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New Approaches to Robust Optimization with Applications

Permanent Link: http://ufdc.ufl.edu/UFE0041990/00001

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Title: New Approaches to Robust Optimization with Applications
Physical Description: 1 online resource (94 p.)
Language: english
Creator: Boyko, Mykyta
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

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Subjects / Keywords: 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: The dissertation focuses on a popular percentile-based risk measure known as Conditional Value at Risk (CVaR). CVaR has emerged from financial applications and allows obtaining optimal solutions that are robust with respect to various uncertainties incorporated into models as random variables. In this work I solve three operations research problems using CVaR-based robustness: multi-sensor scheduling problem, network-flow and system identification for space weather prediction. I develop mathematical programming techniques for solving a class of multi-sensor scheduling problems and formulate robust optimization problems for both deterministic and stochastic cases using linear 0-1 programming techniques. Equivalent formulations are developed in terms of cardinality constraints. I conducted numerical case studies and analyzed the performance of optimization solvers on the problems under consideration. The next problem I consider is robust network flow problem. I propose a new stochastic formulation of minimum cost flow problem aimed at finding network design and flow assignments subject to uncertain factors, such as network component disruptions/failures. I introduced loss function that is proportional to actual loss of flow in the network. In order to quantify the uncertain loss caused by network failures, I utilized CVaR risk measure. The combination of Lagrangian Relaxation and Benders' decomposition is proposed to solve large problems. Predicting geomagnetic activity applied for Dst-index forecasting has been performed through robust identification of discrete dynamic system. The modeling assumes that the state of magnetosphere plasma is determined by solar wind velocity and the magnitude of southern component of magnetic field. The structure of the system is obtained using statistical techniques. The CVar robust deviation measure has been suggested for obtaining system parameters of a stable prediction model.
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 Mykyta Boyko.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Pardalos, Panagote M.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041990:00001

Permanent Link: http://ufdc.ufl.edu/UFE0041990/00001

Material Information

Title: New Approaches to Robust Optimization with Applications
Physical Description: 1 online resource (94 p.)
Language: english
Creator: Boyko, Mykyta
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: 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: The dissertation focuses on a popular percentile-based risk measure known as Conditional Value at Risk (CVaR). CVaR has emerged from financial applications and allows obtaining optimal solutions that are robust with respect to various uncertainties incorporated into models as random variables. In this work I solve three operations research problems using CVaR-based robustness: multi-sensor scheduling problem, network-flow and system identification for space weather prediction. I develop mathematical programming techniques for solving a class of multi-sensor scheduling problems and formulate robust optimization problems for both deterministic and stochastic cases using linear 0-1 programming techniques. Equivalent formulations are developed in terms of cardinality constraints. I conducted numerical case studies and analyzed the performance of optimization solvers on the problems under consideration. The next problem I consider is robust network flow problem. I propose a new stochastic formulation of minimum cost flow problem aimed at finding network design and flow assignments subject to uncertain factors, such as network component disruptions/failures. I introduced loss function that is proportional to actual loss of flow in the network. In order to quantify the uncertain loss caused by network failures, I utilized CVaR risk measure. The combination of Lagrangian Relaxation and Benders' decomposition is proposed to solve large problems. Predicting geomagnetic activity applied for Dst-index forecasting has been performed through robust identification of discrete dynamic system. The modeling assumes that the state of magnetosphere plasma is determined by solar wind velocity and the magnitude of southern component of magnetic field. The structure of the system is obtained using statistical techniques. The CVar robust deviation measure has been suggested for obtaining system parameters of a stable prediction model.
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 Mykyta Boyko.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Pardalos, Panagote M.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041990:00001


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NEWAPPROACHESTOROBUSTOPTIMIZATIONWITHAPPLICATIONSByMYKYTAI.BOYKOADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2010

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c2010MykytaI.Boyko 2

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IdedicatemythesistoparentsIgorandOlga,wifeLidiya,andsonBoris. 3

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ACKNOWLEDGMENTS IamverythankfultomyadvisorandmentorProf.PanosPardalosforhissupportduringmydoctoratestudiesattheUniversityofFlorida.Hisenthusiastichelpandresearchguidancehelpedmedevelopononbothprofessionalandpersonallevels.Iwouldliketoexpressmygratitudetoothermembersofmydoctoratecommittee,Prof.VladimirBoginski,Prof.StanUryasev,andProf.WilliamHagerfortheirvaluablecontributiontomyresearch.IwouldalsoliketoexpressmygreatestappreciationtomycolleaguesfromtheCenterforAppliedOptimization.Intensiveexchangeofideasandjointresearchwithmyfellowgraduatestudentsandpostdocsfromthecenterhelpedmesignicantlyinmywork.Lastbutnotleast,Iwouldliketothankmyfamilyandfriends,whosupportedandencouragedmeinallofmybeginnings. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 LISTOFSYMBOLS .................................... 9 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTION ................................... 12 2STATEOFTHEARTREVIEWOFROBUSTOPTIMIZATION .......... 16 2.1RobustOptimizationFormulations ...................... 16 2.2RelationtoStochasticProgramming ..................... 18 2.3Scenario-basedOptimizationApproach ................... 19 3ROBUSTMULTI-SENSORSCHEDULING ..................... 26 3.1OptimizationTechniquesforSensorNetworks ................ 26 3.1.1PositioningUsingAngleofArrival ................... 28 3.1.2SemideniteProgramming(SDP)forSensorNetworkLocalization 30 3.1.3NetworkInterdiction .......................... 32 3.2RobustSensorsScheduling .......................... 38 3.2.1OptimizationModelsforSensorsScheduling ............ 38 3.2.2Deterministicsetup .......................... 43 3.2.3ProblemSetupunderUncertainty ................... 46 3.3EquivalentFormulationsinCardinalityConstraints ............. 47 3.4SensorSchedulinginNetwork-BasedSettings ............... 53 3.5ComputationalExperiments .......................... 57 4TWOSTAGESTOCHASTICOPTIMIZATIONMODELFORROBUSTNETWORKFLOWDESIGN .................................... 61 4.1ProblemFormulation .............................. 61 4.2DecompositionMethodforNetworkFlowProblem ............. 66 4.3ComputationalExperiments .......................... 75 5ROBUSTSYSTEMIDENTIFICATIONFORSPACEWEATHERFORECASTING 78 5.1ModelingMagnetosphereasaBlackBox .................. 79 5.2RobustModelReconstruction ......................... 83 5.3NonlinearStructureReconstruction ...................... 84 5

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6CONCLUSION .................................... 86 6.1ThesisContribution ............................... 86 6.2FutureWork ................................... 87 REFERENCES ....................................... 89 BIOGRAPHICALSKETCH ................................ 94 6

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LISTOFTABLES Table page 3-1Performanceresultsfordeterministicmodel( 3 )-( 3 ).n-numberofsites;m-numberofsensors.Thenumberofdiscretetimestepsisxed:T=10. .. 57 3-2PerformanceresultsforCVaRtypedeterministicmodel( 3 ).n-numberofsites;m-numberofsensors.Thenumberofdiscretetimestepsisxed:T=10.CVaRcondencelevel=0.9. ...................... 58 3-3PerformanceresultsforCVaRtypestochasticmodel( 3 ).n-numberofsites;m-numberofsensors.Thenumberofdiscretetimestepsisxed:T=10,numberofscenariosS=100.CVaRcondencelevel=0.9. ....... 59 3-4ComparingPSGandCPLEXperformanceforobtainingapproximatesolutionofCVaRtypestochasticproblem( 3 )(n=12sitesandT=10timeperiods). 60 3-5ILOGCPLEXCPUtime(sec)fornetworkdeterministicmodel( 3 )-( 3 ).n-numberofsites;m-numberofsensors.ThenumberofdiscretetimestepsisT=10. ....................................... 60 5-1Leaps-and-boundbasedvariableselectionforxednumberofregressorsk=1,...,8foronestepaheadforecasting(linearmodel).+incolumnskindicatesthatthecorrespondingvariableisaddedtothemodel. .............. 84 5-2Leaps-and-boundbasedvariableselectionforxednumberofregressorsk=1,...,10foronestepaheadforecasting(bilinear).+incolumnskindicatesthatthecorrespondingvariableisaddedtothemodel. .............. 85 7

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LISTOFFIGURES Figure page 2-1GraphicalrepresentationofVaRandCVaRforarandomvalueLspeciedbyprobabilitydensityfunctionfL(x). .......................... 22 2-2GraphicalrepresentationofVaRandCVaRforanitesetofstochasticscenarios(discretecase). .................................... 23 3-1Exampleofapossiblenetwork.Twonodesareconnectedbyanarcifasensorcanmovefromonenodetoanotherinconsequenttimeperiods. ........ 53 3-2Counterexample(m=2):twosensorscannotperformsimultaneousfeasiblemoveduetotheconstraintx1,1+x4,21 ..................... 55 4-1Effectcausedbynetworkfailure.Inbothcasesthenetworklostthesameamountofow. ......................................... 64 4-2Newlossfunctionina6verticenetwork.A-initialnetwork,B-minimalcostowwithoutwithCVaRinobjective,CdeterministiccasewithoutCVaR .... 76 4-3FlowscheduleinthenetworkwithandwithoutconsideringCVaRrobustconstraint.A-initialnetwork,B-minimalcostowwithoutwithoutconsideringreliabilityissues,CandD-CVaRoflossisboundedby5and10respectively. ..... 77 8

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LISTOFSYMBOLS,NOMENCLATURE,ORABBREVIATIONS AoRDAPSGAmericanoptimalDecisionPortfolioSafeguardcard()Cardinalityfunction(numberofnonzerocomponentsofvectorargument)CVaRConditionalValue-at-RiskESetofarcsGNetworkgraphI()Indicatorfunctioninf()InmumLPLinearprogrammingmax()MaximumMILPMixedintegerlinearprogrammingmin()MinimumpdfProbabilitydensityfunctionProbfgProbabilitymeasureofeventIRSetofrealnumberss.t.Subjecttosup()SupremumUAVUnmannedAerialVehicleUUncertaintysetVaRValue-at-RiskVSetofvertices 9

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AbstractofdissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyNEWAPPROACHESTOROBUSTOPTIMIZATIONWITHAPPLICATIONSByMykytaI.BoykoAugust2010Chair:PanosM.PardalosMajor:IndustrialandSystemsEngineeringThedissertationfocusesonapopularpercentile-basedriskmeasureknownasConditionalValueatRisk(CVaR).CVaRhasemergedfromnancialapplicationsandallowsobtainingoptimalsolutionsthatarerobustwithrespecttovariousuncertaintiesincorporatedintomodelsasrandomvariables.InthisworkIsolvethreeoperationsresearchproblemsusingCVaR-basedrobustness:multi-sensorschedulingproblem,network-owandsystemidenticationforspaceweatherprediction.Idevelopmathematicalprogrammingtechniquesforsolvingaclassofmulti-sensorschedulingproblemsandformulaterobustoptimizationproblemsforbothdeterministicandstochasticcasesusinglinear0-1programmingtechniques.Equivalentformulationsaredevelopedintermsofcardinalityconstraints.Iconductednumericalcasestudiesandanalyzedtheperformanceofoptimizationsolversontheproblemsunderconsideration.ThenextproblemIconsiderisrobustnetworkowproblem.Iproposeanewstochasticformulationofminimumcostowproblemaimedatndingnetworkdesignandowassignmentssubjecttouncertainfactors,suchasnetworkcomponentdisruptions/failures.Iintroducedlossfunctionthatisproportionaltoactuallossofowinthenetwork.Inordertoquantifytheuncertainlosscausedbynetworkfailures,IutilizedCVaRriskmeasure.ThecombinationofLagrangianRelaxationandBenders'decompositionisproposedtosolvelargeproblems 10

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PredictinggeomagneticactivityappliedforDst-indexforecastinghasbeenperformedthroughrobustidenticationofdiscretedynamicsystem.Themodelingassumesthatthestateofmagnetosphereplasmaisdeterminedbysolarwindvelocityandthemagnitudeofsoutherncomponentofmagneticeld.Thestructureofthesystemisobtainedusingstatisticaltechniques.TheCVarrobustdeviationmeasurehasbeensuggestedforobtainingsystemparametersofastablepredictionmodel. 11

