecanapproximate(pn)]TJ /F9 11.955 Tf 12.77 0 Td[(p0n)2by2n M21,where2n=p0n(1)]TJ /F9 11.955 Tf 12.77 0 Td[(p0n)isanestimateof2nand21representsachi-squaredrandomvariablewithdegreeoffreedomone(recallthat21canberepresentedasthesquareofastandardnormalrandomvariable).Itfollowsthatwecanapproximatedbysetting dPNn=1p0n(1)]TJ /F9 11.955 Tf 11.95 0 Td[(p0n) M21,1)]TJ /F6 7.97 Tf 6.59 0 Td[(,(4) whenMislarge,where21,1)]TJ /F6 7.97 Tf 6.59 0 Td[(representsthe100(1)]TJ /F4 11.955 Tf 12.2 0 Td[()%(e.g.,=0.05)percentileofthe21randomvariable.Thisapproximationindicatesthatd:=d(M)isanonincreasingfunctionofthesamplesizeMandconvergestozeroasMgoestoinnity. 80

PAGE 81

4.3ReformulationsandSolutionApproaches Inthissection,weprovidetractablereformulationsofbothSUCImodelsdevelopedinSection 4.3.1 .Basedonthereformulations,weproposesolutionapproachesforthemodelsinSection 4.3.2 4.3.1ReformulationofTheDCCandWorst-CaseExpectations Werstderiveareformulationoftheworst-caseexpectationsupP2DEP[h(~R)]forageneralfunctionh():RTB!Roftherandomvector~R.WethenapplythisgeneralresulttoreformulationoftheDCCandtheworst-caseexpectationsinmodels(1-SUCI)and(2-SUCI).Fornotationbrevity,welethnrepresenth(~Rn)forn=1,...,N.Wesummarizeourmainresultbythefollowingtheorem. Theorem6. supP2DEP[h(~R)]equalstheoptimalobjectivevalueofthefollowingsecond-orderconeprogram(SOCP): minu,vdv1+v2+NXn=1)]TJ /F9 11.955 Tf 5.48 -9.69 Td[(u3n)]TJ /F9 11.955 Tf 11.96 0 Td[(u2n)]TJ /F9 11.955 Tf 11.95 0 Td[(p0nu1n (4a)s.t.v2)]TJ /F9 11.955 Tf 11.96 0 Td[(u1nhn,8n=1,...,N, (4b)4v1)]TJ /F9 11.955 Tf 11.96 0 Td[(u2n)]TJ /F9 11.955 Tf 11.95 0 Td[(u3n0,8n=1,...,N, (4c)264u1nu2n375u3n,8n=1,...,N, (4d)v10. (4e) 81

PAGE 82

Proof. Bydenition,wehaveEP[h(~R)]=PNn=1pnhn.ItfollowsthatsupP2DEP[h(~R)]istheoptimalobjectivefunctionofthefollowingoptimizationproblem:supP2DEP[h(~R)]=maxp0NXn=1pnhns.t.NXn=1)]TJ /F9 11.955 Tf 5.48 -9.68 Td[(pn)]TJ /F9 11.955 Tf 11.95 0 Td[(p0n2d,NXn=1pn=1, whichisequivalenttosupP2DEP[h(~R)]=maxp,q0NXn=1pnhns.t.NXn=1qnd,)]TJ /F9 11.955 Tf 5.47 -9.68 Td[(pn)]TJ /F9 11.955 Tf 11.96 0 Td[(p0n2qn,8n=1,...,N, (4)NXn=1pn=1. Weobservethatthequadraticconstraints 4 areequivalentto)]TJ /F9 11.955 Tf 5.47 -9.68 Td[(pn)]TJ /F9 11.955 Tf 11.96 0 Td[(p0n2+1 4qn)]TJ /F3 11.955 Tf 11.96 0 Td[(121 4qn+12,8n=1,...,N. Furthermore,since1 4qn+10duetoqn0foreachn,wehaveconstraints 4 equivalenttothefollowingsecond-orderconeconstraints 264pn)]TJ /F9 11.955 Tf 11.95 0 Td[(p0n1 4qn)]TJ /F3 11.955 Tf 11.96 0 Td[(13751 4qn+1,8n=1,...,N. 82

PAGE 83

ItfollowsthatsupP2DEP[h(~R)]istheoptimalobjectivefunctionofthefollowingSOCP: supP2DEP[h(~R)]=maxp,q0NXn=1pnhn (4a)s.t.NXn=1qnd, (4b)264pn)]TJ /F9 11.955 Tf 11.95 0 Td[(p0n1 4qn)]TJ /F3 11.955 Tf 11.95 0 Td[(13751 4qn+1,8n=1,...,N, (4c)NXn=1pn=1, (4d) whichhasadualproblemasstatedintheSOCP 4a 4e .SinceboththeprimalSOCP 4a 4d andthedualSOCP 4a 4e arestrictlyfeasible,strongdualholdsbetweentheprimalanddualSOCPsandtheproofisthereforecomplete. NowweapplyTheorem 6 toreformulatetheDCCandtheworst-caseexpectationsinmodels(1-SUCI)and(2-SUCI).WestatethemainresultsandprovidedetailedproofsinAppendices B.1 B.3 Proposition12. TheDCC 4 isequivalenttotheconstraintsasfollows: dvC1+vC2+NXn=1)]TJ /F9 11.955 Tf 5.48 -9.69 Td[(uC3n)]TJ /F9 11.955 Tf 11.96 0 Td[(uC2n)]TJ /F9 11.955 Tf 11.96 0 Td[(p0nuC1n)]TJ /F3 11.955 Tf 11.95 0 Td[(1 (4a)vC2)]TJ /F9 11.955 Tf 11.96 0 Td[(uC1n+wn0,8n=1,...,N, (4b)4vC1)]TJ /F9 11.955 Tf 11.95 0 Td[(uC2n)]TJ /F9 11.955 Tf 11.96 0 Td[(uC3n0,8n=1,...,N, (4c)264uC1nuC2n375uC3n,8n=1,...,N, (4d)TXt=1BXb=1rbt TXt=1BXb=1~Rbnt!wn,8n=1,...,N, (4e)vC10,wn2f0,1g. (4f) Proof. SeeAppendix B.1 83

PAGE 84

Proposition13. Theworst-caseexpectationofpenaltycost supP2DEP24 TXt=1BXb=1~Rbt)]TJ /F8 7.97 Tf 17 14.94 Td[(TXt=1BXb=1rbt!++TXt=1BXb=1rbt)]TJ /F3 11.955 Tf 13.31 2.65 Td[(~Rbt+35 equalstheoptimalobjectivevalueofthefollowingoptimizationproblem: minu,v,zdvP1+vP2+NXn=1)]TJ /F9 11.955 Tf 5.48 -9.68 Td[(uP3n)]TJ /F9 11.955 Tf 11.96 0 Td[(uP2n)]TJ /F9 11.955 Tf 11.95 0 Td[(p0nuP1n (4a)s.t.vP2)]TJ /F9 11.955 Tf 11.95 0 Td[(uP1nz0n+TXt=1BXb=1zbnt,8n=1,...,N, (4b)4vP1)]TJ /F9 11.955 Tf 11.96 0 Td[(uP2n)]TJ /F9 11.955 Tf 11.96 0 Td[(uP3n0,8n=1,...,N, (4c)264uP1nuP2n375uP3n,8n=1,...,N, (4d)z0nTXt=1BXb=1~Rbnt)]TJ /F8 7.97 Tf 17 14.94 Td[(TXt=1BXb=1rbt,8n=1,...,N, (4e)zbntrbt)]TJ /F3 11.955 Tf 13.32 2.65 Td[(~Rbnt,8n=1,...,N,8t=1,...,T,8b=1,...,B, (4f)vP10,z0n0,zbnt0,8n=1,...,N,8t=1,...,T,8b=1,...,B. (4g) Proof. SeeAppendix B.2 84

PAGE 85

Proposition14. Theworst-caseexpectationsupP2DEP[Q(y,~R)]equalstheoptimalobjectivevalueofthefollowingoptimizationproblem: minu,vdvS1+vS2+NXn=1)]TJ /F9 11.955 Tf 5.48 -9.69 Td[(uS3n)]TJ /F9 11.955 Tf 11.96 0 Td[(uS2n)]TJ /F9 11.955 Tf 11.95 0 Td[(p0nuS1n (4a)s.t.vS2)]TJ /F9 11.955 Tf 11.95 0 Td[(uS1nq>xn,8n=1,...,N, (4b)4vS1)]TJ /F9 11.955 Tf 11.95 0 Td[(uS2n)]TJ /F9 11.955 Tf 11.96 0 Td[(uS3n0,8n=1,...,N, (4c)264uS1nuS2n375uS3n,8n=1,...,N, (4d)Ly+Wxn~gn,8n=1,...,N, (4e)vS10,xn0,8n=1,...,N. (4f) Proof. SeeAppendix B.3 Wearenowreadytoreformulatemodels(1-SUCI)and(2-SUCI)byusingthereformulationresultsinPropositions 12 14 Proposition15. Themodel(1-SUCI)canbeequivalentlyreformulatedasfollows:z1(d)=miny,x,r,w,u,v,zTXt=1BXb=1Xi2Gb Sbi ybit+S biy bit+Fbiybit+fbit(xbit)+dvP1+vP2+NXn=1)]TJ /F9 11.955 Tf 5.48 -9.69 Td[(uP3n)]TJ /F9 11.955 Tf 11.95 0 Td[(uP2n)]TJ /F9 11.955 Tf 11.95 0 Td[(p0nuP1ns.t. 4)-222()]TJ /F3 11.955 Tf 21.26 0 Td[(1b )]TJ ET 0 0 1 RG 0 0 1 rg BT /F3 11.955 Tf 137.46 -447.53 Td[(4)-222()]TJ /F3 11.955 Tf 21.25 0 Td[(1l 4)-222()]TJ /F3 11.955 Tf 21.26 0 Td[(11a )]TJ ET 0 0 1 RG 0 0 1 rg BT /F3 11.955 Tf 246.68 -447.53 Td[(4)-222()]TJ /F3 11.955 Tf 21.25 0 Td[(11f 4)-222()]TJ /F3 11.955 Tf 21.26 0 Td[(12b )]TJ ET 0 0 1 RG 0 0 1 rg BT /F3 11.955 Tf 365.18 -447.53 Td[(4)-222()]TJ /F3 11.955 Tf 21.26 0 Td[(12g Proposition16. Themodel(2-SUCI)canbeequivalentlyreformulatedasfollows:z2(d)=miny,u,v,xTXt=1BXb=1Xi2Gb Sbi ybit+S biy bit+Fbiybit+dvS1+vS2+NXn=1)]TJ /F9 11.955 Tf 5.48 -9.69 Td[(uS3n)]TJ /F9 11.955 Tf 11.96 0 Td[(uS2n)]TJ /F9 11.955 Tf 11.96 0 Td[(p0nuS1ns.t. 4)-222()]TJ /F3 11.955 Tf 21.25 0 Td[(1b )]TJ ET 0 0 1 RG 0 0 1 rg BT /F3 11.955 Tf 143.29 -558 Td[(4)-222()]TJ /F3 11.955 Tf 21.26 0 Td[(1f 4)-222()]TJ /F3 11.955 Tf 21.25 0 Td[(13b )]TJ ET 0 0 1 RG 0 0 1 rg BT /F3 11.955 Tf 255.52 -558 Td[(4)-222()]TJ /F3 11.955 Tf 21.26 0 Td[(13f ,ybit, ybit,y bit2f0,1g,8b2B,8i2Gb,8t2T. 85

PAGE 86

4.3.2SolutionApproaches Inthissection,weproposeandanalyzesolutionapproachesforthemodels(1-SUCI)and(2-SUCI)basedontheirreformulationsderivedinSection 4.3.1 .First,weobservethatbothreformulationsof(1-SUCI)and(2-SUCI)statedinPropositions 15 16 resultinMISOCPs,whichcanbesolvedbyoff-the-shelfcommercialsoftwaressuchasCPLEX.Atthemeanwhile,weobservethatsolutionofthesemodelsmaynotbeveryefcient(e.g.,underdefaultCPLEX)dueto(i)thereareadditionalbinaryvariableswnassociatedwitheachscenarionformodel(1-SUCI)(seeconstraints 4e 4f ),and(ii)thereformulationof(2-SUCI)incorporatesNsecond-stageproblemsandresultsinalarge-scaleMISOCP.Next,weproposealgorithmstoaddressthesechallengesandtosolvethesemodelsmoreefciently.Morespecically,weprovide(i)strongvalidinequalitiesandacorrespondingseparationalgorithmforthe(1-SUCI)reformulation,and(ii)aBenders'decompositionalgorithmandthecorrespondingBenders'cutsforthe(2-SUCI)reformulation. Proposition17. Givenasetofnonnegativeparametersf^Rn,n=1,...,Ng,dene P:=n(r,w)2R+BN:r^Rnwn,8n=1,...,No.(4) Thenforanysubset N:=fn1,...,nkgNsuchthat^Rnj)]TJ /F17 5.978 Tf 5.75 0 Td[(1^Rnjforj=2,...,k,thestarinequality(see[ 2 ]) rXj2 N^Rnj)]TJ /F3 11.955 Tf 13.32 2.66 Td[(^Rnj)]TJ /F17 5.978 Tf 5.75 0 Td[(1wnj,(4) where^Rn0=0,isvalidforP.Moreover,allstarinequalitiestogetherwiththetrialboundsaresufcienttodescribetheconvexhullofP,i.e.,conv(P)=8<:(r,w)2R+[0,1]N:rXj2 N^Rnj)]TJ /F3 11.955 Tf 13.32 2.65 Td[(^Rnj)]TJ /F17 5.978 Tf 5.75 0 Td[(1wnj,8 NN9=;. Proof. SeeAppendix B.4 86