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CHAPTER1INTRODUCTIONThemainobjectiveofthisdissertationworkistoextendexistingmathematicalprogrammingmodelstostochasticsettings,wheresomeparametersofthemodels,suchasthereliabilityofconnectionsorpenalties,areuncertain.Intherealsettingstypicalforthemajorityofcivilandmilitaryapplications,somecomponentsofthesystemmaybecometemporarilyunavailableduetoweatherconditionsandotherissuesthatcanaffectnormaloperatingprocedures.Theseuncertainfactorscanmakethetraditionaloptimalsolutioninfeasibleandthereforeunrealistic.Thatiswhy,itiscrucialtodevelopmathematicalprogrammingmodels,thatwilltakethesefactorsintoaccount.Theobjectiveoftheapproachsuggestedintheworkistocontrolandrestrictpossiblelossesbyutilizingappropriatequantitativeriskmeasuresandintroducingthemintomathematicalprogrammingmodels.Specically,IproposetousetheconceptofConditionalValue-at-Risk(CVaR)whichhasbeenusedinnancialengineeringapplications,andhasrecentlybeenwidelyappliedtostudyingscheduling,networkoworforecastingproblems.RockafellarandUryasevwerethepioneerstodevelopthegeneralmethodologyofoptimizationofCVaRandapplyittoportfoliooptimizationproblems[ 45 ].Thismethodologyhasbeenverywellreceivedbybothacademiccommunityandpractitioners(asofFebruary2010,GoogleScholarreportsmorethan1000citationsofthisoriginalpaper).ThisdissertationwilldevelopappropriatemathematicalprogrammingformulationsusingCVaRtoachieverobustandefcientperformanceofsensorsscheduling,networkowandpredictionmodels.Ageneralformofatypicaloptimizationproblemaimsatndingtheoptimalsolutionthatremainsfeasibleforallpossibleparametersfromtheuncertaintyset.Thistraditionalapproachtondingrobustsolutionhassomedisadvantages.First,theclassicalrobustproblemismoredifculttosolvethanitsdeterministiccounterpart.Second, 12

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thetraditionalrobustformulationisoftencriticizedforitsconservatism.Indeed,theworstcasescenariomightbeahighlyunlikelyeventand,therefore,itwouldbemorereasonabletoconsiderasetofworstcasescenarios.Inordertoavoidtheconservatismoftraditionalrobustoptimization,wereplaceworstcasescenariowithanaveragedsetofworstcasescenarios.Toaccomplishthis,letusassumethatuncertainparametersaregovernedbysomeprobabilitymeasure.Wewillalsoassociatesomelosswitheachimplementationoftheworstcasescenarios.Inordertoquantifytheuncertainty,wewilluseapopularriskmeasureknownasConditionalValueatRisk(CVaR)intheliterature.CVaRiscloselyrelatedtoawell-knownquantitativeriskmeasurereferredtoasValue-at-Risk(VaR),whichiswidelyusedinnancialengineering.The-CVaRequalstotheexpectationofthe%ofworstcasesofthelosscausedbyuncertaintyfactors.ThisworkcontributestotheareaofrobustoptimizationbycreatingamodelingframeworkforthreeapplicationareasusingCVaR-typerobustconstrains.Alltheproblemswillbemodeledusingalargeyetnalsetofscenarios.Oneoftheapplicationsistherobustsensorsschedulingformultiplesitesurveillance.Thetaskofareasurveillanceisimportantinavarietyofapplicationsinbothmilitaryandciviliansettings.Oneofthemainchallengesthatneedtobeaddressedintheseproblemsisthefactthatthenumberoflocations(sites)thatneedtobevisitedtogatherpotentiallyvaluableinformationisoftenmuchlargerthanthenumberofavailablesurveillancedevices(sensors)usedforcollectinginformation.Undertheseconditions,itisnecessarytoschedulealloftheavailablesensors(thatcanbeinstalled,forinstance,onUnmannedAirVehicles)inordertomaximizetheamountofvaluableinformationcollectedbythesensors.Itispossibletoformulatethisproblemintermsofminimizingtheinformationlossesassociatedwiththefactthatsomelocationsarenotundersurveillanceatcertaintimemoments.Inthesesettings,theinformationlossescanbequantiedasbothxedandvariablelosses,wherexedlosseswouldoccurwhenagivensiteissimplynotundersurveillanceatsometimemoment,whilevariable 13

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losseswoulddependonhowlongasitehasnotbeenpreviouslyvisitedbyasensor.Particularly,considerationofvariablelossesofinformationiscriticalforstrategicallyimportantsitesthatneedtobemonitoredascloselyaspossible.Inaddition,theparametersthatquantifyxedandvariableinformationlossesareinmanycasesuncertainbynature.Inpreviousworksrelatedtothisarea,theuncertaintiesintheseparameterswerenotexplicitlytakenintoaccount(see,e.g.,[ 63 ]).However,itiscruciallyimportanttodevelopefcienttechniquestominimizeorrestricttheinformationlossesunderuncertainty.Thischapterproposesmathematicalprogrammingformulationsthatallowquantifyingandrestrictingtherisksofworst-caselossesassociatedwithuncertainparameters.Themathematicalprogrammingformulationsarerstdevelopedforthedeterministiccase.Thenaturalextensionsoftheseformulationstothestochasticcase(withuncertaininformationlossparameters)aremadebyutilizingquantitativeriskmeasuresthatallowtocontroltheconservativenessoftheoptimalstrategy.Inparticular,thestatisticalconceptreferredtoasConditionalValue-at-Risk(CVaR)isusedintheproposedproblemformulationsunderuncertainty.Usingthesetechniquesallowstoefcientlyincorporateuncertaintiesintotheoptimizationproblemsunderconsideration,aswellastoprovidethemeanstobalancebetweentheoptimalityandtherobustnessofthesolutions.Equivalentreformulationsandextensionsofthegivenproblemsarealsoprovided.Severalnumericalcasestudiesverifytheefciencyofthesuggestedalgorithms.Twosoftwarepackages,IBMCPLEXandAOrDaPSG,areusedtosolvethecasestudies.Thesecondapplicationpresentsanapproachtondinganoptimalrobustnetworkstructurewithrespecttouncertainfactors,suchasdemands,componentfailures,etc.AtwostagestochasticoptimizationproblemisformulatedandBendersdecompositionisproposedforsolvingtheproblemwithalargenumberofsecondstagevariables.TheworkextendsthemodelinitiallysuggestedbyBoginskietalin[ 16 ].Itisassumedthat 14

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anyarcofthenetworkcanbeindependentlydestroyedwiththeknownprobabilityandthelossisdeterminedbytheowalonethefailedarcs.Thepresentworksuggestsabetterlossfunctionwhichisbasedontheamountofowundeliveredtotheconsumersduetotherealizationofafailurescenario.Thatisthelosscanrepresentpenaltynesforfailuretodeliveracontractedamountofproducttoconsumers.Thenetworkisdesignedtoberobustwithrespecttothispenalty.AtwostagestochasticoptimizationproblemisformulatedandBendersdecompositionisproposedforsolvingtheproblemwithalargenumberofsecondstagevariables.Thethirdapplicationwillperformsystemidenticationusingthesoftmarginapproach.Theidentiedlinearandbilinearmodelsaimatforecastingsolarindexactivityforpredictinggeomagneticstorms.Presently,identicationandpredictionmodelsarewidelyusedforstudyinglinearandnon-linearprocessesinthespace.ItispossibletopredicttheEarthmagnetospherebybuildingblackboxmodelwhichrelatedsolarwindparameters(suchasmagneticeldcharacteristics,solarwindspeed,etc)tomeasurablegeophysicsindices(forexampleKporDstindices).Work[ 19 ]suggestedablack-boxtechniqueformodelingcomplexprocessesinmagnetosphere,byassumingthattheyaredescribedbyanunknownbilinearsystem.ACVaR-baseddeviationfunctionissuggestedtouseinordertoprovideabetterstabilityoftheunderlyingdynamicsystem. 15

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CHAPTER2STATEOFTHEARTREVIEWOFROBUSTOPTIMIZATION 2.1RobustOptimizationFormulationsThischapterdescribeshowdatauncertaintycanbeaddressedinoptimizationmodels.Anyreallifeoptimizationproblemcontainsparametersthatarenotexactincertainsense.Theparameterscanbemeasuredwithsomeerrororcausedbychangesofconditionovertime.Theparameterscanbeofarandomnature,forexamplestockquotesorsystemcomponentfailures.Thesimplestsolutionistoconsideramosttypicalvalueofaparameter(i.e.averageorsingleavailableobservation)thataccountsasinglemostlikeaverages.However,thedeviationfromtheaveragecaneasilymaketheprobleminfeasibleorresultinasolutionwhereperturbedparameterswilldiffersignicantlyfromtheinitialsolution.Evensuchnaturalsourceofuncertaintycausedbyapproximatemeasurementorinsufcientknowledgeofthephenomenonofstudycandramaticallyaffectthesolution.Thebook[ 8 ]illustratesitwithanimpressivecasestudy.Theauthorslookintoaconstraintoftherealworldlinearproblemhavingathousandofdecisionvariablesand410constraints.Eachofthe1000410coefcientsoftheproblemissomerealnumberwithseveraldigits.Afterselectingaconstraint,itturnsoutthataverymodestassumptionsuchasthemeasurementerrorfortheconstraintmatrixcoefcientscandeviatewithin0.01%,whichresultsin450%violationoftheconstraintintheworstcase.Moreover,iftheactualparametersareassumedtobeuniformlydistributedonasegmentof0.01%theconstraintisviolatedwithprobability0.5.Thesameconstraintisviolatedby150%withprobability0.18.Eveninsufcientprecisionofdatarepresentationsignicantlyaffectthequalityofthenominalsolution.Ageneralformofatypicaloptimizationproblemcanbeformulatedasfollows Minimizex2IRnf0(x,), (2) s.t.fi(x,)0,i=1,...,m, (2) 16

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wherearetheparametersoftheoptimizationproblem.Traditionally,robustoptimizationparadigmisbasedonthefollowingreasonableassumptionmadefortheparameters2UIRk. Actualdataorparametersofthemodelarerepresentedwiththeso-calleduncertaintysetU. Bythetimeweobtainanoptimalsolutionxwedonotknowtheactualvaluesoftheuncertainparameters. Whenactualparametersarerevealed,theoptimalsolutionxshouldbevalidwithrespecttotheuncertainoutcome. Thedecisionmakercannottolerateanyviolationcausedbyanyuncertainoutcome,i.e.themarginsarehard.Thelatterassumptionisextremelyconservativeandisoneofthereasonsthatcausecriticismofstandardrobusttechniques.Thus,acommonrobustformulationfor( 2 )-( 2 )canbewrittenas Minimizex2IRnmax2Uf0(x,), (2) s.t.fi(x,)0,i=1,...,m,82U. (2) Thelatter( 2 )-( 2 )problemiscalledtherobustcounterpartoftheoriginaluncertainproblem.Thissolutioniscalledrobustsolutiontotheuncertainproblem.TheuncertaintysetUisoftenchoseneitherasapolytoporellipsoidink-dimensionalspace.Thesimplestversionofapolytopcanbeaparallelepipedproducedbyintervaluncertainty U=f[a1,b1][a2,b2]...[ak,bk]g.(2)Thatiseachofthecomponentibelongstoasegment.AnellipsoidaluncertaintyassumesthatvectorofactualparametersliesinsideaunitballwithrespecttosomeMahalanobismetric: U=f2IRkjT1g(2)forsomepositivesemidenitekkmatrix. 17

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2.2RelationtoStochasticProgrammingRobustoptimizationandstochasticoptimizationarecloselyrelateddisciplines.Inparticular,bothofthedisciplinesaddressuncertainties.Whentheuncertainparametersarestochasticbynature,i.e.theyaredescribedbyprobabilisticdistribution,astochasticoptimizationproblemcanbeformulatedforadeterministiccounterpart.Forexample,letbeastochasticparameter.Thenthechanceconstraintcanbeintroducedforthedeterministiccounterpart( 2 )( 2 ).Thisconstraintpracticallyeliminatesthepossibilityofviolations,i.e.theprobabilityofconstraintviolation1isclosetozero.Thatisifxisarobustoptimaldecisionvalueandfistheoptimalvalue, P(f
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computationallyintractable.Forexample,evenforthesimplestcasewhendeterministiccounterpartisLPandparametersi,i=1,...,kareuniformlydistributedonsomeintervals,evaluatingtheprobabilityofconstraintviolationsof( 2 )becomesNP-hard[ 8 ].Thatisgiventheinputofrationalx,rationalintervals[ai,bi],i=1,...,kandrational2(0,1)thereisnoalgorithmthatevaluatestheprobabilitywithaccuracywhoserunningtimeisboundedwithpolynomofthebitsizeoftheinput(unlessP=NP).Moreover,thefunctionP(.)1)]TJ /F5 11.955 Tf 12.56 0 Td[(isgenerallynotconvex,whichmakesevengooddeterministicproblemsunsolvableundertheuncertaintysetup. 2.3Scenario-basedOptimizationApproachFindinganexactsolutionforageneralrobustproblem( 2 )( 2 )isanontrivialtaskthatrequiresspecialtechniquesforeveryparticularcase.Oneoftheapproachesistoallowviolationoftherobustconstraint( 2 )forsomeverysmallfractionofparameter.LetusassumethathasadistributionmeasuredbyaprobabilityfunctionProb.Theaboveassumptionresultsinachance-constrainedoptimizationproblemandiswrittenasfollows Minimizex2IRnf0(x), (2) s.t.Probff(x,)>0g. (2) Theparameter2(0,1)istheprobabilityoftheviolationofconstraints( 2 ).Eveniftheprobabilitydistributionforisknown,solvingsuchaproblemingeneralsetupcouldbeachallenge.AusefulapproachistosampleuncertainparametersimilarlytoMonteCarlosimulationandthensolvetheproblemwhereeachscenarioprovidesasetofconstraints: Minimizex2IRnf0(x), (2) s.t.f(x,s)<0,s=1,...,N. (2) 19