PAGE 87

Remark10. InthedenitionofsetPinProposition 17 ,werewriteconstraints 4e bylettingrrepresentthesumofrbt,andbyletting^Rnrepresentthesumof~Rbntmultipliedby.Notealsothattherepresentationofconv(P)bythestarinequalitiesdoesnotmeanthatwecanrelaxwntobecontinuousvariableswithininterval[0,1]inconstraints 4f ,becauseconstraints 4e 4f arenottheonlyconstraintsinthereformulation.However,therepresentationconclusionreectsthestrengthofthestarinequalities. Wenowpresentaseparationalgorithmtondastarinequality 4 whichismostviolatedbyagivenpoint(r,w)=2conv(P).Algorithm 1 aimstoselectasubset NNandmaximizetheright-handsideofthestarinequality 4 forthegiven(r,w). Algorithm1SeparationAlgorithmforStarInequality 4 1: Initializationofdatastructures:Ptl(n) 0andPost(n) N,8n=0,...,N; 2: Orderf^Rn:n=1,...,Ngsuchthat^R1^R2^RN; 3: forn=Nto0do 4: Ptl(n) maxj=n+1,...,NnPtl(j)+(^Rj)]TJ /F3 11.955 Tf 13.32 2.66 Td[(^Rn)wjo; 5: Post(n) argmaxj=n+1,...,NnPtl(j)+(^Rj)]TJ /F3 11.955 Tf 13.32 2.66 Td[(^Rn)wjo; 6: endfor 7: ifrnwithanarclength(^Rj)]TJ /F3 11.955 Tf 13.55 2.65 Td[(^Rn)wj.Algorithm 1 employsadynamicprogramgoingbackwardsfromnodeNtondadesiredlongestpath,wherepotentialfunctionPtl(n)storesthelengthoflongestpathfromnodentonodeN,andpointerPost(n)storesthenextnodefromnodeninthelongestpath.Hence,Steps 4 and 5 inAlgorithm 1 aretheBellman'sequationusedinthebackwarddynamicprogram, 87

PAGE 88

andAlgorithm 1 iscorrect.Moreover,Algorithm 1 runsinO(N2)timebecauseStep 2 runsinO(NlogN)time(byusing,e.g.,Mergesort[ 21 ]),andSteps 3 6 togetherruninO(N2)time.Wesummarizethediscussionbythefollowingproposition. Proposition18. Givenapoint(r,w),Algorithm 1 canndaviolatedstarinequality 4 ordecidethatnostarinequalitiesareviolatedinpolynomialtime. WeemployaBenders'decompositionalgorithmtosolvemodel(2-SUCI)withamixed-integerprogram(MIP)attherststageandindependentNsecond-orderconesubproblemsatthesecondstage.Letnrepresenttheoptimalobjectivevalueofsubproblemnforn=1,...,N.ThefollowingMIPdescribesacommonmasterproblemattherststage: zMP:=miny,vTXt=1BXb=1Xi2Gb Sbi ybit+S biy bit+Fbiybit+dvS1+vS2+NXn=1n(MP)s.t. 4)-221()]TJ /F3 11.955 Tf 21.25 0 Td[(1b )]TJ ET 0 0 1 RG 0 0 1 rg BT /F3 11.955 Tf 202.59 -313.79 Td[(4)-222()]TJ /F3 11.955 Tf 21.26 0 Td[(1f ,BC(y,v,)0,vS10,ybit, ybit,y bit2f0,1g,8b2B,8i2Gb,8t2T, wherey,v,andaretherst-stagevariables,andBC(y,v)0designatesasetofBenders'cutsobtainedfromsolvingthesubproblems.Givenrst-stageinputsyandv, 88

PAGE 89

thenthsubproblemofmodel(2-SUCI)is: zSPn:=minu,xuS3n)]TJ /F9 11.955 Tf 11.95 0 Td[(uS2n)]TJ /F9 11.955 Tf 11.96 0 Td[(p0nuS1n(SPn(y,v))s.t.uS1n+q>xnvS2, (4a)uS2n+uS3n4vS1, (4b)264uS1nuS2n375uS3n,Wxn~gn)]TJ /F9 11.955 Tf 11.96 0 Td[(Ly, (4c)xn0. WesummarizetheproposedBenders'decompositionalgorithmbyAlgorithm 2 .Inthefollowingproposition,wederivetwotypesofBenders'cutswhichcanbeusedinStep 11 ofAlgorithm 2 Algorithm2ABenders'AlgorithmforSolvingModel(2-SUCI) 1: Initialization:SetUB +1,LB 0,fBC(y,v,)0g ;, 10)]TJ /F12 7.97 Tf 6.59 0 Td[(6; 2: Solve(MP),andobtainanoptimalsolution(y,v,)andtheoptimalobjectivevaluezMP; 3: SetLB zMP; 4: forn=1toNdo 5: Solve(SPn(y,v)),andobtaintheoptimalobjectivevaluezSPn; 6: endfor 7: SetUB minnUB,zMP+PNn=1(zSPn)]TJ /F4 11.955 Tf 11.96 0 Td[(n)o; 8: if(UB)]TJ /F1 11.955 Tf 11.95 0 Td[(LB)=LB100%
PAGE 90

inequalitiesarevalidfor(MP): n+nvS2+4'nvS1+(n)>Ly(n)>~gn,8n=1,...,N, (4a)NXn=1n+nvS2+4'nvS1+(n)>LyNXn=1(n)>~gn. (4b) Proof. Weprovethatvalidityofinequalities 4a ,andthevalidityofinequality 4b followsbysummingalltheinequalities 4a .First,weobservethatthedualofproblem(SPn(y,v))ismaxn,'n,n0)]TJ /F4 11.955 Tf 11.96 0 Td[(nvS2)]TJ /F3 11.955 Tf 11.96 0 Td[(4'nvS1)]TJ /F4 11.955 Tf 11.96 0 Td[(>nLy+>n~gns.t.W>nnq,264n)]TJ /F9 11.955 Tf 11.95 0 Td[(p0n'n)]TJ /F3 11.955 Tf 11.95 0 Td[(1375'n+1, wheren,'n,andnrepresentdualvariablescorrespondingtoconstraints 4a 4b ,and 4c respectively.Hence,inequality 4a followsfromtheweakdualitytheorem. Remark11. Inequalities 4a arestrongerthaninequality 4b asBenders'cuts.However,addingallviolatedinequalities 4a to(MP)inimplementationcouldincreasethesizeof(MP)muchfasterthanaddingaviolatedinequality 4b .Inpractice,onecouldaddallviolatedinequalities 4a inearlieriterationswhenthesizeof(MP)issmalltoreducetheoptimalitygapfaster,andswitchtoaddingviolatedinequalities 4b inlateriterationstokeep(MP)computationallytractable. 4.4ConvergenceAnalysis Inthissection,weanalyzethemonotonicityandconvergencepropertiesofmodels(1-SUCI)and(2-SUCI)whenthedeviationtoleranceddecreasestozero,i.e.,whenthecondencesetDshrinkstothepointestimatorP0.Morespecically,weshowthatlimd!0zi(d)=zi(0)fori=1,2.Thatis,therisk-aversenessofmodels(1-SUCI)and 90

PAGE 91

(2-SUCI)decreasesasDbecomessmallerandsmalleraroundP0,andeventuallyvanishesasDshrinkstobeP0.AsmentionedinSection 4.2.4 ,d(M)isanonincreasingfunctionofthesamplesizeMandconvergestozeroasMgoestoinnity.Thisresultindicatesthatasmoredatawecollect,theproposedmodelsbecomeslessconservativeasthecondencesetDbecomestighter,andtheconservativenessvanishesasthesamplesizegoestoinnity. Werstshowtwolemma,includingageneralconvergenceresultfortheworst-caseexpectationsupP2DEP[h(~R)]forageneralfunctionh()andasimplerreformulationfortheDCC 4 whendissufcientlysmall.Wethenapplytheseresultstomodels(1-SUCI)and(2-SUCI)toshowtheirconvergencerespectively. Lemma6. Theworst-caseexpectationsupP2DEP[h(~R)]decreasesasddecreases.Moreover, limd!0supP2DEP[h(~R)]=EP0[h(~R)]. Proof. Fornotationbrevity,weexplicitlystateDasafunctionofd,i.e.,D:=D(d)=fP:PNn=1(pn)]TJ /F9 11.955 Tf 12.71 0 Td[(p0n)2dg.Wedividetheproofintotwopartsformonotonicityandconvergencerespectively. (Monotonicity)Foranygivend1andd2suchthatd2>d10,weshowthat supP2D(d1)EP[h(~R)]supP2D(d2)EP[h(~R)]. ByTheorem 6 ,wehavethatsupP2D(di)EP[h(~R)]equalstheoptimalobjectivevalueofaSOCP 4a 4e fori=1,2.Sincethefeasibleregionstaysthesameford=d1andd=d2,andforanypoint(u,v)inthefeasibleregiontheobjectivefunctionvaluessatisfy d1v1+v2+NXn=1)]TJ /F9 11.955 Tf 5.48 -9.68 Td[(u3n)]TJ /F9 11.955 Tf 11.96 0 Td[(u2n)]TJ /F9 11.955 Tf 11.95 0 Td[(p0nu1nd2v1+v2+NXn=1)]TJ /F9 11.955 Tf 5.48 -9.68 Td[(u3n)]TJ /F9 11.955 Tf 11.96 0 Td[(u2n)]TJ /F9 11.955 Tf 11.95 0 Td[(p0nu1n(4) becausev10,wehavesupP2D(d1)EP[h(~R)]supP2D(d2)EP[h(~R)]. 91

PAGE 92

(Convergence)ByTheorem 6 andanestablishedcontinuityresultforconvexsemideniteprogram(see,e.g.,Theorem4.1.9in[ 60 ]),supP2D(d)EP[h(~R)]iscontinuousatd=0,i.e., limd!0supP2D(d)EP[h(~R)]=supP2D(0)EP[h(~R)].(4) Butbydenition,supP2D(0)EP[h(~R)]=EP0[h(~R)].Hence,theproofiscomplete. Lemma7. SupposethatN4and1)]TJ /F4 11.955 Tf 12.26 0 Td[(6=PKn=1p0nforanyK=1,...,N.Thenthereexistapositiverealnumber dandanintegern2f1,...,NgsuchthattheDCC 4 isequivalentto TXt=1BXb=1rbtTXt=1BXb=1~Rbnt(4) foranyd2[0, d]. Proof. SeeAppendix B.5 NowweapplyLemma 6 and 7 toprovetheconvergenceofmodels(1-SUCI)and(2-SUCI). Theorem7. SupposethatN4and1)]TJ /F4 11.955 Tf 12.31 0 Td[(6=PKn=1p0nforanyK=1,...,N.Thentheoptimalobjectivevalueofmodel(1-SUCI)z1(d)decreasesasddecreases.Moreover,limd!0z1(d)=z1(0). Proof. Wedividetheproofintotwopartsforthemonotonicityandconvergence,respectively.(Monotonicity)ByProposition 12 andLemma 6 ,thefeasibleregionofmodel(1-SUCI)becomeslargerasddecreases.Also,byProposition 13 andLemma 6 ,theobjectivefunctionofmodel(1-SUCI)decreasesasddecreases.Itfollowsthatz1(d)decreasesasddecrease. 92

PAGE 93

(Convergence)Toshowthatlimd!0z1(d)=z1(0),wecanrestrictthatd dwhichisdenedinLemma 7 .Itfollowsthatwecanreformulatethemodel(1-SUCI)asfollows:z1(d)=minyV(y;d):=TXt=1BXb=1Xi2Gb Sbi ybit+S biy bit+Fbiybit+z(y;d)s.t. 4)-222()]TJ /F3 11.955 Tf 21.25 0 Td[(1b )]TJ ET 0 0 1 RG 0 0 1 rg BT /F3 11.955 Tf 177.62 -110.41 Td[(4)-221()]TJ /F3 11.955 Tf 21.25 0 Td[(1f ,ybit, ybit,y bit2f0,1g,8b2B,8i2Gb,8t2T, wherevaluefunctionsV(y;d)representstheobjectivefunctionofmodel(1-SUCI),andz(y;d)representstheoptimalobjectivevaluewithaxedyandisdenedasz(y;d)=minx,r,u,v,zTXt=1BXb=1Xi2Gb)]TJ /F9 11.955 Tf 5.48 -9.69 Td[(fbit(xbit)+dvP1+vP2+NXn=1)]TJ /F9 11.955 Tf 5.48 -9.69 Td[(uP3n)]TJ /F9 11.955 Tf 11.96 0 Td[(uP2n)]TJ /F9 11.955 Tf 11.96 0 Td[(p0nuP1ns.t. 4)-222()]TJ /F3 11.955 Tf 21.26 0 Td[(1g )]TJ ET 0 0 1 RG 0 0 1 rg BT /F3 11.955 Tf 183.02 -271.78 Td[(4)-222()]TJ /F3 11.955 Tf 21.25 0 Td[(1k 4)-222()]TJ /F3 11.955 Tf 21.25 0 Td[(21 4)-222()]TJ /F3 11.955 Tf 21.25 0 Td[(12b )]TJ ET 0 0 1 RG 0 0 1 rg BT /F3 11.955 Tf 343.88 -271.78 Td[(4)-222()]TJ /F3 11.955 Tf 21.26 0 Td[(12g whereconstraint 4 followsfromareformulationoftheDCC 4 whend< dbyLemma 7 .Hence,z(y;d)istheoptimalobjectivevalueofaSOCP.Byanestablishedcontinuityresultforconvexsemideniteprogram(seeTheorem4.1.9in[ 60 ]),z(y;d)iscontinuousatd=0,i.e.,limd!0z(y;d)=z(y;0)foranyxedy.Itfollowsthatlimd!0V(y;d)=V(y;0)foranyxedy.Hence,forany>0andxedy,thereexistsapositiverealnumberd(y)suchthatjV(y;d))]TJ /F9 11.955 Tf 12.15 0 Td[(V(y;0)j0wehave jV(y;d))]TJ /F9 11.955 Tf 11.96 0 Td[(V(y;0)j<,8y2(y),8d
PAGE 94

Wearenowreadytoshowtheconvergenceresult.Forany>0,welety(d)representanoptimalsolutiontomodel(1-SUCI).Foranyd
PAGE 95

representthesetofallpossibleycombinationsdenedbyconstraints 4b 4f ,thenwehavez2(d1)=miny2(y)V(y;d1)miny2(y)V(y;d2)=z2(d2) forany0d10thereexistsapositiverealnumberd(y)suchthatjV(y;d))]TJ /F9 11.955 Tf 12.5 0 Td[(V(y;0)j0wehave jV(y;d))]TJ /F9 11.955 Tf 11.96 0 Td[(V(y;0)j<,8y2(y),8d0andd
PAGE 96

station.WerunthepowersystemforT=24hwitharenewableenergyutilizationpercentage=85%.WesummarizetheexperimentoutputsinTables 4-1 and 4-2 Totestmodel(1-SUCI),wecompareitwithaclassicalchance-constrainedUCproblem(CCUC)whichdoesnotconsiderdistributionalambiguity.Thatis,in(CCUC),wewilladoptP0astheprobabilitydistributionof~RanddonotincorporatethecondencesetD.Wecomparethesetwomodelsundervariousvaluesofrunningfrom0.10to0.40.Tocomparetheperformanceofthesemodels,werecordthetotalcost(denotedasTCinTable 4-1 )andactualrenewableutilization(denotedasUTLinTable 4-1 ),whichisobtainedfromapost-optimizationMonteCarlosimulationbygeneratingalargenumberofrenewablescenarios.InTable 4-1 ,weobservethatthesolutionobtainedfrommodel(1-SUCI)increasesrenewableutilizationundereachvalueascomparedtotheonefrommodel(CCUC),whileitalsoincursarelativelysmallincrementinthetotalcost.Morespecically,inviewthattherequiredrenewableutilizationissetas=85%,theUCsolutionobtainedfrommodel(1-SUCI)satisestherequirementwhen0.20,whilethesolutionobtainedfrommodel(CCUC)failstosatisfytherequirementonceexceeds0.10. Table4-1. Comparisonofthetotalcostandrenewableenergyutilizationbetweenmodel(1-SUCI)and(CCUC) 1-SUCICCUC1-SUCI(+/-) TC($)UTL(%)TC($)UTL(%)TC(+/-%)UTL(+/-%) =0.10131029388.29130294885.390.563.28=0.15124329286.93123940381.050.316.76=0.20122849386.03121394580.951.205.90=0.25115849682.49115102978.250.655.14=0.30110394881.74108492878.211.754.32=0.35108394878.93107483277.030.852.41=0.40106979077.82106103973.580.825.45 Totestmodel(2-SUCI),wecompareitwithaclassicaltwo-stagestochasticUCproblem(SUC)whichdoesnotconsiderdistributionalambiguity.Thatis,in(SUC),wewilladoptP0astheprobabilitydistributionof~Randdonotincorporatethecondence 96

PAGE 97

setD.Wecomparethesetwomodelsundervariousvaluesofdistancetolerancedrunningfrom0.01to0.20.Tocomparetheperformanceofthesemodels,werecordthetotalcost(denotedasTCinTable 4-2 )andanaveragecost(denotedasSiminTable 4-2 ),whichisobtainedfromapost-optimizationMonteCarlosimulationbygeneratingalargenumberofrenewablescenarios.InTable 4-2 ,weobservethatthetotalcostofmodel(2-SUCI)increasesasdincreases,whileitsaveragecostinsimulationislowerthantheoneassociatedwith(SUC)inallcases.Morespecically,thedifferenceoftheaverageinsimulationreaches1.49%and1.23%whend=0.05andd=0.10respectively.Itindicatesthatmodel(2-SUCI)cangeneratereliableandcost-effectiveUCdecisionswhenthevalueofdissuitablychosen. Table4-2. Comparisonofthetotalcostandtheaveragecostinsimulationbetweenmodel(2-SUCI)and(SUC) 2-SUCISUC2-SUCI(+/-) TC($)Sim($)TC($)Sim($)TC(+/-%)Sim(+/-%) d=0.0111448371163628114059411690390.37-0.46d=0.0511485741151839114059411690390.69-1.49d=0.1011534861154821114059411690391.13-1.23d=0.1511593201160394114059411690391.64-0.74d=0.2011664381158473114059411690392.26-0.91 4.6ConcludingRemarks Inthischapter,weproposeandstudytherisk-aversestochasticunitcommitmentproblems.Beginningwiththedatasamplescollectedfromanambiguousprobabilitydistributionoftherenewableenergy,weconstructacondencesetforthedesiredprobabilitydistributionandformulatetwoSUCImodelsbasedonchanceconstraintandtwo-stageUCmodels,respectively.Wealsoreformulatethemodelstomakethemcomputationallytractableandprovidesolutionapproaches.Finally,weshowthemonotonicityandconvergencepropertiesforbothmodel. 97

PAGE 98

CHAPTER5CONCLUSIONS Inthisdissertation,weproposeandinvestigateseveralrisk-aversestochasticoptimizationmodelsunderuncertainty,whichincorporatethedistributionalambiguityinadata-drivenmanner.Inaddition,weapplytheproposedmodelsinportfoliomanagementandpowersystemoperationalproblems.Possiblefuturedirectionsinclude(i)discoveringmorenonparametricstatisticalestimationwhichleadstotractablereformulations,(ii)extendingandanalyzingtheproposedmodelsinadynamicdecisionmakingenvironment,and(iii)applyingtheproposedmodelindifferentcategoriesofapplications. 98

PAGE 99

APPENDIXADETAILEDPROOFSFORCHAPTER2 A.1S-Lemma WestatetheS-Lemmaasfollows. Lemma8. (S-Lemma,Yakubovich[ 72 ])Letf,g:Rn!Rbequadraticfunctionsandsupposethatthereisanx2Rnsuchthatg(x)<0.Thenthefollowingtwostatementsareequivalent. (i) Thereisnox2Rnsuchthat8>><>>:f(x)<0g(x)0, (ii) Thereisanonnegativenumbery0suchthatf(x)+yg(x)0,8x2Rn. A.2ProofofCorollary 2 Proof. First,weletzDpol1representtheoptimalobjectivevalueoftheoptimizationproblemontheleft-handsideoftheDCC,i.e.,zDpol1=infP2Dpol1Pfa>dP=IXi=1ii,(A{1c) ZRKdP=1,(A{1d) IXi=1i=1,(A{1e) whereconstraints Ab Ac ,and Ae guaranteethat(E[],E[>])2conv(P).Weapplythedualitytheoryforconiclinearprogrammingproblemsanddualizeproblem A 99

PAGE 100

as [Dualpol] zDpol1=maxH,p,q,rr+q(A{2a) s.t.iH+>ip)]TJ /F9 11.955 Tf 11.95 0 Td[(r0,8i=1,...,I,(A{2b) >H+p>+qI[a>1)]TJ /F9 11.955 Tf 11.96 0 Td[(q37526401 2ya1 2ya>1)]TJ /F9 11.955 Tf 11.95 0 Td[(yb375,(A{3a) 264)]TJ /F9 11.955 Tf 9.3 0 Td[(H)]TJ /F12 7.97 Tf 10.5 4.71 Td[(1 2p)]TJ /F12 7.97 Tf 10.49 4.71 Td[(1 2p>1)]TJ /F9 11.955 Tf 11.96 0 Td[(q3752S(K+1)(K+1)+,(A{3b) wherey0.Again,noteherethatstrongdualityholdsforproblems[Primalpol]and[Dualpol]basedonestablishedconiclinearprogrammingtheory.Hence,theDCCinfP2Dpol1Pfa>ip)]TJ /F9 11.955 Tf 11.95 0 Td[(r0,8i=1,...,I,(A{4b) 264)]TJ /F9 11.955 Tf 9.3 0 Td[(H)]TJ /F12 7.97 Tf 10.5 4.71 Td[(1 2p)]TJ /F12 7.97 Tf 10.49 4.71 Td[(1 2p>1)]TJ /F9 11.955 Tf 11.95 0 Td[(q37526401 2ya1 2ya>1)]TJ /F9 11.955 Tf 11.95 0 Td[(yb375,(A{4c) 264)]TJ /F9 11.955 Tf 9.3 0 Td[(H)]TJ /F12 7.97 Tf 10.5 4.71 Td[(1 2p)]TJ /F12 7.97 Tf 10.49 4.7 Td[(1 2p>1)]TJ /F9 11.955 Tf 11.95 0 Td[(q3752S(K+1)(K+1)+,y0.(A{4d) Nowweobservethaty>0,becauseotherwise(i.e.,y=0)wehave264)]TJ /F9 11.955 Tf 9.3 0 Td[(H)]TJ /F12 7.97 Tf 10.49 4.71 Td[(1 2p)]TJ /F12 7.97 Tf 10.49 4.7 Td[(1 2p>1)]TJ /F9 11.955 Tf 11.95 0 Td[(q3752640001375 100

PAGE 101

byconstraint Ac ,andso)]TJ /F9 11.955 Tf 9.3 0 Td[(r+(1)]TJ /F9 11.955 Tf 11.96 0 Td[(q))]TJ /F3 11.955 Tf 27.23 0 Td[(iH)]TJ /F4 11.955 Tf 11.95 0 Td[(>ip+(1)]TJ /F9 11.955 Tf 11.96 0 Td[(q)(foranyi=1,...,I,byconstraint Ab )=264ii>i1375264)]TJ /F9 11.955 Tf 9.3 0 Td[(H)]TJ /F12 7.97 Tf 10.5 4.71 Td[(1 2p)]TJ /F12 7.97 Tf 10.49 4.71 Td[(1 2p>1)]TJ /F9 11.955 Tf 11.96 0 Td[(q375264ii>i13752640001375=1, (A) wherethesecondinequalityholdsbecause264ii>i13750and264)]TJ /F9 11.955 Tf 9.29 0 Td[(H)]TJ /F12 7.97 Tf 10.49 4.71 Td[(1 2p)]TJ /F12 7.97 Tf 10.49 4.71 Td[(1 2p>1)]TJ /F9 11.955 Tf 11.95 0 Td[(q375)]TJ /F15 11.955 Tf 11.96 27.62 Td[(26400013750. Observation A violatesconstraint Aa because2(0,1).Hence,welety=1=yandreplaceH,p,q,r,and(1)]TJ /F9 11.955 Tf 12.04 0 Td[(q)bytheiry-multipliers,i.e.,yH,yp,yq,yr,andy(1)]TJ /F9 11.955 Tf 12.04 0 Td[(q)inLMIs A ,andobtainthereformulationasLMIs 2 A.3ProofofCorollary 3 Proof. First,weletzDmgn1representtheoptimalobjectivevalueoftheoptimizationproblemontheleft-handsideoftheDCC,i.e.,zDmgn1=infP2Dmgn1Pfa>
PAGE 102

dualitytheoryforconiclinearprogrammingproblemsanddualizeproblem A as [Dualmgn] zDmgn1=maxp,h,qKXk=1(pLkLk)]TJ /F9 11.955 Tf 11.96 0 Td[(pUkUk)]TJ /F4 11.955 Tf 11.95 0 Td[(khk)+q(A{7a) s.t.)]TJ /F8 7.97 Tf 16.46 14.95 Td[(KXk=1hk2k+(pL)]TJ /F9 11.955 Tf 11.96 0 Td[(pU)>+qI[a>1)]TJ /F9 11.955 Tf 11.95 0 Td[(q3750, (A) 264)]TJ /F9 11.955 Tf 9.3 0 Td[(H)]TJ /F12 7.97 Tf 10.49 4.71 Td[(1 2(p+ya))]TJ /F12 7.97 Tf 10.49 4.71 Td[(1 2(p+ya)>yb)]TJ /F9 11.955 Tf 11.95 0 Td[(q3750, (A) y0. 102

PAGE 103

We now showthat A issecond-orderconerepresentable,andthereformulationof A followssimilarly: 266666664h1...)]TJ /F12 7.97 Tf 10.49 4.71 Td[(1 2phK)]TJ /F12 7.97 Tf 10.49 4.71 Td[(1 2p>1)]TJ /F9 11.955 Tf 11.96 0 Td[(q3777777750,KXk=1hk2k)]TJ /F8 7.97 Tf 17.13 14.95 Td[(KXk=1pkkK+1+(1)]TJ /F9 11.955 Tf 11.96 0 Td[(q)2K+10,81,...,K,K+12R,8>>>>><>>>>>:hk=0)pk=0,8k=1,...,K,KXk=1hk6=0hkk)]TJ /F15 11.955 Tf 11.95 16.86 Td[(pk 2hkK+12+1)]TJ /F9 11.955 Tf 11.96 0 Td[(q)]TJ /F8 7.97 Tf 18.26 14.95 Td[(KXk=1hk6=0p2k 4hk2K+10,81,...,K,K+12R 103

PAGE 104

,8>>>>>>>>><>>>>>>>>>:hk=0)pk=0,8k=1,...,K,hk0,1)]TJ /F9 11.955 Tf 11.95 0 Td[(q)]TJ /F8 7.97 Tf 18.26 14.95 Td[(KXk=1hk6=0p2k 4hk0,9tk0,8k=1,...,K,suchthat8>>><>>>:p2k4hktk,8k=1,...,K,q+KXk=1tk1,9tk0,8k=1,...,K,suchthat8>>>>>><>>>>>>:264pktk)]TJ /F9 11.955 Tf 11.96 0 Td[(hk375tk+hk,8k=1,...,K,q+KXk=1tk1 (A) Bysubstitutingthesimplication A backintoequation A ,weobtainthereformulationofconstraint A asfollows8>>>>>>>>><>>>>>>>>>:264pLk)]TJ /F9 11.955 Tf 11.95 0 Td[(pUktk)]TJ /F9 11.955 Tf 11.96 0 Td[(hk375tk+hk,8k=1,...,K,KXk=1tk1)]TJ /F9 11.955 Tf 11.96 0 Td[(qtk0,8k=1,...,K. Theformulationofconstraint A canbesimilarlyobtainedasfollows8>>>>>>>>><>>>>>>>>>:264pLk)]TJ /F9 11.955 Tf 11.96 0 Td[(pUk+yaksk)]TJ /F9 11.955 Tf 11.96 0 Td[(hk375sk+hk,8k=1,...,K,q)]TJ /F9 11.955 Tf 11.96 0 Td[(yb+KXk=1sk0sk0,8k=1,...,K. 104