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Ifallconstraintsandtheobjectivefunctionareconvex,theaboveproblemcanbesolvedefciently.Let^xNbetheoptimalsolutionoftheproblem( 2 )-( 2 ).Notethat^xNdependsonNrandomscenariosandconsequentlyisarandomvalueitself.Now,itisimportanttoestimatehowaccuratelytherandomapproximatesolution^xNrepresentstherobustoptimalsolutionof( 2 )-( 2 ).Inotherwords,howmanyscenariosneedtobesampledtoguaranteewithcertaincondencethattheoptimalsolutionviolatesonlyasmallportion(e.g.%)oftheuncertaintysetU.CampiandCalaorehaveprovedthefollowingtheoremin[ 18 ]whichanswersthequestion. Theorem2.1(CampiandCalaore). Fixtworealnumbers2(0,1)(levelparameter)and2(0,1)(condenceparameter).If NN(,)=2 ln1 +2n+2n ln2 (2)then,withprobabilitynosmallerthan1)]TJ /F5 11.955 Tf 12.89 0 Td[(,theoptimalsolution^xNofthescenarioproblemis-levelrobustlyfeasible,thatisProbf2U:f(x,)>0g<.Wecanconcludethatafterrelaxingtraditionalrobustconstraintweinvadeintotheareaofpercentile-basedoptimization.ForfurtherdiscussionletusintroducethenotionofValueatRisk(VaR). Denition1. Givensomecondencelevel2(0,1)theVaRoftherandomvalueLatthecondencelevelalphaisgivenbythesmallestnumberlsuchthattheprobabilitythattherandomvalueLexceedsthresholdlisnotlargerthan(1)]TJ /F5 11.955 Tf 11.96 0 Td[()VaR(L)=inffl2IR:ProbfLlg1)]TJ /F5 11.955 Tf 11.95 0 Td[(g=inffl2IR:FL(l)gVaRinitiallyemergedinnancialengineeringasapopularriskmeasureoftheportfolioofrandomoutcomesecurities.SpeakinginformallyVaRriskmeasure 20

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answersthequestion,Whatisthelowerboundofportfoliolossin(1)]TJ /F5 11.955 Tf 12.09 0 Td[()%worstcasescenarios?UsingthedenitionofVaRwecanwriteaproblemsimilarto( 2 )-(): Minimizex2IRnf0(x), (2) s.t.VaR(f(x,))0, (2) Thatisweexpecttheproblemconstrainttoberobustforalargeportionofstochasticoutcomes.DespitewidepopularityofVaRthereisacriticismofthisriskmeasure.FirstofallVardoesnotcareaboutthetailofthedistributionwhichpotentiallycanresultinunderestimationofrareyethigh-impactlosses,asaresultcouldbeasourceoffalsecondence.Second,VaRriskmeasureisnotsubadditive(nonconvex).TherelationVaR(aX+(1)]TJ /F3 11.955 Tf 11.95 0 Td[(a)Y)aVaR(X)+(1)]TJ /F3 11.955 Tf 11.95 0 Td[(a)VaR(Y),wherea2[0,1]doesnotnecessaryhold.TheabovedescribedprocessmakesincorporatingVaRintooptimizationmodelstobeachallengingprocess.Itisalsounnaturalfrompracticalperspective:theVaR-basedriskofacombinedportfoliocanbelargerthanthesumoftheVaRsofitscomponents.Thiscontradictswidelyrecognizedopinionthatdiversicationdoesnotincreaseinvestmentrisk.LetusintroducethenotionofConditionalValue-at-Risk(CVaR)[ 45 50 ].CVaRiscloselyrelatedtothewell-knownquantitativeriskmeasurereferredtoasValue-at-Risk(VaR).Bydenition,withrespecttoaspeciedprobabilitylevel(1)]TJ /F5 11.955 Tf 13.19 0 Td[()(inmanyapplicationsthevalueof(1)]TJ /F5 11.955 Tf 12.47 0 Td[()issetratherhigh,e.g.95%),the-VaRisthelowestamountsuchthatwithprobability(1)]TJ /F5 11.955 Tf 12.83 0 Td[(),thelosswillnotexceed.Whereasforcontinuousdistributionsthe-CVaRistheconditionalexpectationoflossesabovethatamount.Aswecansee,CVaRisamoreconservativeriskmeasurethanVaR,which 21

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Figure2-1. GraphicalrepresentationofVaRandCVaRforarandomvalueLspeciedbyprobabilitydensityfunctionfL(x). meansthatminimizingorrestrictingCVaRinoptimizationproblemsprovidesmorerobustsolutionswithrespecttotheriskofhighlosses(seegure 2-1 ).Formally,-CVaRofrandomvariableLcanbeexpressedas CVaR(L)=(1)]TJ /F5 11.955 Tf 11.95 0 Td[())]TJ /F7 7.97 Tf 6.58 0 Td[(1Z+1VaR(L)xfL(x)dx,(2)Lisdrivenbydecisionvectorxandthevectorofuncertainparameters.Thenifdistributionofisknown,wecanwriteCVaRofL(x,)asafunctionofdecisionparameterx (x)=CVaR(L(x,))=(1)]TJ /F5 11.955 Tf 11.96 0 Td[())]TJ /F7 7.97 Tf 6.59 0 Td[(1Zy:L(x,y)>VaR(L(x,y)L(x,y)f(y)dy,(2) 22

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Generally,indicatesthelevelofconservatismthedecisionmakeriswillingtoaccept.Thecloserapproachesto1,thenarrowertherangeofworstcasesbecomesinacorrespondingoptimizationproblem. Figure2-2. GraphicalrepresentationofVaRandCVaRforanitesetofstochasticscenarios(discretecase). CVaRisdenedinasimilarwayfordiscreteormixeddistributionsasitisillustratedingure 2-2 ).ThereadercanndtheformaldenitionofCVaRforgeneralcasein[ 46 50 ].RockafellarandUryasev[ 45 54 ]havedemonstratedthatminimizing(x)isequivalenttominimizingthefunction F(x,)=+(1)]TJ /F5 11.955 Tf 11.96 0 Td[())]TJ /F7 7.97 Tf 6.59 0 Td[(1Zy2IRm[L(x,y))]TJ /F5 11.955 Tf 11.95 0 Td[(]+p(y)dy,(2) 23

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where[t]+=8><>:t,t0,0,t<0,andthevariablecorrespondstotheVaRvalue,asitwasintroducedabove.ThesignicanceofthisresultincludesthefactthatthefunctionF(x,)canbefurthersimpliedbysamplingtheprobabilitydistributionofaccordingtoitsc.d.f.F.Ifthesamplinggeneratesacollectionofvectors(scenarios)1,2,...,xiS,thecorrespondingapproximationtoF(x,)is ~F(x,)=+1 S(1)]TJ /F5 11.955 Tf 11.96 0 Td[()SXs=1[L(x,ys))]TJ /F5 11.955 Tf 11.95 0 Td[(]+.(2)Theexpression~F(x,)isconvexandpiecewiselinearwithrespectto.Moreover,ifthefunctionL(x,y)islinearwithrespecttox,~F(x,)canbeeasilyminimizedusingLinearProgrammingtechniques.Letusemphasizethatthesolution^xNofthediscreteproblemisarandomvaluethatdependsontherandomlysampledscenarios.InordertoensurecorrectnessofsuchMonteCarlotypeapproach,itisimportanttoquantifythenumberofscenariostoprovidethesolutionwithcertainlevelofcondencesasitwasdonebyCampiandCalaore[ 18 ]forasimilartypeofproblem.Therststepinthisdirectionwasmadein[ 16 ]whereBoginskietalinvestigatedhowmanyscenariosarerequiredtoensurethatthetruevalueoftheCVaRfunctionF(x,)wascloseenoughtothesample(scenario-based)valuethatwasusedintheLPformulationwithahighcondencelevel.Inparticular,theupperboundrequiredanumberofscenariosontheinputsizewasfoundforarobustnetworkdesignproblem.NowwearereadytoformulatethegeneralframeworkforobtainingrobustsolutionusingCVaRriskmeasure.IfweassociatethemagnitudeofviolationofrobustconstraintwiththelossfunctionL,thanrobustformulationsimilarto( 2 )-( 2.3 )canbewrittenasa 24

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niceconvexprogrammingmodel: Minimizex2IRnf0(x), (2) s.t.CVaRf(x,)0 (2) whichcanbeapproximatedbythelinearprogrammingoptimizationmodelifalloftheobjectiveandconstraintfunctionsarelinearwithrespecttodecisionvariables. 25

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CHAPTER3ROBUSTMULTI-SENSORSCHEDULINGThechapterdevelopsoptimizationframeworkforschedulingseveralsensorsinuncertainenvironments.IhaveappliedCVaRriskmeasuretoachieverobustschedulingsolutionindeterministicandstochasticssettings.Themodelisextendedforthenetworksetup.Theproblems,initiallyformulatedasmixedintegerlinearprogramsarelaterreformulatedintermsofcardinalityfunctionsusedinconstraints.Thetheoremsareformulatedandprovedtojustifysuchtransition.CardinalityformulationshavebeensolvedbyAoRDAPortfolioSafeguardsoftwarepackagethatusedtheheuristicsoptimizedforcardinalityfunctions.Thischapterstartswithacomprehensivesurveyintothestateoftheartsinoptimizationmethodsforsensorsandsensorsnetworks.Thesemethodsareusedforsuchimportanttasksasscheduling,positioning,buildingsensorsnetworktopologiescapabletoprotectthemselvesagainstenemyintrusion,etc. 3.1OptimizationTechniquesforSensorNetworksOrganizingsensorsintothenetworkshasbeenrecentlystudiedintheliterature[ 3 6 27 29 35 36 43 60 64 65 ].Varioussensorsareusedinbothcivilianandmilitarytasks.Sensingdevicescanbedeployedineitherstaticordynamicsettings,wherethepositionsofeachsensorcanbepermanentordynamicallychanging(suchasinthecaseofsensorsinstalledonairvehicles).Multiplesensorsystemsarecommonlyrepresentedasnetworks,sincebesidescollectingimportantinformation,sensingdevicescantransmitandexchangeinformationviawirelesscommunicationbetweensensornodes.Therefore,network(graph)structuresareconvenientandinformativeintermsofefcientrepresentationofthestructureandpropertiesthereof.Toanalyzeandoptimizetheperformanceofsensornetworks,mathematicalprogrammingtechniquesareextensivelyused. 26

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Beforeconsideringaparticularproblemofrobustschedulingforsensors,letusgiveabriefreviewofsomeoftherecentdevelopmentsinmathematicalprogrammingasappliedtosensornetworkresearch.Varioustypesofoptimizationproblemscanbeformulatedandsolvedinthiscontext[ 17 55 60 ].Moreover,thepursuedtaskscanvaryfromoptimizingthenetworkperformancetonetworkinterdiction,wherethegoalistodisruptenemynetworksbyinterferingwithcommunicationnetworkintegrity.Iwilloutlinetheformulationsandbrieydescribethesolutionmethodsusedtotackletheseproblems.Inmanycases,theidentiedoptimizationproblemsarechallengingfromthecomputationalviewpoint,andefcientalgorithmsneedtobedevelopedtoensurethatthenear-optimalsolutionsarefoundquickly.Thisisessentialtoguaranteethatthedecisionsregardingefcientoperationsofsensornetworkscouldbemadeinareal-timemode,whichcanbecrucialinmanyapplications.Moreover,theuncertainfactorsthatcommonlyariseinreal-worldsituationsalsoneedtobeincorporatedintothemathematicalprogrammingproblems,whichmakesthemevenmorechallengingtoformulateandsolve.Inparticular,letusmentionseveralimportantproblemsthathaverecentlybeenaddressedintheliterature.Letusstartthediscussionwiththedescriptionofrecentpromisingdevelopmentsintheareaofsensornetworklocalization,whichallowidentifyingglobalpositionsofallthenodesinanetworkusinglimitedandsometimesnoisyinformation.Itturnsoutthatsemideniteprogrammingtechniquescanbeefcientlyusedtotackletheseproblems.Next,theproblemsofsingleandmultiplesensorschedulingforareasurveillance,includingthesetupsunderuncertaintywillbediscussed.Theissuesofwirelesscommunicationnetworkconnectivityandintegrity,aswellasnetworkinterdictionproblemswillbereviewed. 27

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3.1.1PositioningUsingAngleofArrivalAnimportantclassofproblemsisadeterminingsensorsposition,knownassensorslocalizationintheliterature.Inmanycases,itisimportanttobeawareofthesensor'sphysicalcoordinates.InstallingGPSreceiversineverysensorisnotalwaysoptimalfromthecost-relatedandotherperspectives.Typically,onlyafewnodes(seednodes,landmarks,etc)ofthenetworkareequippedwithGPSandknowtheirphysicallocation.Therestofthenodescanonlycommunicatewithothernodesanddeterminerelativelocationcharacteristicssuchasdistance,angles,etc.Variouslocalizationtechniquesareusedtoobtainlocationofallthesensorsinthenetwork.Awiderangeofsensorsapplicationsrevealsdifferentrequirementstothenetworktopologyidentication[ 33 47 57 59 ].Forexample,networkparameterscanbeinuencedbylandsurface,transmissioncharacteristics,energyconsumptionpolicy,etc.Adhocanddynamicnetworksalsorequireidenticationofnodecoordinates.Utilizingmathematicalprogrammingtechniquesoftenallowstondefcientsolutions.Typically,onlyafewnodes(seednodes,landmarks,etc)ofthenetworkareequippedwithGPSandknowtheirphysicallocation.Therestofthenodescanonlycommunicatewithothernodesanddeterminerelativelocationcharacteristicssuchasdistance,angles,etc.Variouslocalizationtechniquesareusedtoobtainlocationofallthesensorsinthenetwork.Thefollowingmethodassumesthatthenetworkconsistsoftwotypesofnodes:usualandmorecapablenodeswhichknowitsposition.NiculescuandNathproposeamethodaccordingtowhichnodesinanadhocnetworkcollaborateinndingtheirpositionandorientation,assumingthatasmallpartofthenetworkhasapositioncapability.Also,everynodeinthenetworkhasacapabilitytodeterminetheangleofthearrivingsignal(AOA).Eachnodeinthenetworkhasonexedmainaxis(whichmaynotbethesamefordifferentnodes)andisabletomeasureallanglesagainstthisaxis. 28

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Everynodeinanadhocnetworkcancommunicateonlywithitsimmediateneighborswithintheradiorange,anditsneighborsmaynotalwaysbelandmarks,i.e.,thenodesthatknowtheirposition.NiculescuNathproposein[ 42 ]amethodtoforwardorientationinawaythatwouldallowthenodeswhicharenotindirectcontactwithlandmarkstodetermineitsorientationwithrespecttothelandmarks.Orientationmeansbearing,radial,orboth.Bearingisananglemeasurementwithrespecttoanotherobject.Aradialistheangleunderwhichtheobjectisseenfromanotherpoint.Theauthorsexaminetwoalgorithms:DistanceVectorBearing(DV-Bearing),whichallowseachnodetogetabearingtolandmark,andDV-Radial.Thepropagationinbothalgorithmsworksthefollowingway:nodesadjacenttolandmarksdeterminetheirbearing/radialdirectlyfromlandmarkandsendthenetworktheinformationabouttheirposition.DV-Bearingalgorithmworksthefollowingway:nodesA,B,andCareneighborsandcancommunicatewitheachother.SupposethatthenodeAneedstonditsbearingtonodeL,whichisnotwithinradiorangeofnodeAbutwithinradiorangeofnodesBandC.SinceA,B,andCcanlocateeachotherthanthenodeAcandeterminealltheanglesintrianglesABCandBCL.However,thatwouldallowtocalculatetheangleLACandconsequentlythebearingofAtoL,whichisequaltoc+LAC.OncenodeAknowsthreebearingstolandmarks,whicharenotatthesameline,itcancalculateitsownlocationbytriangulation.TheDV-Radialalgorithmworksverysimilarly.TheonlydifferenceisthatnodeAneedstoknownotonlybearingsofnodesBandCtonodeL,butalsotheradialsofBandCfromL.Theknowledgeofradialsimprovesaccuracyofthealgorithm.Whenallanglesaremeasuredagainstthesamedirection(forexample,whencompassisavailable)thenthesetwomethodsbecomeidentical. 29

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3.1.2SemideniteProgramming(SDP)forSensorNetworkLocalizationLetusreviewlocalizationproblemwithprovidedinformationondistancesforanchornodesandunknownsensorsnodes.Supposeweconsiderlocalizationproblemontheplane.Wehavemknownpointsak2IR2,k=1,...,mandnunknownnodesxj2IR2,j=1,...,n.LetusconsiderthreesetsofnodepairsNe,Nl,Nu.ForpairsinNeweknowexactdistancesdkjbetweenakandxjand^dijbetweenxiandxj.Nlisasetofpairswithknownlowerboundsr kjandr ij.Finally,Nuisasetofupperbounds rkjand rij.Naturallyourgoalistominimizeestimationerror,whichimmediatelyleadsustothefollowingnon-convexoptimizationproblem MinimizeX(i,j)2Ne,i
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TheformulatedproblemcanberewrittenbyintroducingmatrixnotationandslackvariablesasMinimizeX(i,j)2Ne,i
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ThelastrelationleadstoastandardSDPformulation:MinimizeX(i,j)2Ne,i
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wirelesscommunicationnetwork.Letusintroduceoptimizationformulationsthatallowtoplacejammingdevicesdeliveringmaximalharmtotheadversesensorsnetwork.Westartbyconsideringadeterministiccasewhennodelocationisknown.Thegoalofjammingistondasetoflocationsforplacingjammingdevicesthatsuppressesthefunctionalityofthenetwork.Letnjammingdevicesbeusedtojammcommunicatingsensors.TheunderlyingassumptionisthatthesensorsandjammerscanbelocatedonaxedsetoflocationsV.Thejammingeffectivenessofdevicejiscalculatedasd:(VV)7!IR,wheredisadecreasingfunctionofthedistancefromthejammingdevicetothenodebeingjammed.ThecumulativelevelofjammingenergyreceivedatnodeiisdenedasQi:=nXj=1dij,wherenisthenumberofjammingdevices.Asaresult,jammingproblemcanbeformulatedastheminimizationofthenumberofjammingdevicesplaced,subjecttoasetofcoveringconstraints:Minimizens.t.QiCi,i=1,2,...,m.Seekingtheoptimalplacementcoordinates(xj,yj),j=1,2,...,nforjammingdevicesgiventhecoordinates(Xi,Yi),i=1,2,...,mleadstonon-convexformulationsformostfunctionsd.Thus,integerprogrammingmodelsfortheproblemareproposed.AxedsetN=f1,2,...,ngofpossiblelocationsforthejammingdevicesandasetofcommunicationnodesareintroducedbyCommanderetalin[ 22 ].Denethedecision 33

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variablexjas xj=8>><>>:1,ifajammingdeviceisinstalledatlocationj,0,otherwise.(3)Thenwehavetheoptimalnetworkcoveringformulationgivenas MinimizenXj=1cjxj (3) s.t. (3) nXj=1dijxjCi,i=1,2,...,m, (3) xj2f0,1g,j=1,2,...,n, (3) Inthisinstancetheobjectiveistominimizethecostofjammingdevicesusedwhileachievingsomeminimumlevelofcoverageateachnode.Ifcj=1,thenumberofjammersisminimized.Ifthegoalistosuppresssensorscommunicationswecanminimizejammingcostwithrespecttotherequiredlevelofconnectivityindex.Communicationbetweennodesiandjisassumedtobedestroyedifatleastoneofthenodesisjammed.Further,letyij:=1ifthereexistsapathfromnodeitonodejinthejammednetworkandletzi=1 34

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beanindicatorthati-thnodeisjammed.Thiscanbeformulatedas MinimizenXj=1cjxj (3) s.t.mXj=1j6=iyijL,8i2M, (3) M(1)]TJ /F3 11.955 Tf 11.95 0 Td[(zi)>Si)]TJ /F3 11.955 Tf 11.96 0 Td[(Ci)]TJ /F3 11.955 Tf 21.91 0 Td[(Mzi,8i2M, (3) yijisconsistentwiththenetworkandzi (3) xj2f0,1g,8j2N, (3) zi2f0,1g,8i2M, (3) yij2f0,1g,8i,j2M, (3) whereSi:=Pnj=1dijxjdenotethecumulativelevelofjammingatnodei,M2IRissomelargeconstant.Thisproblemcanbeformulatedasamixedintegerlinearproblem.Thejusticationisprovidedin[ 22 ].Finally,Commanderetalprovidespercentilebasedformulationfordeterministicjammingproblems.Supposeitisdeterminedthatjammingsomefraction2(0,1)ofthenodesissufcientforeffectivelydismantlingthenetwork.Thiscanbeaccomplishedbyincluding-VaRconstraintsintheoriginalmodel.Lety:M7!f0,1gbeanindicatorthatequalstooneifnodeiisjammed(yi=1).Tondtheminimumnumberofjammingdevicesthatwillallowcovering100%ofthenetworknodeswithprescribedlevelsofjammingCi,wemustsolvethefollowingintegerprogram 35

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MinimizenXj=1cjxj (3) s.t.mXi=1yim, (3) nXj=1dijxjCiyi,i=1,2,...,m, (3) xj2f0,1g,j=1,2,...,n, (3) yi2f0,1g,i=1,2,...,m. (3) The-CVaRoptimizationmodelfornetworkcoveringcanbeformulatedasamixedintegerlinearprogramusingstandardlinearizationframework: MinimizenXj=1cjxj (3) s.t.+1 (1)]TJ /F5 11.955 Tf 11.95 0 Td[()mmXi=1maxCmin)]TJ /F6 7.97 Tf 18.31 14.95 Td[(nXj=1xjdij)]TJ /F5 11.955 Tf 11.96 0 Td[(,00, (3) 2R, (3) xj2f0,1g. (3) TheVaRandCVaRmodelscanalsobewrittenforconnectivitysuppressionmodelsinthesimilarfashion.Wereferthereaderto[ 22 ]fordetails.Thedeterministicformulationsofthewirelessnetworkjammingproblemareextendedin[ 21 ]totacklethestochasticjammingproblemformulationsusingpercentiletypeconstraints.Theseformulationsconsiderthecasewhentheexacttopologyofthenetworktobejammedisnotknownbutweknowthedistributionofnetworkparameters.Sincetheexactlocationsofthenetworknodesareunknown,itisassumedthatasetofintelligencedatahasbeencollectedandfromthatasetSofthemostlikelyscenarioshavebeencompiled.Scenarios2Scontainsboththenodelocations 36

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f(s1,s1),(s2,s2),...,(sm,sm)gandthesetofjammingthresholdsfCs1,Cs2,...,Csmg.Foreachscenarios2S,thesetMs=f1,2,...,msgneedstobejammed.Takingintoaccountallofthescenarioswecanwriteaprogramfornodecoveringproblem: MinimizenXk=1ckxk, (3) s.t.nXk=1dsikxkCsi,i=1,2,...,ms,s=1,2,...,S, (3) xk2f0,1g,k=1,2,...,n, (3) Itisunlikelytondthesolutionthatcanprovideeffectivejammingstrategyforallscenarios.Therefore,thenotionofpercentile-basedriskmeasurescanbeutilizedtodevelopformulationsoftherobustjammingproblemsincorporatingtheseriskconstraints.RobustnodecoveringproblemwithValue-at-Riskconstraintscanbeformulatedas MinimizenXk=1ckxk, (3) s.t.nXk=1dsikxkCsisi,8s2S,8i2Ms, (3) msXi=1sims,8s2S, (3) xk2f0,1g,8k2N, (3) si2f0,1g,8s2S,8i2Ms, (3) Thelossfunctioncanbeconsideredasthedifferencebetweentheenergyrequiredtojamnetworknodei,namelyCsi,andthecumulativeamountofenergyreceivedatnodeiduetoxovereachscenario.Withthistherobustnodecoveringproblemwith 37

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CVaRconstraintsisformulatedasfollows. MinimizenXk=1ckxk, (3) s.t.s+1 (1)]TJ /F5 11.955 Tf 11.95 0 Td[()ms (3) msXi=1max(Csmin)]TJ /F6 7.97 Tf 18.31 14.95 Td[(nXk=1dsikxk)]TJ /F5 11.955 Tf 11.96 0 Td[(s,0)0,8s2S, (3) xk2f0,1g,8k2N, (3) s2R,8s2S. (3) TheCVaRconstraint( 3 )impliesthatforthe(1)]TJ /F5 11.955 Tf 10.25 0 Td[()100%oftheworst(least)coverednodes,theaveragevalueoff(x)islessthanorequalto0. 3.2RobustSensorsSchedulingSurveillanceisanimportanttaskthatcanbeeffectivelyperformedbyanintelligentnetworkofsensors.Forexample,satellitescanbeequippedwithcamerastomonitorEarthsurfacefordifferentevents,suchasforestre,bordercrossingorenemyhostileactivity.Anotherexampleistrafcmonitoringattheroadsandintersections.Manyscienticpublicationshaverecentlyappearedintheliteratureduetoincreasinginterestintheproblemofndingoptimalscheduleforsensors[ 20 30 32 44 52 58 ].Mostcommontechnologicalandbudgetconstraintisalownumberofsensorstomonitoralltheobjectsofinterestsimultaneously.Thus,ndingtheschedulethatreducespotentiallossoflimitedobservationsisataskofhighimportance. 3.2.1OptimizationModelsforSensorsSchedulingThissectionintroducesageneralmathematicalframeworkformulti-sensorschedulingproblems.Beforeintroducingmymodel,IprovideabriefoverviewoftheresearchconductedbyJavuzandJeffcoat.Initially,weutilizetheconceptsintroducedin[ 63 ]thatweredevelopedforadeterministiccaseofasingle-sensorschedulingproblem.Wethengeneralizeandextendtheseformulationstothemorerealisticcases 38