PAGE 105

Hence,theDCCinfP2Dmgn1Pfa>0,becauseotherwise(i.e.,y=0)wehave264)]TJ /F9 11.955 Tf 9.3 0 Td[(H)]TJ /F12 7.97 Tf 10.49 4.71 Td[(1 2(pL)]TJ /F9 11.955 Tf 11.96 0 Td[(pU))]TJ /F12 7.97 Tf 10.5 4.71 Td[(1 2(pL)]TJ /F9 11.955 Tf 11.96 0 Td[(pU)>1)]TJ /F9 11.955 Tf 11.95 0 Td[(q3752640001375 by A ,anditfollowsthatKXk=1(pUkUk)]TJ /F9 11.955 Tf 11.95 0 Td[(pLkLk+khk)+(1)]TJ /F9 11.955 Tf 11.95 0 Td[(q)=264DiagL(L)>1375264)]TJ /F9 11.955 Tf 9.3 0 Td[(H)]TJ /F12 7.97 Tf 10.5 4.7 Td[(1 2(pL)]TJ /F9 11.955 Tf 11.96 0 Td[(pU))]TJ /F12 7.97 Tf 10.49 4.71 Td[(1 2(pL)]TJ /F9 11.955 Tf 11.95 0 Td[(pU)>1)]TJ /F9 11.955 Tf 11.96 0 Td[(q375+(U)]TJ /F4 11.955 Tf 11.96 0 Td[(L)>pU264L(L)>13752640001375=1, whereDiag:=diag(1,...,K),andsoconstraint Aa isviolatedbecause2(0,1).Hence,welety=1=y0andreplacepU,pL,h,t,s,and(1)]TJ /F9 11.955 Tf 12.8 0 Td[(q)withtheiry-multipliers,i.e.,ypU,ypL,yh,yt,ys,andy(1)]TJ /F9 11.955 Tf 12.03 0 Td[(q),respectively.Thisreplacementshowsthatconstraints A areequivalenttoconstraints 2 ,andcompletestheproof. 105

PAGE 106

A.4ProofofProposition 1 Proof. FollowingTheorem 1 ,weobservethattheoptimizationproblemunderDCCcanbereformulatedasaconvexsemideniteprogramasfollows:z(int(C))=minx2X (x)(A{13a) s.t.2642001375264G)]TJ /F9 11.955 Tf 9.3 0 Td[(p)]TJ /F9 11.955 Tf 9.3 0 Td[(p>1)]TJ /F9 11.955 Tf 11.95 0 Td[(r375+264001375264Hpp>q375y,(A{13b) 264G)]TJ /F9 11.955 Tf 9.3 0 Td[(p)]TJ /F9 11.955 Tf 9.3 0 Td[(p>1)]TJ /F9 11.955 Tf 11.95 0 Td[(r37526401 2a1 2a>y+a>)]TJ /F3 11.955 Tf 12.04 2.66 Td[(b375,(A{13c) 264G)]TJ /F9 11.955 Tf 9.3 0 Td[(p)]TJ /F9 11.955 Tf 9.3 0 Td[(p>1)]TJ /F9 11.955 Tf 11.95 0 Td[(r3752S(K+1)(K+1),264Hpp>q3752S(K+1)(K+1),y0,(A{13d) whereaandbareafnefunctionsofxasdiscussedinSection 2.1.2 .Foranygiven>0,letz(int(C+))representtheoptimalobjectivevalueofproblem Aa Ad withscalarbreplacedbyb+.Itfollowsthat z(int(C))z(C)z(int(C+)),(A) becausePf2int(C)gPf2CgPf2int(C+)g, whereC+:=f2RK:a>b+g.Byanestablishedcontinuityresultforconvexsemideniteprogram(seeTheorem4.1.9inShapiroandScheinberg[ 60 ]),z(int(C+))iscontinuousat=0,i.e.,lim#0z(int(C+))=z(int(C)),becauseXisaconvexset, ()isaconvexfunction,andtheSlaterconditionholdsforproblem Aa Ad byassumption.Therefore,theproofiscompleteinviewof A Remark12. ItcanbeshownthatasufcientconditionoftheSlaterconditionforproblem Aa Ad isasfollows: thereexists 0<, suchthatwithreplacedby 106

PAGE 107

0 thereexistsafeasiblesolution(x0,G0,H0,p0,r0,q0,y0)toproblem Aa Ad with y0>0. A.5ProofofProposition 2 Proof. First,wereformulatethenonlinearDCCinfP2D1P>M+c>+d01)]TJ /F4 11.955 Tf 10.6 0 Td[(byusingCorollary 1 andProposition 1 ,andobtainLMIs 2a 2b ,and 2d Second,toensureaconservativeapproximation,weensurethat>M+c>+da>i)]TJ /F3 11.955 Tf 12.93 2.65 Td[(biforall2RKandalli=1,...,m,whichimpliesthateachquadratic>M+c>+d)]TJ /F3 11.955 Tf 12.94 0 Td[(a>i+biisnonnegativeonRK.Hence,thisstatementcanbereformulatedasLMIs 2c A.6ProofofProposition 4 Proof. WeestablishtheequivalencebetweenWVaRconstraint 2 andthesingleDCCinfP2D1Pfa>bg1)]TJ /F4 11.955 Tf 11.95 0 Td[(asfollows:inf`2R:Pfa>`g1)]TJ /F4 11.955 Tf 11.96 0 Td[(b,8P2D1 (A),Pfa>bg1)]TJ /F4 11.955 Tf 11.96 0 Td[(,8P2D1 (A),infP2D1Pfa>bg1)]TJ /F4 11.955 Tf 11.96 0 Td[(. Weremarktheequivalencebetween A and A asfollows.DenotesetL:=f`2R:Pfa>`g1)]TJ /F4 11.955 Tf 12.76 0 Td[(gin A .First,if A isvalid,thenbis nosmallerthan theinmumofsetLandthusb2LbecausethefunctionF(`):=Pfa>`gisnondecreasingandright-continuous.Itthenfollowsthat A holds.Second,if A isvalid,thenb2L,andhence A holdsbythedenitionofinmum.TheproofiscompletebynoticingTheorem 1 andProposition 1 A.7ProofofLemma 2 Proof. (i) Bydenition,isasupremumoflinearfunctionsandhenceconvex. (ii) Foranyx1,x22Rsuchthatx1
PAGE 108

Also,since(t)=+1fort<0,wehave(x)=supt2Rfxt)]TJ /F4 11.955 Tf 11.95 0 Td[((t)g=supt0fxt)]TJ /F4 11.955 Tf 11.95 0 Td[((t)g, andso(x1)=supt0fx1t)]TJ /F4 11.955 Tf 11.95 0 Td[((t)gsupt0fx2t)]TJ /F4 11.955 Tf 11.95 0 Td[((t)g=(x2). (iii) Since(1)=0,wehave(x)=supt0fxt)]TJ /F4 11.955 Tf 11.96 0 Td[((t)gx. (iv) Weprovebycontradiction.Supposethat(x)=montheinterval[a,b]and(y)=m06=mforsomey0,wehave(z0+z)z0+zand(z0)z0byproperty(iii)inLemma 2 .Itfollowsthat(z0+z)+(1)]TJ /F4 11.955 Tf 11.96 0 Td[()(z0)(z0+z)+(1)]TJ /F4 11.955 Tf 11.95 0 Td[()z0)(z0+z)+(1)]TJ /F4 11.955 Tf 11.96 0 Td[()(z0)z0+z)]TJ /F9 11.955 Tf 11.96 0 Td[(d(sinced>0))(z0+z))]TJ /F9 11.955 Tf 11.96 0 Td[(z0)]TJ /F4 11.955 Tf 11.96 0 Td[(z+d(1)]TJ /F4 11.955 Tf 11.96 0 Td[()((z0+z))]TJ /F4 11.955 Tf 11.95 0 Td[((z0)))(z0+z))]TJ /F9 11.955 Tf 11.95 0 Td[(z0)]TJ /F4 11.955 Tf 11.95 0 Td[(z+d (z0+z))]TJ /F4 11.955 Tf 11.95 0 Td[((z0)1)]TJ /F4 11.955 Tf 11.96 0 Td[(. 108

PAGE 109

A.9ProofofProposition 5 Proof. First,since(x)=(x)]TJ /F3 11.955 Tf 11.96 0 Td[(1)2,wehave(x)=8><>:)]TJ /F3 11.955 Tf 9.3 0 Td[(1,ifx)]TJ /F3 11.955 Tf 21.92 0 Td[(2,1 4x2+x,ifx)]TJ /F3 11.955 Tf 21.92 0 Td[(2. Hence,m ()=)]TJ /F3 11.955 Tf 9.3 0 Td[(2and m()=+1.Second,wesolve the probleminfz>0,z0+z)]TJ /F12 7.97 Tf 13.17 0 Td[(2(z0+z))]TJ /F9 11.955 Tf 11.95 0 Td[(z0)]TJ /F4 11.955 Tf 11.96 0 Td[(z+d (z0+z))]TJ /F4 11.955 Tf 11.95 0 Td[((z0)=infz>0,z0)]TJ /F12 7.97 Tf 13.18 0 Td[(2(z0))]TJ /F9 11.955 Tf 11.96 0 Td[(z0+(1)]TJ /F4 11.955 Tf 11.96 0 Td[()z+d (z0))]TJ /F4 11.955 Tf 11.96 0 Td[((z0)]TJ /F9 11.955 Tf 11.95 0 Td[(z) tooptimality,wherewemakeatransformbyreplacingz0byz0)]TJ /F9 11.955 Tf 12.96 0 Td[(z.Weletf(z0,z)representtheobjectivefunctionanddiscussthefollowingcases: (1) Ifz0)]TJ /F9 11.955 Tf 11.96 0 Td[(z)]TJ /F3 11.955 Tf 21.91 0 Td[(2,then(z0)]TJ /F9 11.955 Tf 11.95 0 Td[(z)=)]TJ /F3 11.955 Tf 9.3 0 Td[(1and(z0)=1 4z20+z0.Itfollowsthatf(z0,z)=)]TJ /F12 7.97 Tf 6.67 -4.98 Td[(1 4z20+z0)]TJ /F9 11.955 Tf 11.96 0 Td[(z0+(1)]TJ /F4 11.955 Tf 11.95 0 Td[()z+d )]TJ /F12 7.97 Tf 6.68 -4.97 Td[(1 4z20+z0)]TJ /F3 11.955 Tf 11.95 0 Td[(()]TJ /F3 11.955 Tf 9.3 0 Td[(1)=1 4z20+(1)]TJ /F4 11.955 Tf 11.96 0 Td[()z+d )]TJ /F12 7.97 Tf 6.68 -4.98 Td[(1 2z0+12, andso@f(z0,z) @z0=1 2z0)]TJ /F3 11.955 Tf 11.96 0 Td[((1)]TJ /F4 11.955 Tf 11.96 0 Td[()z)]TJ /F9 11.955 Tf 11.95 0 Td[(d )]TJ /F12 7.97 Tf 6.67 -4.98 Td[(1 2z0+13. Sincez0z)]TJ /F3 11.955 Tf 11.96 0 Td[(2,z0)]TJ /F3 11.955 Tf 21.92 0 Td[(2and<1=2byassumption,wehave(1=2)z0)]TJ /F3 11.955 Tf 11.95 0 Td[((1)]TJ /F4 11.955 Tf 11.96 0 Td[()z)]TJ /F9 11.955 Tf -438.19 -23.91 Td[(d()]TJ /F3 11.955 Tf 11.88 0 Td[(1=2)z)]TJ /F9 11.955 Tf 11.88 0 Td[(d)]TJ /F3 11.955 Tf 11.88 0 Td[(1<0and1 2z0+10.Hence,@f(z0,z)=@z0<0foranyxedzanditisoptimaltochoosez0=z)]TJ /F3 11.955 Tf 11.95 0 Td[(2.Itfollowsthatinfz>0,z0)]TJ /F12 7.97 Tf 13.18 0 Td[(2f(z0,z)=infz>0f(z)]TJ /F3 11.955 Tf 11.96 0 Td[(2,z)=infz>04(d+1)1 z2)]TJ /F3 11.955 Tf 11.96 0 Td[(41 z+1. Therefore,itisoptimaltochoosez=2(d+1)=andinfz>0,z0)]TJ /F12 7.97 Tf 13.18 0 Td[(2f(z0,z)=1)]TJ /F4 11.955 Tf 21.03 8.09 Td[(2 d+1. (2) Ifz0)]TJ /F9 11.955 Tf 11.34 0 Td[(z)]TJ /F3 11.955 Tf 21.91 0 Td[(2,then(z0)]TJ /F9 11.955 Tf 11.35 0 Td[(z)=1 4(z0)]TJ /F9 11.955 Tf 11.35 0 Td[(z)2+(z0)]TJ /F9 11.955 Tf 11.35 0 Td[(z)and(z0)=1 4z20+z0.Itfollowsthatf(z0,z)=)]TJ /F12 7.97 Tf 6.67 -4.98 Td[(1 4z20+z0)]TJ /F9 11.955 Tf 11.96 0 Td[(z0+(1)]TJ /F4 11.955 Tf 11.95 0 Td[()z+d )]TJ /F12 7.97 Tf 6.68 -4.97 Td[(1 4z20+z0)]TJ /F15 11.955 Tf 11.95 9.68 Td[()]TJ /F12 7.97 Tf 6.68 -4.97 Td[(1 4(z0)]TJ /F9 11.955 Tf 11.96 0 Td[(z)2+(z0)]TJ /F9 11.955 Tf 11.96 0 Td[(z)=1 4z20+(1)]TJ /F4 11.955 Tf 11.96 0 Td[()z+d 1 2zz0+z)]TJ /F12 7.97 Tf 13.15 4.71 Td[(1 4z2, 109

PAGE 110

andso@f(z0,z) @z0=z)]TJ /F9 11.955 Tf 5.48 -9.69 Td[(z20+(4)]TJ /F9 11.955 Tf 11.95 0 Td[(z)z0)]TJ /F3 11.955 Tf 11.96 0 Td[(4(1)]TJ /F4 11.955 Tf 11.95 0 Td[()z)]TJ /F3 11.955 Tf 11.96 0 Td[(4d 8)]TJ /F12 7.97 Tf 6.67 -4.98 Td[(1 2zz0+z)]TJ /F12 7.97 Tf 13.15 4.71 Td[(1 4z22. Forxedz,weset@f(z0,z)=@z0=0andobtainz0=(z)]TJ /F3 11.955 Tf 11.96 0 Td[(4)p z2+8(1)]TJ /F3 11.955 Tf 11.95 0 Td[(2)z+16(d+1) 2. Sincez0z)]TJ /F3 11.955 Tf 11.96 0 Td[(2,weruleoutthenegativerootandsoz0=(z)]TJ /F3 11.955 Tf 11.96 0 Td[(4)+p z2+8(1)]TJ /F3 11.955 Tf 11.96 0 Td[(2)z+16(d+1) 2 isastationarypointoff(z0,z)withzxed andthecorrespondingobjectivevalue f(z0,z)=1 2s 16(d+1)1 z2+8(1)]TJ /F3 11.955 Tf 11.96 0 Td[(2)1 z+1+1 21)]TJ /F3 11.955 Tf 11.95 0 Td[(41 z. Nowwerstshowthatz0isanoptimalsolutionforinfz0z)]TJ /F12 7.97 Tf 6.59 0 Td[(2f(z0,z)withzxed .Wecomparethevalueoff(z0,z)withf(+1,z)andf(z)]TJ /F3 11.955 Tf 12.12 0 Td[(2,z)because+1andz)]TJ /F3 11.955 Tf 12.12 0 Td[(2aretheendpointsofthefeasibleregionofz0.Weobservethatf(+1,z)=+1.Also,wehavef(z)]TJ /F3 11.955 Tf 11.96 0 Td[(2,z)=1 4(z)]TJ /F3 11.955 Tf 11.95 0 Td[(2)2+(1)]TJ /F4 11.955 Tf 11.96 0 Td[()z+d 1 2z(z)]TJ /F3 11.955 Tf 11.95 0 Td[(2)+z)]TJ /F12 7.97 Tf 13.15 4.7 Td[(1 4z2=4(d+1)1 z2)]TJ /F3 11.955 Tf 11.95 0 Td[(41 z+1, andf(z)]TJ /F3 11.955 Tf 12.46 0 Td[(2,z)f(z0,z).Toseethat,wecomparethevaluesoff(z)]TJ /F3 11.955 Tf 12.46 0 Td[(2,z)andf(z0,z)bythefollowinginequalities,wheretheinequalitiesbelowimplythoseabove.f(z)]TJ /F3 11.955 Tf 11.96 0 Td[(2,z)f(z0,z)(8(d+1)1 z2)]TJ /F3 11.955 Tf 11.96 0 Td[(81 z+2s 16(d+1)1 z2+8(1)]TJ /F3 11.955 Tf 11.96 0 Td[(2)1 z+1+1)]TJ /F3 11.955 Tf 11.96 0 Td[(41 z("8(d+1)1 z2+4(1)]TJ /F3 11.955 Tf 11.96 0 Td[(2)1 z+1#216(d+1)1 z2+8(1)]TJ /F3 11.955 Tf 11.95 0 Td[(2)1 z+1(161 z22(d+1)1 z+(1)]TJ /F3 11.955 Tf 11.96 0 Td[(2)20. 110

PAGE 111

Hence,infz0z)]TJ /F12 7.97 Tf 6.59 0 Td[(2f(z0,z)=f(z0,z)withzxed. Therefore,wehave infz>0,z0z)]TJ /F12 7.97 Tf 6.59 0 Td[(2f(z0,z)=infz>01 2p 16(d+1)z2+8(1)]TJ /F3 11.955 Tf 11.96 0 Td[(2)z+1+1 2(1)]TJ /F3 11.955 Tf 11.96 0 Td[(4z), wherewehave1=zreplacedbyz.Similarly,weset@f(z0,z) @z=8(d+1)z+2(1)]TJ /F3 11.955 Tf 11.95 0 Td[(2) p 16(d+1)z2+8(1)]TJ /F3 11.955 Tf 11.95 0 Td[(2)z+1)]TJ /F3 11.955 Tf 11.96 0 Td[(2=0, andobtainz=p d2+4d()]TJ /F4 11.955 Tf 11.96 0 Td[(2))]TJ /F3 11.955 Tf 11.96 0 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[(2)d 4d(d+1). Therefore,wehavef(z0,z)=1)]TJ /F4 11.955 Tf 11.96 0 Td[(+p d2+4d()]TJ /F4 11.955 Tf 11.95 0 Td[(2))]TJ /F3 11.955 Tf 11.96 0 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[(2)d 2(d+1). Again,weshallcomparethevalueoff(z0,z)withf(z0,+1)andf(z0,0)since+1and0aretheendpointsofthefeasibleregionofz.Weobservethatf(z0,+1)=+1andf(z0,0)=1f(z0,z),andhenceinfz>0,z0z)]TJ /F12 7.97 Tf 6.59 0 Td[(2f(z0,z)=1)]TJ /F4 11.955 Tf 11.95 0 Td[(+p d2+4d()]TJ /F4 11.955 Tf 11.96 0 Td[(2))]TJ /F3 11.955 Tf 11.95 0 Td[((1)]TJ /F3 11.955 Tf 11.95 0 Td[(2)d 2(d+1). Finally,wecomparetheoptimalvalueoff(z0,z)inthetwocases.Weclaimthattheoptimalvalueobtainedinthelattercaseissmaller(andhencegloballyoptimal).Toseethat,wecomparethetwovaluesbythefollowinginequalities,wheretheinequalitiesbelowimplythoseabove.1)]TJ /F4 11.955 Tf 21.03 8.09 Td[(2 d+11)]TJ /F4 11.955 Tf 11.96 0 Td[(+p d2+4d()]TJ /F4 11.955 Tf 11.96 0 Td[(2))]TJ /F3 11.955 Tf 11.96 0 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[(2)d 2(d+1)(d+2)]TJ /F3 11.955 Tf 11.95 0 Td[(22p d2+4d()]TJ /F4 11.955 Tf 11.96 0 Td[(2)((d+2)]TJ /F3 11.955 Tf 11.95 0 Td[(22)2d2+4d()]TJ /F4 11.955 Tf 11.96 0 Td[(2)(42()]TJ /F3 11.955 Tf 11.96 0 Td[(1)20. 111

PAGE 112

Therefore,theperturbedrisklevelis0=)]TJ /F15 11.955 Tf 13.15 18.53 Td[(p d2+4d()]TJ /F4 11.955 Tf 11.96 0 Td[(2))]TJ /F3 11.955 Tf 11.95 0 Td[((1)]TJ /F3 11.955 Tf 11.95 0 Td[(2)d 2d+2. A.10ProofofProposition 6 Proof. First,Since(x)=jx)]TJ /F3 11.955 Tf 11.95 0 Td[(1j,wehave(x)=8>>>><>>>>:)]TJ /F3 11.955 Tf 9.3 0 Td[(1,ifx<)]TJ /F3 11.955 Tf 9.3 0 Td[(1,x,if)]TJ /F3 11.955 Tf 9.3 0 Td[(1x1,+1,ifx>1. Hence,m ()=)]TJ /F3 11.955 Tf 9.3 0 Td[(1and m()=1.Second,wesolve the probleminfz>0,)]TJ /F12 7.97 Tf 6.58 0 Td[(1z0+z1f(z0,z):=(z0+z))]TJ /F9 11.955 Tf 11.96 0 Td[(z0)]TJ /F4 11.955 Tf 11.95 0 Td[(z+d (z0+z))]TJ /F4 11.955 Tf 11.96 0 Td[((z0) tooptimality.Wediscussthefollowingcases: (1) Ifz0)]TJ /F3 11.955 Tf 21.92 0 Td[(1,then(z0)=)]TJ /F3 11.955 Tf 9.3 0 Td[(1and(z0+z)=z0+z.Itfollowsthatf(z0,z)=(1)]TJ /F4 11.955 Tf 11.95 0 Td[()z+d z0+z+1. Noteherethatforanygivenz,f(z0,z)isanonincreasingfunctionofz0,duetothefactthatz0+z+10.Meanwhile,z0+z1 .Hence,itisoptimaltochoosez0=minf1)]TJ /F9 11.955 Tf 11.95 0 Td[(z,)]TJ /F3 11.955 Tf 9.3 0 Td[(1gandsof(z0,z)=8><>:(1)]TJ /F6 7.97 Tf 6.59 0 Td[()z+d 2,ifz2,(1)]TJ /F6 7.97 Tf 6.59 0 Td[()z+d z,ifz2. Therefore,f(z0,z)isnonincreasingonzontheinterval(0,2]andnondecreasingonzontheinterval[2,+1),andsof(z0,z)=1)]TJ /F4 11.955 Tf 11.96 0 Td[(+d 2. 112

PAGE 113

(2) If)]TJ /F3 11.955 Tf 9.3 0 Td[(1z01,then(z0)=z0.Also,wehavez2 and(z0+z)=z0+z because)]TJ /F3 11.955 Tf 9.29 0 Td[(1z0+z1.Hence,f(z0,z)=(1)]TJ /F4 11.955 Tf 11.95 0 Td[()z+d z=1)]TJ /F4 11.955 Tf 11.95 0 Td[(+d z1)]TJ /F4 11.955 Tf 11.95 0 Td[(+d 2, andthelowerboundisattainedatz=2.Therefore,f(z0,z)=1)]TJ /F4 11.955 Tf 11.96 0 Td[(+d 2. Tosumup,wehave1)]TJ /F4 11.955 Tf 11.95 0 Td[(0=f(z0,z)=1)]TJ /F4 11.955 Tf 11.96 0 Td[(+d 2,orequivalently0=)]TJ /F8 7.97 Tf 13.15 4.71 Td[(d 2. A.11ProofofProposition 7 Proof. Wedividetheproofintotwoparts.Intherstpart,weshowthattheperturbedrisklevel 0=1)]TJ /F3 11.955 Tf 19.44 0 Td[(infx2(0,1)ne)]TJ /F8 7.97 Tf 6.58 0 Td[(dx1)]TJ /F6 7.97 Tf 6.59 0 Td[()]TJ /F3 11.955 Tf 11.96 0 Td[(1 x)]TJ /F3 11.955 Tf 11.95 0 Td[(1o.(A) Inthesecondpart,weshowhowtocompute0byusingbisectionlinesearch. (Risklevel)First,since(x)=xlogx)]TJ /F9 11.955 Tf 12.96 0 Td[(x+1,wehave(x)=ex)]TJ /F3 11.955 Tf 12.96 0 Td[(1.Hence,m ()=and m()=+1.Second,wesolve the probleminfz>0f(z0,z):=(z0+z))]TJ /F9 11.955 Tf 11.96 0 Td[(z0)]TJ /F4 11.955 Tf 11.96 0 Td[(z+d (z0+z))]TJ /F4 11.955 Tf 11.95 0 Td[((z0)=infz>0ez+(d)]TJ /F4 11.955 Tf 11.95 0 Td[(z)]TJ /F9 11.955 Tf 11.95 0 Td[(z0)]TJ /F3 11.955 Tf 11.96 0 Td[(1)e)]TJ /F8 7.97 Tf 6.59 0 Td[(z0 ez)]TJ /F3 11.955 Tf 11.96 0 Td[(1 tooptimality.Since@f @z0=)]TJ /F9 11.955 Tf 10.5 8.09 Td[(d)]TJ /F4 11.955 Tf 11.95 0 Td[(z)]TJ /F9 11.955 Tf 11.96 0 Td[(z0 ez0(ez)]TJ /F3 11.955 Tf 11.95 0 Td[(1), wehavez0=d)]TJ /F4 11.955 Tf 11.96 0 Td[(zbysetting@f=@z0=0,andsoinfz>0f(z0,z)=infz>0f(z0,z)=infz>0ez)]TJ /F9 11.955 Tf 11.95 0 Td[(ez)]TJ /F8 7.97 Tf 6.59 0 Td[(d ez)]TJ /F3 11.955 Tf 11.96 0 Td[(1=infz>01)]TJ /F9 11.955 Tf 11.95 0 Td[(e)]TJ /F8 7.97 Tf 6.59 0 Td[(d(1=ez)1)]TJ /F6 7.97 Tf 6.59 0 Td[( 1)]TJ /F3 11.955 Tf 11.96 0 Td[((1=ez)=infx2(0,1)e)]TJ /F8 7.97 Tf 6.58 0 Td[(dx1)]TJ /F6 7.97 Tf 6.59 0 Td[()]TJ /F3 11.955 Tf 11.96 0 Td[(1 x)]TJ /F3 11.955 Tf 11.95 0 Td[(1, (A) whereequation A followsbyreplacing(1=ez) with x.Therefore,wehaveprovedequation A 113

PAGE 114

(Computation)Wecompute0bysearchingtheoptimalsolutiontotheminimizationprobleminfx2(0,1)ne)]TJ /F8 7.97 Tf 6.58 0 Td[(dx1)]TJ /F6 7.97 Tf 6.59 0 Td[()]TJ /F3 11.955 Tf 11.95 0 Td[(1 x)]TJ /F3 11.955 Tf 11.96 0 Td[(1o. First,bydenoting1)]TJ /F4 11.955 Tf 11.96 0 Td[(0=infx2(0,1)h(x),wehave h0(x)=1)]TJ /F9 11.955 Tf 11.95 0 Td[(e)]TJ /F8 7.97 Tf 6.59 0 Td[(dx1)]TJ /F6 7.97 Tf 6.59 0 Td[()]TJ /F9 11.955 Tf 11.96 0 Td[(e)]TJ /F8 7.97 Tf 6.59 0 Td[(d(1)]TJ /F4 11.955 Tf 11.95 0 Td[()x)]TJ /F6 7.97 Tf 6.58 0 Td[( (x)]TJ /F3 11.955 Tf 11.95 0 Td[(1)2,8x2(0,1). Itisclearthat(x)]TJ /F3 11.955 Tf 12.24 0 Td[(1)2decreasesasxincreases.Meanwhile,sincex<1andx)]TJ /F6 7.97 Tf 6.59 0 Td[()]TJ /F12 7.97 Tf 6.59 0 Td[(1>x)]TJ /F6 7.97 Tf 6.59 0 Td[(,wehave (1)]TJ /F9 11.955 Tf 11.95 0 Td[(e)]TJ /F8 7.97 Tf 6.59 -.01 Td[(dx1)]TJ /F6 7.97 Tf 6.58 .01 Td[()]TJ /F9 11.955 Tf 11.95 0 Td[(e)]TJ /F8 7.97 Tf 6.58 -.01 Td[(d(1)]TJ /F4 11.955 Tf 11.95 0 Td[()x)]TJ /F6 7.97 Tf 6.59 -.01 Td[()0x=e)]TJ /F8 7.97 Tf 6.59 -.01 Td[(d(1)]TJ /F4 11.955 Tf 11.95 0 Td[()(x)]TJ /F6 7.97 Tf 6.59 -.01 Td[()]TJ /F12 7.97 Tf 6.58 -.01 Td[(1)]TJ /F9 11.955 Tf 11.96 0 Td[(x)]TJ /F6 7.97 Tf 6.59 -.01 Td[()>0. Therefore,h0(x)increaseasxincreasesin(0,1),andhencethefunctionh(x)isconvexoverxin(0,1).Becauselimx!0+h0(x)=andlimx!1)]TJ /F9 11.955 Tf 8.25 5.81 Td[(h0(x)=+1,wehave: theinmumofh(x)isattainedintheinterval(0,1).(A) Wecancomputetheoptimalxbyforcing 1)]TJ /F9 11.955 Tf 11.96 0 Td[(e)]TJ /F8 7.97 Tf 6.59 0 Td[(d(x)1)]TJ /F6 7.97 Tf 6.59 0 Td[()]TJ /F9 11.955 Tf 11.96 0 Td[(e)]TJ /F8 7.97 Tf 6.59 0 Td[(d(1)]TJ /F4 11.955 Tf 11.95 0 Td[()(x))]TJ /F6 7.97 Tf 6.59 0 Td[( (x)]TJ /F3 11.955 Tf 11.96 0 Td[(1)2=0, i.e.,(x)=e)]TJ /F8 7.97 Tf 6.58 0 Td[(dx+e)]TJ /F8 7.97 Tf 6.59 0 Td[(d(1)]TJ /F4 11.955 Tf 10.73 0 Td[().Theintersectionoffunction s xande)]TJ /F8 7.97 Tf 6.58 0 Td[(dx+e)]TJ /F8 7.97 Tf 6.59 0 Td[(d(1)]TJ /F4 11.955 Tf 10.73 0 Td[()canbeeasilycompute d by a bisectionlinesearch.Finally,toachieveaccuracy,i.e.,j^x)]TJ /F9 11.955 Tf 12.84 0 Td[(xj,oftheincumbentprobingvalue^x,weonlyhavetoconductSstepsofbisection,suchthat2)]TJ /F8 7.97 Tf 6.59 0 Td[(S.ItfollowsthatSlog2(1 ). A.12ProofofProposition 8 Proof. Wedividetheproofintotwoparts.Weshowthemonotonicityintherstpart,andtheconvergenceinthesecondpart. (Monotonicity)Supposethatwearegivend1>d2>0.First,sinceisnondecreasingbyproperty(ii)inLemma 2 ,wehave(z0+z)(z0)foranygivenz0andz>0.Second,itcanbeshownthat(z0+z)>(z0),becauseotherwisetheDCC 114