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ofmulti-sensorschedulingproblems,includingthesetupsinuncertainenvironments.Assumethattherearemsensorsandnsitesthatneedtobeobservedateverydiscretetimemomentt=1,...,T.Weassumethatasensorcanobserveonlyonesiteatonepointoftimeandimmediatelyswitchtoanothersiteatthenexttimemoment.Sincemisusuallysignicantlysmallerthann,wehavebreachesinsurveillancethatcancausealossofpotentiallyvaluableinformation.Ourgoalistobuildastrategythatoptimizesapotentiallossassociatedwithnotobservingcertainsitesatsometimemoments.Furtheronweintroducethebinarydecisionvariables xi,t=8><>:1,ifi-thsiteisobservedattimet,0,otherwise,(3)andintegervariablesyi,tthatdenotethelasttimesiteiwasvisitedasoftheendoftimet,i=1,...,n,t=1,...,T,m
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Timeyi,tisequaltothetimewhenthesiteiwaslastvisitedbyasensorbytimet.Thisconditionissetbythefollowingconstraints: 0yi,t)]TJ /F3 11.955 Tf 11.96 0 Td[(yi,t)]TJ /F7 7.97 Tf 6.59 0 Td[(1txi,t,8i=1,...,n,8t=1,...,T, (3) txi,tyi,tt,8i=1,...,n,8t=1,...,T, (3) Itshouldbenotedthattheaboveconstraintsautomaticallyensurethatthefeasiblevaluesofyi,tareinteger.Itiseasytoverifybyconsideringpossiblevaluesofbinaryvariablesxi,t.Therefore,inthefollowingmathematicalprogrammingproblems,wecansetthevariablesyi,t2IR.Theinclusionoftheseconstraintswillmakethesevariablesintegerinanyfeasiblesolution.Thisenablesustodecreasethenumberofinteger(binary)variablesintheconsideredproblems.Consequently,usingthenotationC=maxi,tfai(1)]TJ /F3 11.955 Tf 11.59 0 Td[(xi,t)+bi,t(t)]TJ /F3 11.955 Tf 11.59 0 Td[(yi,t)gandstandardlinearizationtechniques,wecanformulatethemulti-sensorschedulingoptimizationprobleminthedeterministicsetupasthefollowingmixedintegerlinearprogram: MinimizeC (3) s.t.Cai(1)]TJ /F3 11.955 Tf 11.95 0 Td[(xi,t)+bi,t(t)]TJ /F3 11.955 Tf 11.95 0 Td[(yi,t),8i=1,...,n,8t=1,...,T, (3) nXi=1xi,tm,8t=1,...,T, (3) 0yi,t)]TJ /F3 11.955 Tf 11.95 0 Td[(yi,t)]TJ /F7 7.97 Tf 6.58 0 Td[(1txi,t,8i=1,...,n,8t=1,...,T, (3) txi,tyi,tt,8i=1,...,n,8t=1,...,T, (3) yi,0=0,8i=1,...,n, (3) xi,t2f0,1g,8i=1,...,n,8t=1,...,T, (3) yi,t2IR,8i=1,...,n,8t=0,...,T. (3) SingleSensorSchedulingproblemisNP-hard[ 62 ]Therefore,variousgreedyheuristicsareused.TheideabehindgreedyalgorithmsuggestedbyJavuzandJeffcoatin[ 62 ]issimple.Attimet=1wendthesitewiththesmallestpenalty.Then,atnext 40

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timeperiodwendanothersitewiththesmallestpenalty.ThissequenceisrepeatedforallTtimeintervals.Thus,thecomplexityofsuggestedapproachisO(nT).YavuzandJeffcoathasalsosuggestedthelookaheadmodicationofgreedyheuristicwhichtakesmorecomputationaltime.However,computationalexperimentsdemonstratethatthesolutionisimprovedcomparedtotheinitialsimplegreedyoptimization.Thestochasticnatureofschedulingsurveillancereducespredictabilityofsensorsbehaviorand,asaresul,tplaysanimportantroleformilitarytasks.HereJavuzandJeffcoatassumedthatsiteswerechosenrandomlybasedonprobabilitypijoftransitionfromi-thsitetoj-th.Then,sensorschedulingcanbeconsideredasaMarkovchainstochasticprocessandcharacterizedbysteadystateprobabilitiesi.Thegoalofstochasticapproachistondsuchsteadystateprobabilitiesthatminimizemaximumloss.Letribethevisitperiodofsitei.Then,thepenaltyofinformationlossatsiteiisai+(ri)]TJ /F4 11.955 Tf 13.07 0 Td[(1)bi,tattimet.Letusconsiderasufcientlysmallplanninghorizonwithtime-invariantsitedynamics.Thisallowsustoreducebi,ttobianddenotethisapproachasstatic.Visitingsiteiforeveryri>0periodsisequaltospendingi=1=riofthesensor'stimeatthesitei.TheoptimalscheduleisachievedwhenPii=1;i.e.,alltheavailabletimeisutilized.Also,thesensorneverstaysatanysitefortwoconsecutiveperiodsoftime.Thusri2(ori0.5)shouldbesatisedforeachsite.Then,thenon-linearmodelforobtainingoptimalstationaryprobabilitiesisformulatedas Minimizemaxifai+1 i)]TJ /F4 11.955 Tf 11.96 0 Td[(1big (3) s.t.nXi=1i=1, (3) i0.5,8i=1,...,n (3) i2R,8i=1,...,n (3) (3) 41

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Aheuristicsolvesthisnonlinearcontinuousproblem.Utilizingconstraints( 3 ),JavuzandJeffcoatdenealowerboundontheobjectivefunctionvaluewithCL=maxifai+big.Then,wesetC=maxifai+1 )]TJ /F4 11.955 Tf 11.96 0 Td[(1big=CLandcalculateri=(C)]TJ /F3 11.955 Tf 11.96 0 Td[(ai)=bi+1andi=1=riforalli.Notethati=0.5forthesiteswithai+bi=Candi<0.5fortherestconstraints.Ifthedeterminedprobabilitiesadduptoonethenwehavefoundtheoptimalsolutionandcanterminate.Ifsumofprobabilitiesislessthanone,thesolutionisoptimalagainandwecanshiftsome-suptomakePii=1.InthecasewhenPii>1thefoundCisinfeasibleandwecanapplyiterativeprocedures,suchasbisection,tondsuchCthatPii=1.Thestaticapproachminimizesaveragepenalty,determinedbysteadyprobabilities,anddoesnotaddressthecasesoflonglastingabsenceatasite.Takingintoaccountthefactthatsomerandomoutcomesmayresultinextremelylongpenalties,itisreasonabletoincreasetheprobabilitiesofvisitingthesitesthatwerevisitedalongtimeago.Ontheotherhand,probabilityofobservingtherecentlyvisitedsitesshouldbedecreased.Recallthatyi,trepresentthelasttimewhenthesiteiwasvisitedbythetimet.Theprobabilityofvisitingasitemustdependonthedifferencet)]TJ /F3 11.955 Tf 11.9 0 Td[(yi,t.Thusitwillincreaseprobabilityofvisitingoverduesites. 42

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Tocreateapreferenceforvisitingsiteiattimet,thefollowingadjustmentfactorsareproposedqi,t=it)]TJ /F3 11.955 Tf 11.95 0 Td[(yi,t rikItisbiggerthan1foroverduesitesandlessthan1forthesitesthathavebeenvisitedwithintheirexpectedvisitingperiods.Theparameterkisauser-denedparameter,whichdeterminestheweightoftheadjustmentfactor.Theprobabilitiesofvisitingeachsiteatspecictimepointtarebasedonprevioushistoryandcanbecomputedaspi=qi=Q,whereQ=Pni=1qi.Finally,ahybridmethodisbasedonacombinationofthegreedyalgorithmandthestochasticmethod,discussedabove.Therststepcalculatesthepenaltyofnotvisitingsitei:ci=ai+bi,t(t)]TJ /F3 11.955 Tf 12.55 0 Td[(yi,t).Thenextstepcalculatespreferencevaluestovisiteachsite:qi=ci cmaxk,wherecmax=maxifcig.Finally,theprobabilitiesofvisitareequalto:pi=qi=Q,whereQ=Pni=1qi. 3.2.2DeterministicsetupThesimplestcaseistomodelonesensorthatobservesagroupofsitesatdiscretetimepoints.Somephysicalsystemsrequirevirtuallyzerotimeforchangingasitebeingobserved.Forexample,thetimeofacamerarefocusinginstalledonasatelliteisnegligiblysmall.ThisassumptionleadstothemodelproposedbyYavuzandJeffcoatin[ 63 ].AssumethatweneedtoobservensitesduringTtimeperiods.Duringeveryperiodasensorisallowedtowatchonlyatoneofnsites.Theschedulingdecisioncanbemodeledusingbinaryvariablesxi,t xi,t=8><>:1,ifi-thsiteisobservedattimet,0,otherwise,(3)tisadiscreetvariableandt=1,2,...,T.Ifasiteiisnotobservedforsomeperiodoftime,itleadstothepenaltythatisproportionaltothetimeofnotobserving 43

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thissite.Thispenaltycanbemodeledusinganothergroupofdecisionvariables.Letyi,tdenotethetimeoflastvisitingsiteibeforetimemomentt.Letusnotethatvariablesxi,tcompletelydeterminevaluesofyi,t.Fixedpenaltyaiandvariablepenaltybitareassociatedwithsiteiattimemomentt.Thus,thepenaltyattimetassociatedwithsiteiis ai(1)]TJ /F3 11.955 Tf 11.95 0 Td[(xi,t)+bi,t(t)]TJ /F3 11.955 Tf 11.96 0 Td[(yi,t).(3)[ 63 ]suggestsminimizingmaximumlossoverallsitesandtimeintervals.Thustheobjectivefunctionisdenedasmaxi,tfai+bi,t(t)]TJ /F3 11.955 Tf 12.23 0 Td[(yi,t)g.Thisobjectivefunctioncanbelinearizedandconsequentlytheproblemlooksasfollowing MinimizeC (3) s.t.Cai(1)]TJ /F3 11.955 Tf 11.95 0 Td[(xi,t)+bi,t(t)]TJ /F3 11.955 Tf 11.95 0 Td[(yi,t),8i=1,...,n,8t=1,...,T, (3) nXi=1xi,t1,8t=1,...,T, (3) 0yi,t)]TJ /F3 11.955 Tf 11.95 0 Td[(yi,t)]TJ /F7 7.97 Tf 6.58 0 Td[(1txi,t,8i=1,...,n,8t=1,...,T, (3) txi,tyi,tt,8i=1,...,n,8t=1,...,T, (3) yi,0=0,8i=1,...,n, (3) xi,t2f0,1g,8i=1,...,n,8t=1,...,T, (3) yi,t2IR,8i=1,...,n,8t=0,...,T. (3) Constraints( 3 )ensurethatthesensorvisitsonlyonesiteatatime.Constraints( 3 )-( 3 )setthedependenceyi,tonxi,t.Thatisyi,tissettotifandonlyifthesensorisobservingsiteiattimetotherwiseyi,t=yi,t)]TJ /F7 7.97 Tf 6.58 0 Td[(1.Foreverysiteiandeverytimemomentt,wecancalculatethepenaltyassociatedwiththelasttimeasensorvisitedthissite(seeformula( 3 )).Letuspick(1)]TJ /F5 11.955 Tf 12.8 0 Td[()%ofworstcasesamongthesenTpenaltyvalues.Theninsteadofminimizingthemaximumloss,wecanminimizetheaveragelosstakenoverthese(1)]TJ /F5 11.955 Tf 13.03 0 Td[()%percentofworst-casepenaltyvalues.Althoughthisformulationisdeterministic,wewill 44

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demonstratethatitisequivalenttocomputing(1)]TJ /F5 11.955 Tf 12.16 0 Td[()ConditionalValue-atRisk(CVaR)forasetofrandomoutcomeshavingequalprobabilitiespi,t=1 nT.Thus,wecangeneralizeourformulationandwritetheobjectivefunctionforourproblemas Minimizex,yCVaR[L(x,y,i,t))],(3)where L(x,y,i,t)=ai(1)]TJ /F3 11.955 Tf 11.96 0 Td[(xi,t)+bi,t(t)]TJ /F3 11.955 Tf 11.95 0 Td[(yi,t)(3)Theparticularextremecasewhen!1correspondstominimizingmaximumpenaltyoverallt-sandi-s.Thiscasecorrespondstotheproblem( 3 )-( 3 ).Theotherextremecase=0givesaveragetakenoveralltimepointsandsites(ifweassumeuniformdistribution).Inthelattercasewecareaboutaverageloss.Besides,thereisahighchanceofnotpayingenoughattentiontoparticularbadoutcomes.Usingthegeneralapproachoutlinedinchapter2informulas( 2 )-( 2 ),ourproblemisnowformulatedasfollows: Minimizex,y,+1 1)]TJ /F5 11.955 Tf 11.96 0 Td[(Xi,tpi,t[ai(1)]TJ /F4 11.955 Tf 11.96 0 Td[(xi,t)+bi,t(t)]TJ /F4 11.955 Tf 11.96 0 Td[(yi,t))]TJ /F5 11.955 Tf 11.95 0 Td[(]+ (3) s.t.constraints( 3{59 )-( 3{64 ),2IR, (3) wherethevaluesofpi,tcanallbesetequalto1=nTasindicatedinthebeginningofthissection.Furthermore,thisproblemformulationcanbeeasilytransformedintoalinearmixedintegerproblembyintroducingasetofarticialvariableszi,tthatwillleadtothefollowingproblemwithasetofnTadditionalconstraints. 45

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Minimizex,y,+1 nT(1)]TJ /F5 11.955 Tf 11.95 0 Td[()Xi,tyi,t (3) s.t.ai(1)]TJ /F3 11.955 Tf 11.96 0 Td[(xi,t)+bi,t(t)]TJ /F3 11.955 Tf 11.95 0 Td[(yi,t))]TJ /F5 11.955 Tf 11.96 0 Td[(yi,t (3) yi,t0 (3) constraints( 3{59 )-( 3{64 ),2IR, (3) 3.2.3ProblemSetupunderUncertaintyToextendthedeterministicproblemformulationstoamorerealisticsetup,wherethevaluesofthepenaltyparametersareuncertain,IproposeanewCVaR-basedformulationofmulti-sensorschedulingproblems.Inthissetup,assumethatthexedandvariablepenaltyvaluesaiandbi,tarerandomvariableswithgivenjointdistributions.Further,wecanconsiderasetofpenaltyvalues(asi,bsi,t),s=1,...,ScorrespondingtoSdiscretesamples(orscenarios)asanapproximationofthejointdistribution.Thenforeachs=1,...,Sthelossfunctioncanbewrittenas: L(x,y,i,t,s)=asi(1)]TJ /F3 11.955 Tf 11.96 0 Td[(xi,t)+bsi,t(t)]TJ /F3 11.955 Tf 11.96 0 Td[(yi,t).(3)Itisappropriatetoconsider(1)]TJ /F5 11.955 Tf 12.02 0 Td[()%ofworst-casepenaltiesoverallindicesi,t,s.Wecanthenchoseameasureoflossasanaverageoverthese(1)]TJ /F5 11.955 Tf 12.21 0 Td[()%worstcasesandminimizetheaverage.Namely,weminimize CVaR[L(x,y,i,t,s)].(3) 46

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Usingtheapproachdescribedintheprevioussection,weobtainthefollowingrobustoptimizationproblemthatexplicitlytakesintoaccounttheuncertainpenaltyparameters: Minimizex,y,+1 nTS(1)]TJ /F5 11.955 Tf 11.95 0 Td[()Xi,t,s)]TJ /F4 11.955 Tf 10.46 -9.69 Td[(asi(1)]TJ /F4 11.955 Tf 11.95 0 Td[(xi,t)+bsi,t(t)]TJ /F4 11.955 Tf 11.95 0 Td[(yi,t))]TJ /F5 11.955 Tf 11.96 0 Td[(+ (3) s.t.constraints( 3{59 )-( 3{64 ),( 3{68 ).Asmentionedabove,thisformulationcanbelinearizedbyintroducingextravariablesandconstraints,andthelinearmixedintegerformulationisprovidedbelow. Minimizex,y,+1 nTS(1)]TJ /F5 11.955 Tf 11.96 0 Td[()Xi,t,syi,t,s (3) s.t.asi(1)]TJ /F3 11.955 Tf 11.96 0 Td[(xi,t)+bsi,t(t)]TJ /F3 11.955 Tf 11.95 0 Td[(yi,t))]TJ /F5 11.955 Tf 11.96 0 Td[(yi,t,s (3) yi,t,s0 (3) constraints( 3{59 )-( 3{64 ),( 3{68 ). 3.3EquivalentFormulationsinCardinalityConstraintsInthissection,weshowthatthedevelopedlinearmixedintegerprogrammingproblemscanbeequivalentlyreformulatedasproblemswithcardinalityconstraints.Asitwillbediscussedlater,solvingtheseequivalentreformulationscanprovidebettercomputationalspeedandperformanceinndingnear-optimalsolutionsoftheconsideredproblems.Itshouldbenotedthatduetothehighdimensionalityandcomplexityoftheseproblems,itisoftenimpossibletondexactoptimalsolutionsinareasonabletime;however,itisoftenusefulinpracticetoutilizeheuristictechniquesthatcanndnear-optimalsolutionsfast.Thereexistheuristics[ 38 ]aswellassoftwarepackages[ 1 ]whichcansolveoptimizationproblemsformulatedintermsofcardinalityconstraints.Cardinalityfunction 47

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simplyequalstothenumberofnon-zerocomponentsofitsvectorargument.Moreformally,forx=(x1,...,xn)T2IRncard(x)=nXi=1I(xi),whereIisanindicatorfunctiondenedas: I(z)=8><>:1,z6=0;0,otherwise.(3)Thissectionpresentsaproblemformulatedintermsofcardinalityfunction.Thisnewproblemisequivalenttotheinitialformulation( 3 )-( 3 )thatcanbewrittenasProblem(I): Minimizemaxi,tfai(1)]TJ /F4 11.955 Tf 11.96 0 Td[(xi,t)+bi,t(t)]TJ /F4 11.955 Tf 11.95 0 Td[(yi,t)gs.t.nXi=1xi,tm,8t=1,...,T, (3) 0yi,t)]TJ /F3 11.955 Tf 11.95 0 Td[(yi,t)]TJ /F7 7.97 Tf 6.58 0 Td[(1txi,t,8i=1,...,n,8t=1,...,T, (3) txi,tyi,tt,8i=1,...,n,8t=1,...,T, (3) yi,0=0,8i=1,...,n, (3) xi,t2f0,1g,8i=1,...,n,8t=1,...,T, (3) yi,t2IR,8i=1,...,n,8t=0,...,T. (3) Thenewproblemcanbeformulatedas: 48

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Problem(II): Minimizemaxi,tfai(1)]TJ /F4 11.955 Tf 11.96 0 Td[(xi,t)+bi,t(t)]TJ /F4 11.955 Tf 11.95 0 Td[(yi,t)gs.t.card(~xt)m,8t=1,...,T, (3) 0yi,t)]TJ /F3 11.955 Tf 11.95 0 Td[(yi,t)]TJ /F7 7.97 Tf 6.58 0 Td[(1txi,t,8i=1,...,n,8t=1,...,T, (3) yi,tt,8i=1,...,n,8t=1,...,T, (3) yi,0=0,8i=1,...,n, (3) 0xi,t1,8i=1,...,n,8t=1,...,T, (3) yi,t2IR,8i=1,...,n,8t=0,...,T. (3) where~xt=(x1,t,...,xN,t)T.Thefollowingtheoremprovidestherelationbetweenthetwoproblems Theorem3.1. Thesetofoptimalsolutionsofproblem(I)belongstothesetofoptimalsolutionsofproblem(II).Moreover,ifapoint(xII,yII)isanoptimalsolutionfor(II),theoptimalsolutionof(I)(xI,yI)canbeconstructedasxIi,t=dxIIi,te,yIi,t=maxtfxIi,g.Inordertoprovethetheoremwewilluseanauxiliaryformulation.Problem(III): 49

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Minimizemaxi,tfai(1)]TJ /F4 11.955 Tf 11.96 0 Td[(xi,t)+bi,t(t)]TJ /F4 11.955 Tf 11.95 0 Td[(yi,t)gs.t.card(~xt)m,8t=1,...,T,0yi,t)]TJ /F3 11.955 Tf 11.95 0 Td[(yi,t)]TJ /F7 7.97 Tf 6.58 0 Td[(1txi,t,8i=1,...,n,8t=1,...,T,yi,tt,8i=1,...,n,8t=1,...,T,yi,0=0,8i=1,...,n,0xi,t1,8i=1,...,n,8t=1,...,T,yi,t2IR,8i=1,...,n,8t=0,...,T.Denotebyz(I),z(II)andz(III)theoptimalobjectivevaluesofproblems(I)-(III)consequently. Lemma1. Foreveryoptimalsolutionof(III)thereexistsasolutionthatwillbebothfeasibleandoptimalin(I)and(III)(i.e.formulations(I)and(III)areequivalentinthissense). Proof. Equations( 3 )-( 3 )enforcethatyi,t=maxtfxi,g.Ifthereexistsanoptimalsolutionfor(III)suchthatxi,t=1butyi,t
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Thus,foreveryoptimalsolutionof(III)thereexistsasolutionthatwillbebothfeasibleandoptimalin(I)and(III)thatmeansthatthesetwoformulationsareequivalent. ProofofTheorem 3.1 .Letusconsidersomeoptimalsolutionof(II)x0i,t,y0i,tandbuildanewsolutionxi,t=Ifx0i,t>0g;yi,t=y0i,t.Thissolutionwillstillbefeasibleandoptimalfor(II)sinceitwillnotincreasetheobjectivevalue.Obviously,thissolutionwillbefeasibleandoptimalfor(III)(z(II)z(III)).AccordingtoLemma 1 foreveryoptimalsolutionof(III)thereexistsasolutionxi,t=xi,t,yi,t=maxtfxi,gthatwillbebothfeasibleandoptimalfor(I)and(III).Thissolutionwillbeintegerandfeasiblefor(II).Since8i,tyi,tyi,t,theobjectivevaluewillnotincreaseand,therefore,thissolutionwillalsobeoptimalfor(II).Thus,therealwaysexiststheoptimalintegersolutionfor(II)thatwillbealsooptimalfor(I)Inordertopreventthesolutionof(II)frombeingnon-integral,weaddapenaltytotheobjectiveof(II):Problem(IV): Minimizemaxi,tfai(1)]TJ /F4 11.955 Tf 11.96 0 Td[(xi,t)+bi,t(t)]TJ /F4 11.955 Tf 11.96 0 Td[(yi,t)g+(mT)]TJ /F14 11.955 Tf 11.95 11.35 Td[(Xi,txi,t)s.t.constraints( 3{86 )-( 3{91 ),where>0. Corollary1. Problems(I)and(IV)havethesamesetofoptimalvaluesoffxi,tg.Similartheoremscanbeprovenfortheotherformulations,namelypercentiledeterministicandstochasticsetups.Fordeterministiccaseproblem(Ia)whichisequivalentto( 3 )isrelatedtoreformulatedintermsofcardinalityproblem(IIa): 51

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Problem(Ia): MinimizeCVaRfai(1)]TJ /F4 11.955 Tf 11.95 0 Td[(xi,t)+bi,t(t)]TJ /F4 11.955 Tf 11.95 0 Td[(yi,t)gs.t.constraints( 3{80 )-( 3{85 ).Problem(IIa): MinimizeCVaRfai(1)]TJ /F4 11.955 Tf 11.95 0 Td[(xi,t)+bi,t(t)]TJ /F4 11.955 Tf 11.95 0 Td[(yi,t)gs.t.constraints( 3{86 )-( 3{91 ). Theorem3.2. Thesetofoptimalsolutionsofproblem(Ia)belongstothesetofoptimalsolutionsofproblem(IIa).Moreoverifapoint(xII,yII)isanoptimalsolutionfor(IIa),theoptimalsolutionof(Ia)(xI,yI)canbeconstructedasxIi,t=dxIIi,te,yIi,t=maxtfxIi,g.Forthestochasticcaseproblem(Ib),whichisequivalentto( 3 )isrelatedtothereformulatedintermsofcardinalityproblem(IIb):Problem(Ib): MinimizeCVaRfasi(1)]TJ /F4 11.955 Tf 11.96 0 Td[(xi,t)+bsi,t(t)]TJ /F4 11.955 Tf 11.96 0 Td[(yi,t)gs.t.constraints( 3{80 )-( 3{85 ).Problem(IIb): MinimizeCVaRfasi(1)]TJ /F4 11.955 Tf 11.96 0 Td[(xi,t)+bsi,t(t)]TJ /F4 11.955 Tf 11.96 0 Td[(yi,t)gs.t.constraints( 3{86 )-( 3{91 ). Theorem3.3. Thesetofoptimalsolutionsofproblem(Ib)belongstothesetofoptimalsolutionsofproblem(IIb).Moreoverifapoint(xII,yII)isanoptimalsolutionfor(IIb),the 52