PAGE 115

willbeviolated(theproofissimilartotheoneforTheorem 2 onrulingoutthecasez0=z0,m ()z0+z m()n(z0+z))]TJ /F9 11.955 Tf 11.95 0 Td[(z0)]TJ /F4 11.955 Tf 11.95 0 Td[(z+d1 (z0+z))]TJ /F4 11.955 Tf 11.95 0 Td[((z0)o1)]TJ /F3 11.955 Tf 41.78 0 Td[(infz>0,m ()z0+z m()n(z0+z))]TJ /F9 11.955 Tf 11.96 0 Td[(z0)]TJ /F4 11.955 Tf 11.95 0 Td[(z+d2 (z0+z))]TJ /F4 11.955 Tf 11.96 0 Td[((z0)o. Therefore,thevalueof0increasesasddecreases. (Convergence)Nowsupposethatd#0andwewanttoprovethat0".Since1)]TJ /F4 11.955 Tf 12.2 0 Td[(0canbedenedasinfz>0,m ()z0+z m()f(z0,z):=n(z0+z))]TJ /F9 11.955 Tf 11.95 0 Td[(z0)]TJ /F4 11.955 Tf 11.95 0 Td[(z+d (z0+z))]TJ /F4 11.955 Tf 11.95 0 Td[((z0)o, and1)]TJ /F4 11.955 Tf 11.95 0 Td[(0decreasesasddecreases,wehavelimd#0infz>0,m ()z0+z m()f(z0,z)=infz>0,m ()z0+z m()infd>0f(z0,z)=infz>0,m ()z0+z m()n(z0+z))]TJ /F9 11.955 Tf 11.96 0 Td[(z0)]TJ /F4 11.955 Tf 11.95 0 Td[(z (z0+z))]TJ /F4 11.955 Tf 11.96 0 Td[((z0)o. Weshowthatforz0=0,infz>0f(0,z)1)]TJ /F4 11.955 Tf 12.25 0 Td[(andhencetheconclusionfollows.First,weobservethat(0)=0,because(0)=supxf)]TJ /F4 11.955 Tf 15.27 0 Td[((x)g=)]TJ /F3 11.955 Tf 11.29 0 Td[(infx(x)andx=1isaglobalminimizerforbyassumption.Itfollowsthatf(0,z)=(z))]TJ /F4 11.955 Tf 11.96 0 Td[(z (z)=1)]TJ /F4 11.955 Tf 20.11 8.09 Td[(z (z). Second,sincex=1istheuniqueminimizerfor,wehave()0(0)=argmaxxf)]TJ /F4 11.955 Tf 15.28 0 Td[((x)g=1byapropertyofconvexconjugates.Hence,wehave(z)=z+o(z) basedonTaylorseries ,wherelimz#0o(z)=z=0.Itfollowsthat(z) z!1asz#0, 115

PAGE 116

andsof(0,z)=1)]TJ /F4 11.955 Tf 20.1 8.08 Td[(z (z)!1)]TJ /F4 11.955 Tf 11.95 0 Td[(asz#0, notingthatm ()0basedonthefactthat(x)xforx>0(seeproperty(iii)inLemma 2 ). Therefore,infz>0f(0,z)1)]TJ /F4 11.955 Tf 12.68 0 Td[(isproved,andtheproofiscompletebynotingthatdtendstozeroasNtendstoinnitybyassumption. A.13ProofofProposition 9 Proof. First,wedenef(d):=p d2+4d()]TJ /F4 11.955 Tf 11.96 0 Td[(2))]TJ /F3 11.955 Tf 11.95 0 Td[((1)]TJ /F3 11.955 Tf 11.95 0 Td[(2)d 2d+2. FromProposition 5 ,wehave0=)]TJ /F9 11.955 Tf 11.95 0 Td[(f(d). Weshowthatf(d)decreasesasddecreases.Tothisend,wehave f0(d)=1 2(d+1)2"(22)]TJ /F3 11.955 Tf 11.95 0 Td[(2+1)d+2()]TJ /F4 11.955 Tf 11.95 0 Td[(2) p d2+4d()]TJ /F4 11.955 Tf 11.95 0 Td[(2))]TJ /F3 11.955 Tf 11.95 0 Td[((1)]TJ /F3 11.955 Tf 11.95 0 Td[(2)#.(A) Toshowthatf0(d)>0,wecomparef0(d)andzerobythefollowinginequalities,wheretheinequalitiesbelowimplythoseabove.f0(d)>0((22)]TJ /F3 11.955 Tf 11.95 0 Td[(2+1)d+2()]TJ /F4 11.955 Tf 11.95 0 Td[(2) p d2+4d()]TJ /F4 11.955 Tf 11.95 0 Td[(2)>(1)]TJ /F3 11.955 Tf 11.96 0 Td[(2)((22)]TJ /F3 11.955 Tf 11.96 0 Td[(2+1)d+2()]TJ /F4 11.955 Tf 11.96 0 Td[(2)>(1)]TJ /F3 11.955 Tf 11.96 0 Td[(2)p d2+4d()]TJ /F4 11.955 Tf 11.95 0 Td[(2)((22)]TJ /F3 11.955 Tf 11.95 0 Td[(2+1)d+2()]TJ /F4 11.955 Tf 11.95 0 Td[(2)2>(1)]TJ /F3 11.955 Tf 11.96 0 Td[(2)2d2+4d()]TJ /F4 11.955 Tf 11.96 0 Td[(2)(42(1)]TJ /F4 11.955 Tf 11.96 0 Td[()2(d+1)2>0. Hence,f0(d)>0andso0increasesasddecreases.Furthermore,sincelimd#0f(d)=0,wehavelimd#00=)]TJ /F3 11.955 Tf 11.95 0 Td[(limd#0f(d)=, 116

PAGE 117

andso0increasestoasddecreasestozero. Second,sinced=d(N)byassumption,wehaveVoD=d0 dN=d0 dddd(N) dN=f0(d)dd(N) dN. (A) Theproofiscompletebysubstitutingthedenitionoff0(d)in A intoequation A A.14ProofofProposition 10 Proof. Wedividetheproofintothreeparts.Weshowtheconvergenceintherstpart,developtherelationshipbetween0anddinthesecondpart,andcomputethevalueofdata in thelastpart. (Convergence)FromProposition 7 ,weknowthat 1)]TJ /F4 11.955 Tf 11.95 0 Td[(0=infx2(0,1)ne)]TJ /F8 7.97 Tf 6.58 0 Td[(dx1)]TJ /F6 7.97 Tf 6.59 0 Td[()]TJ /F3 11.955 Tf 11.96 0 Td[(1 x)]TJ /F3 11.955 Tf 11.95 0 Td[(1o.(A) Itisclearthate)]TJ /F8 7.97 Tf 6.58 0 Td[(d"1asd#0,andsowehave 0=1)]TJ /F3 11.955 Tf 19.43 0 Td[(infx2(0,1)ne)]TJ /F8 7.97 Tf 6.58 0 Td[(dx1)]TJ /F6 7.97 Tf 6.59 0 Td[()]TJ /F3 11.955 Tf 11.96 0 Td[(1 x)]TJ /F3 11.955 Tf 11.95 0 Td[(1ox??1)]TJ /F3 11.955 Tf 19.44 0 Td[(infx2(0,1)nx1)]TJ /F6 7.97 Tf 6.59 0 Td[()]TJ /F3 11.955 Tf 11.96 0 Td[(1 x)]TJ /F3 11.955 Tf 11.96 0 Td[(1o=1)]TJ /F3 11.955 Tf 11.96 0 Td[((1)]TJ /F4 11.955 Tf 11.96 0 Td[()=, whichprovesthat0increasestoasddecreasestozero. (Relationshipbetween0andd)FromProposition 7 ,weknowthattheoptimalvalueoftheembeddedoptimizationprobleminequality A canbeattainedbysomex2(0,1) (basedonclaim A intheproofofProposition 7 ) ,whichisthestationarypointoftheobjectivefunction.Itfollowsthat 8>><>>:e)]TJ /F16 5.978 Tf 5.76 0 Td[(dx1)]TJ /F21 5.978 Tf 5.76 0 Td[()]TJ /F12 7.97 Tf 6.58 0 Td[(1 x)]TJ /F12 7.97 Tf 6.59 0 Td[(1=1)]TJ /F4 11.955 Tf 11.96 0 Td[(0x=e)]TJ /F8 7.97 Tf 6.58 0 Td[(dx+e)]TJ /F8 7.97 Tf 6.58 0 Td[(d(1)]TJ /F4 11.955 Tf 11.95 0 Td[() (A) 117

PAGE 118

Solv ing thisnonlinearequationsystem, wereformulatetherstequationandthensubstitute thesecondequationintotherst asfollows :e)]TJ /F8 7.97 Tf 6.59 0 Td[(dx)]TJ /F3 11.955 Tf 11.95 0 Td[((1)]TJ /F4 11.955 Tf 11.95 0 Td[(0)xx=0x)e)]TJ /F8 7.97 Tf 6.59 0 Td[(dx)]TJ /F3 11.955 Tf 11.95 0 Td[((1)]TJ /F4 11.955 Tf 11.95 0 Td[(0)xe)]TJ /F8 7.97 Tf 6.59 0 Td[(dx+e)]TJ /F8 7.97 Tf 6.59 0 Td[(d(1)]TJ /F4 11.955 Tf 11.96 0 Td[()=0e)]TJ /F8 7.97 Tf 6.59 0 Td[(dx+e)]TJ /F8 7.97 Tf 6.59 0 Td[(d(1)]TJ /F4 11.955 Tf 11.96 0 Td[())(x)]TJ /F3 11.955 Tf 11.96 0 Td[(1)(1)]TJ /F4 11.955 Tf 11.95 0 Td[(0)x)]TJ /F4 11.955 Tf 11.96 0 Td[(0(1)]TJ /F4 11.955 Tf 11.96 0 Td[()=0. Rulingoutthesolutionx=1,wehavex=0(1)]TJ /F6 7.97 Tf 6.59 0 Td[() (1)]TJ /F6 7.97 Tf 6.58 0 Td[(0)2(0,1).Finally,wesubstitutethesolutionofxback in tothesecondequationin A andobtain e)]TJ /F8 7.97 Tf 6.59 0 Td[(d=x(x+1)]TJ /F4 11.955 Tf 11.96 0 Td[()=)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(0)]TJ /F3 11.955 Tf 5.48 -9.69 Td[((1)]TJ /F4 11.955 Tf 11.95 0 Td[(0)(1)]TJ /F4 11.955 Tf 11.96 0 Td[()1)]TJ /F6 7.97 Tf 6.59 0 Td[(.(A) Finally,bytakingthenaturallogarithmonbothsidesofequation A ,weobtainthat d=log 0+(1)]TJ /F4 11.955 Tf 11.96 0 Td[()log1)]TJ /F4 11.955 Tf 11.96 0 Td[( 1)]TJ /F4 11.955 Tf 11.96 0 Td[(0.(A) (Valueofdata)Fromequation A wehavedd d0=)]TJ /F4 11.955 Tf 11.89 8.09 Td[( 0+1)]TJ /F4 11.955 Tf 11.96 0 Td[( 1)]TJ /F4 11.955 Tf 11.95 0 Td[(0=0)]TJ /F4 11.955 Tf 11.95 0 Td[( 0(1)]TJ /F4 11.955 Tf 11.96 0 Td[(0). Itiseasytoobservethatddd0isamonotonefunctionof0andddd06=0.Hence,wehaved0 dd=1.dd d0=0(1)]TJ /F4 11.955 Tf 11.95 0 Td[(0) 0)]TJ /F4 11.955 Tf 11.95 0 Td[(. Therefore,VoD=d0 dddd(N) dN=0(1)]TJ /F4 11.955 Tf 11.96 0 Td[(0) 0)]TJ /F4 11.955 Tf 11.96 0 Td[(d0(N). 118