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optimalsolutionof(Ib)(xI,yI)canbeconstructedasxIi,t=dxIIi,te,yIi,t=maxtfxIi,g. 3.4SensorSchedulinginNetwork-BasedSettings Figure3-1. Exampleofapossiblenetwork.Twonodesareconnectedbyanarcifasensorcanmovefromonenodetoanotherinconsequenttimeperiods. First,letusdiscussaspecialcaseofonesensor(m=1)togiveanideaofthismodelingapproach.Inthecasewhensurveillancerequiressensorstophysicallymovefromonesitetoanother,theirtransitionabilitiesarelimitedwithdistanceorotherconstraint(forexample,amountaincanbeanaturalobstacleforUAVtomovebetweensites).Inthiscase,eachsitecanbemodeledasanodeofanetworkG=(V,E).Wheneverthereisnoarcbetweentwonodesiandj,weaddtheinequality xi,t+xj,t+11,(3) 53

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thatprohibitstheinfeasiblemovei!jinconsequenttimeperiodstandt+1.Formulations( 3 )-( 3 ),( 3 )and( 3 )canbeslightlymodiedtoobtaincorrespondingformulationsforonesensor(m=1).Ifwedemandthesensortostartandcomebacktoadepotlocatedatthecertainsite,wecanoptionallysettheinitial(i02f1,...,ng)andnal(iT2f1,...,ng)locationsofthesensor.Thus,forthespecialcasewhenm=1problem( 3 )-( 3 )canbeformulatedonthenetwork: Minimizemaxi,tfai(1)]TJ /F4 11.955 Tf 11.95 0 Td[(zi,t)+bi,t(t)]TJ /F4 11.955 Tf 11.95 0 Td[(yi,t)g (3) s.t.constraints( 3{59 )-( 3{64 ),(m=1),xi,t+xj,t+11whenever(i,j)62E,8t=1,...T,i,j=1,...,n (3) xi0,1=1,wherei02f1,...,ngistheinitiallocationofthesensor (3) xiT,1=1,whereiT2f1,...,ngisthenallocationofthesensor (3) Inthisformulationconstraints( 3 )prohibitinfeasiblemovesbetweennotconnectednodes.( 3 )and( 3 )setinitialandnaldestinationforthesensor.Theotherformulations,namelydeterministic( 3 )andstochastic( 3 ),canbeeasilyadaptedfornetworkcase(m=1)inthesamewaybyaddingnetworkconstraints( 3 )-( 3 )totheexistingsetsofconstraints.Thisapproach,however,maynotbeeasilyextendedforthecasesoftwoormoresensors.Ifwesimplyadd( 3 )-( 3 )toexistingnon-networkformulations,wecanarriveatthesituationwhichpreventsfeasiblemoveswhentwoormoresensorsareinvolved.Figure 3-2 providesacounterexample.Lettwosensorsattimemomentt=1belocatedatnodes1and3.Althoughtheycouldmovetonodes2and4respectivelyatthenexttimepointt=2,theconstraintx1,1+x2,41wouldprohibitthismove.Toavoid 54

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Figure3-2. Counterexample(m=2):twosensorscannotperformsimultaneousfeasiblemoveduetotheconstraintx1,1+x4,21 suchasituationweneedtoaddonemoreindexfordecisionvariablex: xi,t,k=8><>:1,sensorkissurveillingsiteiattimet;0,otherwise.(3)WecanensurethateverysensorisassignedtoasiteateverytimeperiodTwiththeconstraint: nXi=1xi,t,k=1,8k=1,...,m,8t=1,...,T.(3)Letusintroducezi,tindicatingwhethersiteiisobservedattimet,namely zi,t=8><>:1,ifanysensorissurveillingsiteiattimet;0,otherwise.(3)Variableszi,tandxi,t,kcanberelatedwiththeconstraint zi,tnXi=1xi,t,kmzi,t,(3)whichstatesthatsiteiisbeingobservedattimet(zi,t=1)ifandonlyifatleastonesensorispresentatsitei(Pni=1xi,t,k>0). 55

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Thelossfunctioniswrittenas L(x,y,i,t)=asi(1)]TJ /F3 11.955 Tf 11.95 0 Td[(zi,t)+bsi,t(t)]TJ /F3 11.955 Tf 11.96 0 Td[(yi,t).(3)Ifwewanttominimizethemaximumlossusingtheformulatedaboveconstraintsandsimilarlytothenon-networksetup,wecanreformulatethedeterministicmaximumlossminimizationproblem( 3 )-( 3 )onnetworkasfollows: Minimizemaxi,tfai(1)]TJ /F4 11.955 Tf 11.96 0 Td[(zi,t)+bi,t(t)]TJ /F4 11.955 Tf 11.95 0 Td[(yi,t)g (3) s.t.nXi=1xi,t,k=1,8k=1,...,m,8t=1,...,T, (3) zi,tmXk=1xi,t,kmzi,t,8i=1,...,n,8t=1,...,T, (3) 0yi,t)]TJ /F3 11.955 Tf 11.95 0 Td[(yi,t)]TJ /F7 7.97 Tf 6.58 0 Td[(1tzi,t,8i=1,...,n,8t=1,...,T, (3) tzi,tyi,tt,8i=1,...,n,8t=1,...,T, (3) yi,0=0,8i=1,...,n, (3) xi,t,k+xj,t+1,k1,whenever(i,j)62E,8t=1,...T,i,j=1,...,n,k=1,...,m (3) xi0,k,1,k=1,wherei0,k2f1,...,ngistheinitiallocationofsensork (3) xiT,k,T,k=1,whereiT,k2f1,...,ngisthenallocationofsensork, (3) xi,t,k2f0,1g,8i=1,...,n,8t=1,...,T,8k=1,...,m, (3) yi,t2IR,8i=1,...,n,8t=0,...,T, (3) zi,t2f0,1g,8i=1,...,n,8t=1,...,T. (3) FormulationsforCVaRstochasticanddeterministiccasesaswellasthelinearizedformulationcanbeobtainedthesamewayasinnonnetworkcase. 56

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3.5ComputationalExperiments Table3-1. Performanceresultsfordeterministicmodel( 3 )-( 3 ).n-numberofsites;m-numberofsensors.Thenumberofdiscretetimestepsisxed:T=10. n=8n=9n=10n=11n=12 m=1cplexvalue320330330332332psgvalue375376376370375%14.7%12.2%12.2%10.3%11.5%time:cplex/psg31.2/2.279.1/2.4134.9/2.5167.7/2.7198.9/2.7 m=2cplexvalue240245250260265psgvalue305310304310310%21.3%21%17.8%16.1%14.5%time:cplex/psg38.4/2.2142.7/2.3928/2.52042.7/2.65898.9/2.8 m=3cplexvalue206210215217224psgvalue233250265256275%11.6%16%18.9%15.2%18.5%time:cplex/psg30.7/2.339.3/2.481.1/2.51003.6/2.79317.9/2.9 m=4cplexvalue190194196200200psgvalue215217242237242%11.6%10.6%19%15.6%17.4%time:cplex/psg6/2.376.5/2.564.6/2.6231.4/2.7589.3/2.8 m=5cplexvalue183185188190190psgvalue196202215217220%6.6%8.4%12.6%12.4%13.6%time:cplex/psg1.4/2.32.2/2.538/2.651.9/2.768/3 m=6cplexvalue165170170183185psgvalue185185197195200%10.8%8.1%13.7%6.2%7.5%time:cplex/psg1.1/2.41.5/2.52.9/2.729.1/2.8123/3 m=7cplexvalue155160163168170psgvalue160171185190188%3.1%6.4%11.9%11.6%9.6%time:cplex/psg0.3/2.30.9/2.51.8/2.8113.2/2.925.2/3 Thecomputationalexperimentswereperformedonthetestproblemsusingtwocommercialoptimizationsoftwaresolvers:ILOGCPLEX[ 2 ]andAOrDAPSG[ 1 ].Theperformanceofthesolversiscomparedintables 3-1 3-3 (eachtablecorrespondstooneofthethreeproblemformulations).ItcanbeobservedthatCPLEXndsexactsolutions,however,ittakestoomuchtimeforlargeinstances,especiallyfortheproblemsunderuncertainty.PSGallowssacricingqualityfortime,i.e.theobtainedsolutionsfor 57

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Table3-2. PerformanceresultsforCVaRtypedeterministicmodel( 3 ).n-numberofsites;m-numberofsensors.Thenumberofdiscretetimestepsisxed:T=10.CVaRcondencelevel=0.9. n=8n=9n=10n=11n=12 m=1cplexvalue307.1318.4317318.7318psgvalue360.1357.9356.8357.6353.7%14.7%11%11.2%10.9%10.1%time:cplex/psg45.5/2.899.3/374/2.962.8/3.1130.9/3 m=2cplexvalue231.9241.1245.2--psgvalue297299297.5293.5291.6%21.9%19.4%17.6%--time:cplex/psg1072.8/310924.6/3.116212.1/3.1-/3.2-/3.2 m=3cplexvalue198.6205.3---psgvalue223.6237.3245255.4260.9%11.2%13.5%---time:cplex/psg1910.7/2.845540.4/2.8-/2.9-/3-/3.4 m=4cplexvalue187.9190.4---psgvalue211.4207.2230218.2221.8%11.1%8.1%---time:cplex/psg7989.2/3.223852.7/3-/3.3-/3.4-/3.2 m=5cplexvalue173.3179.3---psgvalue193.4192.6197.8207.7210%10.4%6.9%---time:cplex/psg7672.6/2.925593/2.8-/3.3-/3.3-/3.7 m=6cplexvalue161.1165.3---psgvalue179.5182.8185.2195.2190.9%10.2%9.5%---time:cplex/psg2250.3/2.836618.1/3.3-/3.4-/2.9-/3.4 m=7cplexvalue142.4----psgvalue150.9166.6173.4179183.8%5.6%----time:cplex/psg6640.9/2.965122.8/3.3-/3-/3-/3.7 cardinalityformulationsarenotgloballyoptimal,butthecomputationaltimeisnegligiblysmall.Thenumericalexperimentsshowthatlocalsolutionsdifferfromglobalonesin10-20%formostcases.Table 3-4 comparesperformanceofCPLEXandPSGinndingapproximatesolutionsforstochasticcase(n=12sitesandT=10timeperiods).WestoppedCPLEXwhenitfoundtheobjectiveassmallasthePSGobjectivevalue(andrecordedthecomputationtime).ItappearsthatPSGoutperformsCPLEXforproblemswitha 58

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Table3-3. PerformanceresultsforCVaRtypestochasticmodel( 3 ).n-numberofsites;m-numberofsensors.Thenumberofdiscretetimestepsisxed:T=10,numberofscenariosS=100.CVaRcondencelevel=0.9. n=8n=9n=10n=11n=12 m=1cplexvalue-----psgvalue394398.7390.1395.6394.9%-----time:cplex/psg-/25.9-/28-/35.8-/40.2-/47.7 m=2cplexvalue-----psgvalue297.9304321.7324.4318.6%-----time:cplex/psg-/32.7-/38.6-/51.4-/67.8-/76 m=3cplexvalue-----psgvalue260.4260.1262.4264.4269%-----time:cplex/psg-/38.4-/45.4-/56.3-/71.4-/84.6 m=4cplexvalue-----psgvalue229.4231.6242.9236.3250.8%-----time:cplex/psg-/47.2-/60.2-/72.5-/84.6-/98.4 m=5cplexvalue-----psgvalue209.1212.4220.8224.6230.1%-----time:cplex/psg-/56.1-/72.3-/83.7-/96.6-/119.5 m=6cplexvalue-----psgvalue193.2199.9209.2210.3218.7%-----time:cplex/psg-/51.8-/67.3-/86.1-/112.9-/121.4 m=7cplexvalue-----psgvalue164.4186.3187.6198.6204.4%-----time:cplex/psg-/45-/66.5-/89.8-/111.3-/133.5 largenumberofstochasticscenarioswhiletheyhavesimilarperformanceforsmallsizeproblems.Therefore,basedonthesizeoftheproblemanduserrequirements,onecandeterminetheappropriateequivalentproblemformulationandtheoptimizationsolverthatcanbeusedtondanoptimaloranear-optimalsolution.Weperformedexperimentsonnetworkformulationusingthenetworkprovidedongure 3-1 .Table 3-5 providesCPUtimesinsecondsforobtainingtheexactsolution 59