PAGE 119

APPENDIXBDETAILEDPROOFSFORCHAPTER4 B.1ProofofProposition 12 Proof. Bydenitionofprobabilitymeasure,wehaveinfP2DP(TXt=1BXb=1rbtTXt=1BXb=1~Rbt)1)]TJ /F4 11.955 Tf 11.96 0 Td[(,infP2DEP(I TXt=1BXb=1rbtTXt=1BXb=1~Rbt!)1)]TJ /F4 11.955 Tf 11.96 0 Td[(, whereI(A)representstheindicatorfunctionofeventA,i.e.,I(A)=1ifAhappensandI(A)=0otherwise.ItfollowsthattheDCC 4 isequivalentto supP2DEP()]TJ /F9 11.955 Tf 9.3 0 Td[(I TXt=1BXb=1rbtTXt=1BXb=1~Rbt!))]TJ /F3 11.955 Tf 11.95 0 Td[(1, which,fromTheorem 6 ,isfurtherequivalentto dvC1+vC2+NXn=1)]TJ /F9 11.955 Tf 5.48 -9.68 Td[(uC3n)]TJ /F9 11.955 Tf 11.96 0 Td[(uC2n)]TJ /F9 11.955 Tf 11.96 0 Td[(p0nuC1n)]TJ /F3 11.955 Tf 11.95 0 Td[(1vC2)]TJ /F9 11.955 Tf 11.96 0 Td[(uC1n+I TXt=1BXb=1rbtTXt=1BXb=1~Rbnt!0,8n=1,...,N, (B) 4vC1)]TJ /F9 11.955 Tf 11.95 0 Td[(uC2n)]TJ /F9 11.955 Tf 11.96 0 Td[(uC3n0,8n=1,...,N,264uC1nuC2n375uC3n,8n=1,...,N,vC10. Toreformulateconstraints B ,weintroduceanadditionalbinaryvariablewnforeachscenariontorepresenttheindicatorfunction.Theremainingtaskistoshowthatconstraints B canbereformulatedasconstraints 4b and 4e byusingbinaryvariableswn.Toseethis,wediscussthefollowingcases. Case1. whenPTt=1PBb=1rbt
PAGE 120

Case2. whenPTt=1PBb=1rbtPTt=1PBb=1~Rbnt,theindicatorfunctionequalsoneandmeanwhilewncanbezerooronebyconstraints 4e .Butduetoconstraints 4b ,wncanbesettoequaloneatoptimality. Hence,wehavewn=IPTt=1PBb=1rbtPTt=1PBb=1~Rbntandsoconstraints B equivalenttoconstraints 4b and 4e ,whichcompletestheproof. B.2ProofofProposition 13 Proof. Leth(~R)=PTt=1PBb=1~Rbt)]TJ /F15 11.955 Tf 11.96 8.97 Td[(PTt=1PBb=1rbt++PTt=1PBb=1rbt)]TJ /F3 11.955 Tf 13.31 2.66 Td[(~Rbt+,thenbyTheorem 6 theworst-caseexpectationofpenaltycostsupP2DEP[h(~R)]equalstheoptimalobjectivevalueofproblem 4a 4e withhn= TXt=1BXb=1~Rbnt)]TJ /F8 7.97 Tf 16.99 14.94 Td[(TXt=1BXb=1rbt!++TXt=1BXb=1rbt)]TJ /F3 11.955 Tf 13.32 2.65 Td[(~Rbnt+. Sincehnisapiecewiselinearfunctionof~Rn,itisequivalenttoreplaceconstraints 4b byconstraints 4b 4e 4f andadditionalvariablesz0nandzbnt.Therefore,theproofiscomplete. B.3ProofofProposition 14 Proof. Leth(~R)=Q(y,~R),thenbyTheorem 6 theworst-caseexpectationofpenaltycostsupP2DEP[h(~R)]equalstheoptimalobjectivevalueofproblem 4a 4e withhn=minxn0q>xns.t.Ly+Wxn~gn foreachn=1,...,N,wherexnandgnrepresentthedecisionvariablesandright-handsidesofsecond-stageproblem 4a 4e correspondingtorandomvectorrealization~Rn,respectively.Itremainstoprovethatconstraints 4b and 4e areequivalentto 120

PAGE 121

constraintsvS2)]TJ /F9 11.955 Tf 11.95 0 Td[(uS1nminxn0q>xn,8n=1,...,N. (B)s.t.Ly+Wxn~gn, Tothatend,werstobservethatconstraints 4b and 4e implyconstraints B because B incorporatesaminimizationproblemoverq>xn.Second,weletxnrepresentanoptimalsolutiontotheincorporatedoptimizationprobleminconstraints B .Thenvariables(vS2,uS1n,xn)satisfyconstraints 4b and 4e .Itfollowsthatconstraints B alsoimplyconstraints 4b and 4e ,whichcompletestheproof. B.4ProofofProposition 17 Proof. Werstprovethevalidityofthestarinequality,andthenshowtherepresentationofconv(P).Noteherethatthevalidityproofissimilartotheonein[ 2 ],andwepresentadetailedproofinthischapterforcompleteness. (Validity)Foranysubset N=fn1,...,nkgNsuchthat^Rnj)]TJ /F17 5.978 Tf 5.76 0 Td[(1^Rnjforj=2,...,k,wediscusstwocases.First,ifwnj=0foreachj=1,...,k,thenthestarinequality 4 holdsbecauser0.Second,ifthereexistsj2f1,...,kgsuchthatwnj=1,weletjrepresentthelargestindexwithwnj=1,i.e.,j=argmaxfj=2,...,k:wnj=1g.Itfollowsthatr^Rnj (B)=jXj=1^Rnj)]TJ /F3 11.955 Tf 13.32 2.66 Td[(^Rnj)]TJ /F17 5.978 Tf 5.75 0 Td[(1+kXj=j+1^Rnj)]TJ /F3 11.955 Tf 13.32 2.66 Td[(^Rnj)]TJ /F17 5.978 Tf 5.75 0 Td[(1wnj (B)jXj=1^Rnj)]TJ /F3 11.955 Tf 13.32 2.66 Td[(^Rnj)]TJ /F17 5.978 Tf 5.75 0 Td[(1wnj+kXj=j+1^Rnj)]TJ /F3 11.955 Tf 13.31 2.66 Td[(^Rnj)]TJ /F17 5.978 Tf 5.76 0 Td[(1wnj=Xj2 N^Rnj)]TJ /F3 11.955 Tf 13.32 2.66 Td[(^Rnj)]TJ /F17 5.978 Tf 5.75 0 Td[(1wnj, (B) whereinequality B isduetothedenitionofP,equality B isbecausewnj=0forallj=j+1,...,k,andinequality B isdueto(i)^Rnj^Rnj)]TJ /F17 5.978 Tf 5.76 0 Td[(1forj2 N,and(ii)wnj1.Hence,thestarinequality 4 alsoholdsinthiscase. 121

PAGE 122

(Representation)WedenepolyhedronQ:=8<:(r,w)2R+[0,1]N:rXj2 N^Rnj)]TJ /F3 11.955 Tf 13.32 2.65 Td[(^Rnj)]TJ /F17 5.978 Tf 5.75 0 Td[(1wnj,8 NN9=;. Toshowthatconv(P)=Q,weprovethatforanyc:=(cr,cw)2RN+1suchthatc6=0,anyoptimalsolution(r,w)totheproblemminfcrr+PNn=1cwnwn:(r,w)2Pgsatisesastarinequality 4 forsome NNoratrivialboundatequality.First,weobservethatPandQhavethesamerecessionconef(a,0):a0g,andsowecanassumecr0withoutlossofgeneralitybecauseotherwisetheoptimizationproblemisunbounded.Second,ifcr=0,thenthereexistsann2f1,...,Ngsuchthatcwn6=0becausec6=0.Itisclearthatatoptimalitywn=0ifcwn>0,orotherwisewn=1ifcwn<0.Hence,anyoptimalsolutionsatiseseithertrivialboundwn0orwn1atequality.Third,ifcr>0,thenatoptimalityr=maxf^Rn:wn=1g.Wediscussthefollowingcases. Case1. Ifwn=0foreachn=1,...,N,thenr=0andthecorrespondingoptimalsolutionsatisestrivialboundr0atequality. Case2. Ifw6=0,thenweletnrepresenttheindexwhichattainsthemaximumforr,i.e.,n:=argmaxf^Rn:wn=1g.Itfollowsthatwn=0if^Rn>^Rn,andwn=1if^Rn^Rnandcwn<0.Finally,wedene N=fng[fn2f1,...,Ng:^Rn^Rnandcwn<0g,andthecorrespondingoptimalsolutionsatisesthestarinequalityrPj2 N(^Rnj)]TJ /F3 11.955 Tf 13.31 2.65 Td[(^Rnj)]TJ /F17 5.978 Tf 5.76 0 Td[(1)wnjatequality. B.5ProofofLemma 7 Proof. First,wedene^Rn:=PTt=1PBb=1~Rbnt,^R0=0,andassumethat^R1^RNwithoutlossofgeneralitybecausewecansortf^Rn:n=1,...,Ngtobeso.Since1)]TJ /F4 11.955 Tf 10.54 0 Td[(6=PKn=1p0nforanyK=1,...,N,wealsodenen=minfK2f1,...,Ng:PKn=1p0n>1)]TJ /F4 11.955 Tf 12.42 0 Td[(g.Inaddition,wedene d:=minf4 Np2min,4 Nd2mingwherepmin=minn=1,...,Nfp0nganddmin=Pnn=1p0n)]TJ /F3 11.955 Tf 11.95 0 Td[((1)]TJ /F4 11.955 Tf 11.95 0 Td[(). 122

PAGE 123

Second,foranygivenfrbt:t2T,b2Bg,wedenem(r):=maxfn2f0,...,Ng:PTt=1PBb=1rbt^Rng.Noteherethatm(r)iswell-denedbecause^Rn0foreachn=1,...,N,andrbt0foreacht2Tandb2B. Third,wecomputetheleft-handside(LHS)oftheDCC 4 ,i.e.,theworst-caseprobabilitybound.Bythedenitionofm(r),wehaveLHS:=infP2DP(TXt=1BXb=1rbtTXt=1BXb=1~Rbt)=infP2DNXn=1I(TXt=1BXb=1rbt^Rn)=minp0m(r)Xn=1pns.t.NXn=1pn=1,NXn=1(pn)]TJ /F9 11.955 Tf 11.95 0 Td[(p0n)2d. Form(r)=1,...,N)]TJ /F3 11.955 Tf 11.96 0 Td[(1,wesolvethecorrespondingoptimizationproblemasfollows: LHS=m(r)Xn=1p0n+minm(r)Xn=1n (Ba)s.t.NXn=1n=0, (Bb)NXn=12nd, (Bc)n)]TJ /F9 11.955 Tf 21.91 0 Td[(p0n,8n=1,...,N (Bd)=m(r)Xn=1p0n)]TJ /F15 11.955 Tf 12.62 19.42 Td[(r m(r)(N)]TJ /F3 11.955 Tf 13.8 0 Td[(m(r))d N. (Be) 123

PAGE 124

Toseewhyequality Be holds,wesolvearelaxationofproblem Ba Bd byrelaxingconstraints Bd asfollows:minm(r)Xn=1ns.t.NXn=1n=0,NXn=12nd,=max0,2Rminm(r)Xn=1n+ NXn=12n)]TJ /F9 11.955 Tf 11.96 0 Td[(d!+ NXn=1n!=max0,2Rminm(r)Xn=12n+(+1)n+NXn=m(r)+1(2n+n))]TJ /F4 11.955 Tf 11.96 0 Td[(d=max>0,2R)]TJ /F3 11.955 Tf 12.34 8.09 Td[(m(r)(+1)2 4)]TJ /F3 11.955 Tf 13.15 8.09 Td[((N)]TJ /F3 11.955 Tf 13.8 0 Td[(m(r))2 4)]TJ /F4 11.955 Tf 11.96 0 Td[(d=)]TJ /F3 11.955 Tf 20.15 0 Td[(min>0,2RN2+2m(r)+m(r) 4+d=)]TJ /F3 11.955 Tf 15.27 0 Td[(min>0m(r)(N)]TJ /F12 7.97 Tf 7.88 0 Td[(m(r)) N 4+d=)]TJ /F15 11.955 Tf 11.95 19.41 Td[(r m(r)(N)]TJ /F3 11.955 Tf 13.8 0 Td[(m(r))d N, whosecorrespondingoptimalsolutionisn=)]TJ /F15 11.955 Tf 9.3 13.6 Td[(q (N)]TJ /F12 7.97 Tf 7.89 0 Td[(m(r))d Nm(r)forn=1,...,m(r)andn=q m(r)d N(N)]TJ /F12 7.97 Tf 7.88 0 Td[(m(r))forn=m(r)+1,...,N.Theoptimalsolutionsatisesconstraints Bd becausen0)]TJ /F9 11.955 Tf 21.92 0 Td[(p0nforeachn=m(r)+1,...,Nandn=)]TJ /F15 11.955 Tf 9.3 21.67 Td[(s (N)]TJ /F3 11.955 Tf 13.8 0 Td[(m(r))d Nm(r))]TJ /F15 11.955 Tf 25.25 21.11 Td[(s 1 m(r))]TJ /F3 11.955 Tf 14.71 8.09 Td[(1 Np d)]TJ /F3 11.955 Tf 26.44 8.09 Td[(2pmin p N)]TJ /F9 11.955 Tf 25.24 0 Td[(pmin)]TJ /F9 11.955 Tf 25.24 0 Td[(p0n foreachn=1,...,m(r).Itfollowsthattheworst-caseprobabilityboundsatisesequality Be form(r)=1,...,N)]TJ /F3 11.955 Tf 12.55 0 Td[(1.Buttheequality Be clearlyholdswhen(i)m(r)=0,becauseLHS=0whenPTt=1PBb=1rbt^Rnforalln=1,...,N,and(ii)m(r)=N, 124

PAGE 125

becauseLHS=PNn=1p0n=1whenPTt=1PBb=1rbt^RN.Hence,wehave LHS=m(r)Xn=1p0n)]TJ /F15 11.955 Tf 15.27 19.42 Td[(r m(r)(N)]TJ /F3 11.955 Tf 13.81 0 Td[(m(r))d N,8m(r)=0,...,N.(B) Fourth,weprovetheequivalencebetweentheDCC 4 andconstraint 4 ford2[0, d].Fornecessity,supposethatagivensolutionfrbt:t2T,b2BgsatisestheDCC 4 anddoesnotsatisfyconstraint 4 .Itfollowsthatm(r)
PAGE 126

REFERENCES [1] P.Artzner,F.Delbaen,J.-M.Eber,andD.Heath.Coherentmeasuresofrisk.MathematicalFinance,9(3):203,1999. [2] A.Atamturk,G.L.Nemhauser,andM.Savelsbergh.Themixedvertexpackingproblem.MathematicalProgramming,89(1):35,2000. [3] R.Barth,H.Brand,P.Meibom,andC.Weber.Astochasticunit-commitmentmodelfortheevaluationoftheimpactsofintegrationoflargeamountsofintermittentwindpower.InternationalConferenceonProbabilisticMethodsAppliedtoPowerSystems,pages1,2006. [4] E.Beale.Onminimizingaconvexfunctionsubjecttolinearinequalities.JournaloftheRoyalStatisticalSociety.SeriesB(Methodological),pages173,1955. [5] A.Ben-Tal,D.denHertog,A.DeWaegenaere,B.Melenberg,andG.Rennen.Robustsolutionsofoptimizationproblemsaffectedbyuncertainprobabilities.ManagementScience,59(2):341,2013. [6] A.Ben-Tal,A.Goryashko,E.Guslitzer,andA.Nemirovski.Adjustablerobustsolutionsofuncertainlinearprograms.MathematicalProgramming,99(2):351,2004. [7] A.Ben-TalandA.Nemirovski.Robustsolutionsoflinearprogrammingproblemscontaminatedwithuncertaindata.MathematicalProgramming,88(3):411,2000. [8] D.BertsimasandV.Goyal.Onthepowerandlimitationsofafnepoliciesintwo-stageadaptiveoptimization.Mathematicalprogramming,134(2):491,2012. [9] D.Bertsimas,E.Litvinov,X.Sun,J.Zhao,andT.Zheng.Adaptiverobustoptimizationforthesecurityconstrainedunitcommitmentproblem.IEEETransac-tionsonPowerSystems,28(1):52,2013. [10] D.BertsimasandM.Sim.Thepriceofrobustness.OperationsResearch,52(1):35,2004. [11] J.BirgeandF.Louveaux.Introductiontostochasticprogramming.Springer,1997. [12] F.BouffardandF.Galiana.Stochasticsecurityforoperationsplanningwithsignicantwindpowergeneration.InPowerandEnergySocietyGeneralMeeting-ConversionandDeliveryofElectricalEnergyinthe21stCentury,Pittsburgh,PA,2008. [13] G.C.Calaore.Ambiguousriskmeasuresandoptimalrobustportfolios.SIAMJournalonOptimization,18(3):853,2007. 126

PAGE 127

[14] G.C.CalaoreandM.C.Campi.Uncertainconvexprograms:Randomizedsolutionsandcondencelevels.MathematicalProgramming,102(1):25,2005. [15] G.C.CalaoreandM.C.Campi.Thescenarioapproachtorobustcontroldesign.IEEETransactionsonAutomaticControl,51(5):742,2006. [16] G.C.CalaoreandL.ElGhaoui.Ondistributionallyrobustchance-constrainedlinearprograms.JournalofOptimizationTheoryandApplications,130(1):1,2006. [17] A.CharnesandW.W.Cooper.Deterministicequivalentsforoptimizingandsatiscingunderchanceconstraints.OperationsResearch,11(1):18,1963. [18] A.Charnes,W.W.Cooper,andG.H.Symonds.Costhorizonsandcertaintyequivalents:anapproachtostochasticprogrammingofheatingoil.ManagementScience,4(3):235,1958. [19] W.Chen,M.Sim,J.Sun,andC.P.Teo.FromCVaRtouncertaintyset:Implicationsinjointchanceconstrainedoptimization.OperationsResearch,58(2):470,2010. [20] X.ChenandY.Zhang.Uncertainlinearprograms:Extendedafnelyadjustablerobustcounterparts.OperationsResearch,57(6):1469,2009. [21] T.Cormen,C.Leiserson,R.Rivest,andC.Stein.IntroductiontoAlgorithms.MITpress,2001. [22] G.Dantzig.Linearprogrammingunderuncertainty.Managementscience,1(3-4):197,1955. [23] E.DelageandY.Ye.Distributionallyrobustoptimizationundermomentuncertaintywithapplicationtodata-drivenproblems.OperationsResearch,58(3):595,2010. [24] F.Delbaen.Coherentriskmeasuresongeneralprobabilityspaces.InAdvancesinFinanceandStochastics,pages1.Springer,2002. [25] L.DevroyeandL.Gyor.NonparametricDensityEstimation:The`1View.JohnWiley&SonsInc,1985. [26] L.ElGhaoui,M.Oks,andF.Oustry.Worst-casevalue-at-riskandrobustportfoliooptimization:Aconicprogrammingapproach.OperationsResearch,51(4):543,2003. [27] E.ErdoganandG.Iyengar.Ambiguouschanceconstrainedproblemsandrobustoptimization.MathematicalProgramming,107(1):37,2006. [28] S.Fink,C.Mudd,K.Porter,andB.Morgenstern.Windenergycurtailmentcasestudies.NationalRenewableEnergyLaboratoryReport,October2009. 127

PAGE 128

[29] H.FollmerandA.Schied.Stochasticnance:Anintroductionindiscretetime.TheMathematicalIntelligencer,26(4):67,2004. [30] X.Guan,P.Luh,J.Yan,andJ.Amal.Anoptimization-basedmethodforunitcommitment.InternationalJournalofElectricalPower&EnergySystems,14(1):9,1992. [31] K.Isii.Theextremaofprobabilitydeterminedbygeneralizedmoments(I)boundedrandomvariables.AnnalsoftheInstituteofStatisticalMathematics,12(2):119,1960. [32] R.Jiang,J.Wang,andY.Guan.Robustunitcommitmentwithwindpowerandpumpedstoragehydro.IEEETransactionsonPowerSystems,27(2):800,2012. [33] P.KallandS.Wallace.Stochasticprogramming.JohnWileyandSons,1994. [34] S.Kucukyavuz.Onmixingsetsarisinginchance-constrainedprogramming.MathematicalProgramming,132(1):31,2010. [35] M.A.Lejeune.Pattern-basedmodelingandsolutionofprobabilisticallyconstrainedoptimizationproblems.OperationsResearch,60(6):1356,2012. [36] S.Lindenberg.20%windenergyby2030:Increasingwindenergy'scontributiontou.s.electricitysupply.TechnicalReportDOE/GO-102008-2567,U.S.DepartmentofEnergy,July2008. [37] J.Luedtke.Abranch-and-cutdecompositionalgorithmforsolvinggeneralchance-constrainedmathematicalprograms.Preprintavailableatwww.optimization-online.org,2011. [38] J.LuedtkeandS.Ahmed.Asampleapproximationapproachforoptimizationwithprobabilisticconstraints.SIAMJournalonOptimization,19(2):674,2008. [39] J.Luedtke,S.Ahmed,andG.L.Nemhauser.Anintegerprogrammingapproachforlinearprogramswithprobabilisticconstraints.MathematicalProgramming,122(2):247,2010. [40] B.L.MillerandH.M.Wagner.Chanceconstrainedprogrammingwithjointconstraints.OperationsResearch,13(6):930,1965. [41] A.NemirovskiandA.Shapiro.Scenarioapproximationsofchanceconstraints.InG.CalaoreandF.Dabbene,editors,ProbabilisticandRandomizedMethodsforDesignunderUncertainty,pages3.Springer,2006. [42] A.NemirovskiandA.Shapiro.Convexapproximationsofchanceconstrainedprograms.SIAMJournalonOptimization,17(4):969,2007. 128

PAGE 129

[43] U.Ozturk,M.Mazumdar,andB.Norman.Asolutiontothestochasticunitcommitmentproblemusingchanceconstrainedprogramming.IEEETransac-tionsonPowerSystems,19(3):1589,2004. [44] B.Pagnoncelli,S.Ahmed,andA.Shapiro.Sampleaverageapproximationmethodforchanceconstrainedprogramming:theoryandapplications.JournalofOptimizationTheoryandApplications,142(2):399,2009. [45] A.Papavasiliou,S.Oren,andR.O'Neill.Reserverequirementsforwindpowerintegration:ascenario-basedstochasticprogrammingframework.IEEETransac-tionsonPowerSystems,26(4):2197,2011. [46] L.Pardo.StatisticalInferenceBasedonDivergenceMeasures,volume185.CRCPress,2006. [47] E.Parzen.Onestimationofaprobabilitydensityfunctionandmode.TheAnnalsofMathematicalStatistics,33(3):1065,1962. [48] J.Pinter.Deterministicapproximationsofprobabilityinequalities.MathematicalMethodsofOperationsResearch,33(4):219,1989. [49] I.PolikandT.Terlaky.AsurveyoftheS-Lemma.SIAMReview,49(3):371,2007. [50] A.Prekopa.Onprobabilisticconstrainedprogramming.InProceedingsofthePrincetonSymposiumonMathematicalProgramming,pages113.Citeseer,1970. [51] A.Prekopa.StochasticProgramming.Springer,1995. [52] R.RockafellarandS.Uryasev.Conditionalvalue-at-riskforgenerallossdistributions.JournalofBanking&Finance,26(7):1443,2002. [53] R.T.RockafellarandS.Uryasev.Optimizationofconditionalvalue-at-risk.JournalofRisk,2:21,2000. [54] M.Rosenblatt.Remarksonsomenonparametricestimatesofadensityfunction.TheAnnalsofMathematicalStatistics,27(3):832,1956. [55] P.Ruiz,C.Philbrick,E.Zak,K.Cheung,andP.Sauer.Uncertaintymanagementintheunitcommitmentproblem.IEEETransactionsonPowerSystems,24(2):642,2009. [56] T.Santoso,S.Ahmed,M.Goetschalckx,andA.Shapiro.Astochasticprogrammingapproachforsupplychainnetworkdesignunderuncertainty.EuropeanJournalofOperationalResearch,167(1):96,2005. [57] S.Sen,R.Doverspike,andS.Cosares.Networkplanningwithrandomdemand.TelecommunicationSystems,3(1):11,1994. 129

PAGE 130

[58] A.Shapiro.Ondualitytheoryofconiclinearproblems.InSemi-InniteProgram-ming,pages135.KluwerAcademicPublishers,2000. [59] A.Shapiro,D.Dentcheva,andA.Ruszczynski.Lecturesonstochasticprogram-ming:modelingandtheory,volume9.SIAM,2009. [60] A.ShapiroandK.Scheinberg.Dualityandoptimalityconditions.InH.Wolkowicz,R.Saigal,andL.Vandenberghe,editors,HandbookofSemideniteProgramming,volume27ofInternationalSeriesinOperationsResearch&ManagementScience,pages67.Springer,2000. [61] J.Smith.GeneralizedChebychevinequalities:Theoryandapplicationsindecisionanalysis.OperationsResearch,43(5):807,1995. [62] A.L.Soyster.Convexprogrammingwithset-inclusiveconstraintsandapplicationstoinexactlinearprogramming.OperationsResearch,21(5):1154,1973. [63] A.Street,F.Oliveira,andJ.Arroyo.Contingency-constrainedunitcommitmentwithn-Ksecuritycriterion:arobustoptimizationapproach.IEEETransactionsonPowerSystems,26(3):1581,2011. [64] S.Takriti,J.Birge,andE.Long.Astochasticmodelfortheunitcommitmentproblem.IEEETransactionsonPowerSystems,11(3):1497,1996. [65] A.Tuohy,P.Meibom,E.Denny,andM.O'Malley.Unitcommitmentforsystemswithsignicantwindpenetration.IEEETransactionsonPowerSystems,24(2):592,2009. [66] L.Vandenberghe,S.Boyd,andK.Comanor.GeneralizedChebyshevboundsviasemideniteprogramming.SIAMReview,49(1):52,2007. [67] J.Wang,M.Shahidehpour,andZ.Li.Security-constrainedunitcommitmentwithvolatilewindpowergeneration.IEEETransactionsonPowerSystems,23(3):1319,2008. [68] Q.Wang,Y.Guan,andJ.Wang.Achance-constrainedtwo-stagestochasticprogramforunitcommitmentwithuncertainwindpoweroutput.IEEETransactionsonPowerSystems,27(1):206,2012. [69] S.Wang,S.Shahidehpour,D.Kirschen,S.Mokhtari,andG.Irisarri.Short-termgenerationschedulingwithtransmissionandenvironmentalconstraintsusinganaugmentedlagrangianrelaxation.IEEETransactionsonPowerSystems,10(3):1294,1995. [70] B.Whittle,J.Yu,S.Teng,andJ.Mickey.ReliabilityunitcommitmentinERCOT.InProceedingsoftheIEEEPowerEngineeringSocietyGeneralMeeting,Montreal,Quebec,2006. 130

PAGE 131

[71] H.Xu,C.Caramanis,andS.Mannor.Optimizationunderprobabilisticenvelopeconstraints.OperationsResearch,60(3):682,2012. [72] V.Yakubovich.S-procedureinnonlinearcontroltheory.VestnikLeningradUniversity,4:73,1977. [73] H.Yu,C.Chung,K.Wong,andJ.Zhang.Achanceconstrainedtransmissionnetworkexpansionplanningmethodwithconsiderationofloadandwindfarmuncertainties.IEEETransactionsonPowerSystems,24(3):1568,2009. [74] S.Zymler,D.Kuhn,andB.Rustem.Distributionallyrobustjointchanceconstraintswithsecond-ordermomentinformation.MathematicalProgramming,137(1-2):167,2013. 131

PAGE 132

BIOGRAPHICALSKETCH RuiweiJiangwasborninXiamen,FujianinChinain1986.HeistheonlychildofQingxinYeandJiguangJiang.RuiweiearnedhisBachelordegreeinIndustrialEngineeringfromTsinghuaUniversity,Beijingin2009.RuiweijoinedtheDepartmentofIndustrialandSystemsEngineeringattheUniversityofFlorida(UF)inAugust2009,startinghisdoctoralstudyunderthesupervisionofDr.YongpeiGuan.HereceivedhisPh.D.fromtheUniversityofFloridainthesummerof2013.Followinggraduation,hejoinedtheDepartmentofSystemsandIndustrialEngineeringattheUniversityofArizonaasafacultymember.Ruiweiwonthe2013CommitteeofStochasticProgrammingBestStudentPaperAwardasthebestpaperintheory.Hewasalsooneoftherecipientsofthe2010,2011,and2012UFOutstandingAcademicAchievementsAward,theOfceofResearchTravelGrants,theGraduateStudentCouncilTravelGrants,andthe2013MixedIntegerProgrammingWorkshopStudentTravelAward. 132