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Table3-4. ComparingPSGandCPLEXperformanceforobtainingapproximatesolutionofCVaRtypestochasticproblem( 3 )(n=12sitesandT=10timeperiods). PSGValuePSGTime(sec)CPLEXTime(sec) m=1389.210332m=2318.532110185m=3276.332133230m=4246.648136247m=5229.234208210m=6218.166158330m=7204.1201260m=8193.199191220m=9180.976155250m=10164.746191280m=11146.607180200 Table3-5. ILOGCPLEXCPUtime(sec)fornetworkdeterministicmodel( 3 )-( 3 ).n-numberofsites;m-numberofsensors.ThenumberofdiscretetimestepsisT=10. n=6n=7n=8n=9n=10n=11n=12 m=10.310.770.762.160.900.501.21m=23.963.968.0015.85238.42174.93315.76m=30.9321.4822.209.17340.86350.852037.73m=40.200.310.7434.2614.45926.64436.80m=50.321.792.6231.4091.3862.203359.69m=60.581.794.3669.7619.05238.59m=72.124.0822.8719.5747.20m=80.796.297.1438.69m=91.173.196.81m=100.792.15m=111.94 forthedeterministicnetworkcase( 3 )-( 3 )inILOGCPLEX.Computationwasperformedforanumberofsitesfrom6to12and10discretetimesteps. 60

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CHAPTER4TWOSTAGESTOCHASTICOPTIMIZATIONMODELFORROBUSTNETWORKFLOWDESIGNOptimizationonnetworksisprobablytheoldesttypeofproblemsstudiedintheeldofoperationresearch.Hundredsofformulationsandapplicationshavebeendevelopedoverthedecades.Theinterestinoptimizationcanbeexplainedbyitsnumerousapplicationsintransportation,energy,computing,andmanyotherengineeringdisciplines[ 4 10 24 25 51 ].Inordertomodelrealworldproblemsmoreefciently,manyoftheseapplicationsneedtoaddressuncertainties,adverseactionsofenemiesagainstthenetworkaswellastheircombination.Thelattersignicantlycomplicatesthenetworkproblemsduetotheincreaseinthecomputationalcomplexityoftheresultingoptimizationproblems.Someoftheproblemsdiscussedintheliteratureincludethecaseswhenthelengthofarcisarandomvariable[ 23 ],arccapacitiesarerandom[ 28 ],anarcissubjecttofailure[ 26 ],robustowunderassumptionthattheworstoutcomehappens[ 11 ].Thischaptersolvesanetworkowproblemwheretheconditionalexpectationofworstcollateralowlossisconstrained.Iintroducealossfunctionthatcharacterizesthetotallossofowthathappensinthenetworkasaresultofarcfailure.Then,acombinationofBenders'decompositionandLagrangianrelaxationisusedtosolvethetwostagestochasticformulation. 4.1ProblemFormulationLetanetworkberepresentedbyadirectedgraphG=(V,E)andeacharc(i,j)2Ehaveanassociatedcostci,jperunitofowpassedalonethearcfromnodeitoj.Letmaximumcapacityvalueui,jdenotethemaximumamountofowthatcanbepushedalonethearc(i,j).Letdidenotethedemandorsupplyforeachnodei2V.Weassumethatpositivevaluesofdirepresenttheexcessofoworsupplyandnegativevaluesrepresentdemands.Thewell-knowndeterministicminimumcostnetworkow 61

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problemcanbeformulatedasthefollowinglinearproblem:MinimizeX(i,j)2Eci,jxi,j, (4)s.t.Xfj:(i,j)2Egxi,j)]TJ /F14 11.955 Tf 23.66 11.36 Td[(Xfj:(j,i)2Egxj,i=di,8i2V, (4)0xi,jui,j,8i,j2V. (4)Here,weminimizethetotalcostoftransferringtheowxi,jinthenetworkGprovidedthatalldemandsaresatisedandnospillageoccurs(constraints( 4 )).Boginskietalsuggestedin[ 16 ]arobuststochasticformulationofthisprobleminthecasewheneveryarc(i,j)ofthenetworkarcscanbeindependentlydestroyedwithknownprobabilitiespi,jforeacharc(i,j).LetavectorofBernoullirandomvariablesyrepresenttheuncertainarcfailuresinthenetwork:yi,j=8><>:1,withprobabilitypi,j;0,withprobability1)]TJ /F3 11.955 Tf 11.95 0 Td[(pi,j.Inordertoapproximatethestochasticoutcome,itwassuggestedtogenerateanitesetofscenarios: ysi,j=8><>:1,ifarc(i,j)failsunderscenarios,0,otherwise.(4)Thefunctionrepresentingthelossoftakingdecisionxunderrealizationofrandomvariableyiscalledlossfunctionandisdenedin[ 16 ]as L(x,y)=X(i,j)2Exi,jyi,j.(4)Here,thelossisdeterminedbytheowschedulexi,jandstochasticoutcomeyi,j.Thelossisequaltotheamountofowbeingpassedthroughthedisruptedarcs. 62

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TheCVaRminimizationproblemcanbewrittenasalinearprogramusingstandardtechniquesdescribedinchapter2:MinimizeX(i,j)2Eci,jxi,j, (4)s.t.Xfj:(i,j)2Egxi,j)]TJ /F14 11.955 Tf 23.66 11.35 Td[(Xfj:(j,i)2Egxj,i=di,8i2V, (4)0xi,jui,j,8i,j2V. (4)+1 S(1)]TJ /F5 11.955 Tf 11.95 0 Td[()SXs=1tsC (4)tsX(i,j)2Exi,jysi,j)]TJ /F5 11.955 Tf 11.96 0 Td[(,8s=1,2,...,S, (4)2IR. (4)Thelossfunctionusedinthisformulationhascertaindisadvantages.Particularly,forthecasesrepresentedingure 4-1 thelossesL()=4,ifthearc(1,2)failsandL()=7,ifthearcs(1,2),(2,3),(2,4),and(2,5)fail.Itcanbeseen,however,thatbothcasesareequivalentinasensethatthereisnoowinthenetwork.Asaresult,thelossisactuallythesame.Anotheressentialfactorshouldbeconsidered.Itisnotespeciallycriticalwhicharcsfailed.Whatmatters,however,iswhetherwesatisfythedemandofconsumers.Moreover,notallconsumersareequalintherealnetworks.Forexample,ifwedeliverelectricityinapowernetworkgrid,itiscrucialtoensureuninterruptablesupplyofhospitalsandotherstrategicobjects,whilestreetlightinghasamuchlowerdegreeofimportance.Takingintoaccounttheabovearguments,wesuggestthatlossshouldbedeterminedbytheamountofundeliveredowcausedbyarcfailures.Letusintroduce 63

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Figure4-1. Effectcausedbynetworkfailure.Inbothcasesthenetworklostthesameamountofow. variablesxsi,jrepresentingtheactualpossibleowaftersomearcsfailedinthescenariosandrsiisanabsolutevalueofaremainingsupply/demandforthenodei.Naturally,fortransitionnodesitheremainingbalanceisalwayszero.Clearly,xsi,jisdeterminedbythedecisionvectorxandtheoutcomevectory.Thedependenceissetbythefollowingsetofconstraints:0xsi,jxi,j(1)]TJ /F3 11.955 Tf 11.95 0 Td[(ysi,j),8i,j,s; (4)Xfj:(i,j)2Egxsi,j)]TJ /F14 11.955 Tf 23.66 11.36 Td[(Xfj:(j,i)2Egxsj,i=sign(di)rsi8s,i; (4)0rsijdisj (4)Constraints( 4 )ensurethattheremainingowxsi,jinthescenariosdoesnotexceedtheinitiallyscheduledowxi,j.( 4 )ensuresthatinowandoutowarebalancedineverynodei.( 4 )guaranteesthatactualinowandoutowinthenetworkdonotexceedthedemandandsupply.Thedifference)]TJ /F3 11.955 Tf 9.3 0 Td[(di)]TJ /F3 11.955 Tf 11.95 0 Td[(rsiiscomputedforeachoftheconsumingnodes,i.e.di<0andcharacterizedbythedecitofowintendedfortherecipientatnodei.Naturally,ourgoalistondsuchaschedule 64

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thatminimizesthelossofapartialorcompletefailuretosatisfythedemand.Letusassociatesomepenaltyfactorliwitheachconsumingnodei(di<0).Then,thelossfunctionisdeterminedasasolutionofthefollowinglinearoptimizationproblem:L(x,y)=minxs,rss.t.( 4{12 ){( 4{14 )Xfi:di<0gli()]TJ /F3 11.955 Tf 9.3 0 Td[(di)]TJ /F3 11.955 Tf 11.95 0 Td[(rsi),Dependingonthecontext,thepenaltyfactorli0canbeamonetaryforfeitforinabilitytomeetcontractobligationorsomeempiricalimportancecoefcientofthedestinationnode.Clearly,thefunctionL(x,y)isapiecewiselinearfunctionwithrespecttoeitherdecisionxorstochasticoutcomeysincealltherelationsaresetbyminimizinglinearobjectivefunctionsubjecttolinearconstraintswhichlinearlydependontheparametersxandy.Usingtheintroduceddenitionsofloss,therobustnetworkowproblemcanbeformulatedasMinimizeX(i,j)2Eci,jxi,j, (4)s.t.constratints( 4{2 )-( 4{3 ),( 4{12 )-( 4{14 );CVaRL(x,y)C. (4)Here,Cintheconstraint( 4 )isthepredenedmaximallevelof-CVaRoftheloss.TheaboveformulationcanbetransformedintothefollowinglinearproblemusingthestandardtechniqueforlinearizingCVaRfunction: 65

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Minimizex,xs.rsX(i,j)2Eci,jxi,j, (4)s.t.Xfj:(i,j)2Egxi,j)]TJ /F14 11.955 Tf 23.67 11.36 Td[(Xfj:(j,i)2Egxj,i=di,8i2V, (4)0xi,jui,j,8i,j2V. (4)0xsi,j(1)]TJ /F3 11.955 Tf 11.95 0 Td[(ysi,j)xi,j,8i,j,s; (4)Xfj:(i,j)2Egxsi,j)]TJ /F14 11.955 Tf 23.67 11.35 Td[(Xfj:(j,i)2Egxsj,i=sign(di)rsI,8s,i,; (4)0rsijdij; (4)+1 S(1)]TJ /F5 11.955 Tf 11.96 0 Td[()SXs=1tsC; (4)tsXfi:di<0gli)]TJ /F2 11.955 Tf 5.48 -9.68 Td[()]TJ /F3 11.955 Tf 9.3 0 Td[(di)]TJ /F3 11.955 Tf 11.96 0 Td[(rsixsj,i)]TJ /F5 11.955 Tf 11.95 0 Td[(,ts0,8s=1,2,...,S; (4),ts,xij,xsi,j,rsk2IR,8(i,j)2E,8k2V,8s=1,2,...,S. (4) 4.2DecompositionMethodforNetworkFlowProblemTheformulatedproblem( 4 )-( 4 )isapartialcaseofatwostagestochasticproblemthatcanbewrittenasfollows:Minimizexf(x)+g(x,y),wherethedeterministicpartf(x)canbecomputedwhenthedecisionistakenandthesocalledsecondstagepartg(x,y)thatdependsontheinitialdecisionx,stochasticoutcomeandtheresponsevariableythatisdeterminedonthesecondstageafterarandomscenariohadbeenimplemented.Inthecasewheng(x,y)islinearwithrespecttoxandyandtherelationsbetweenxandyaresetbylinearconstraint,wecanusesamplingandnallyarriveatthelinearprogramhavingblock-ladderstructure: 66

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V=MinimizecTx+fT1y1+fT2y1,+...+fTkyks.t.Ax=b,B1x+D1y1=d1,B2x+D2y2=d2,.........Bkx+D1y1=dk.(4)Forexample,inournetworkowxvariablescorrespondtotheinitialowdesignandsecondstagerespondvariablesyiaretheremainingowsandbalances.Duetopotentiallylargeamountofscenariosthesizeofthelinearprogramcanbehugeand,therefore,theproblemmaynotbetractable.Thereareseveraltechniquesaimedtodealwithsuchaspecialstructureandlargeamountofvariables.Theyaretheinternalpointmethods,columngenerationalgorithms,etc.WewilluseBendersdecompositionmethodthathasbeenaclassicalmethodforsolvinglargescalestochasticprogrammingproblems.ItwasoriginallydesignedbyBenders[ 9 ]tosolvemix-integerlinearproblems,andlaterextendedtononlinearprograms.Ithasbeenrealizedthatthealgorithmtstheneedsofstochasticoptimizationwhentheproblemisdecomposableintodeterministic(rststage)andstochastic(secondstage)parts.Theproblem( 4 )canberewrittenasafollowingMasterProblem(MP):MP:V=minxcTx+PKj=1zi(x),s.t.Ax=b,x0,where 67

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P2i:zi(x)=minyifTiyi,s.t.Diyi=di)]TJ /F3 11.955 Tf 11.95 0 Td[(Bix,yi0,isasecondstageproblemthatcorrespondstoscenarioi,whentherststagedecisionvariablesxarexed.Byutilizingnotionofduality,wecancomputezi(x)viathedualofP2iwhereparameterxismovedtoobjectivefunction.LetusdenotethedualasD2iasD2i:zi(x)=maxpipi(di)]TJ /F3 11.955 Tf 11.96 0 Td[(Dix),s.t.DTipifi.Denotethefeasiblesetofthedualproblemas D2=fpijDTipifig.(4)Let p1i,...,pIii,(4)betheextremepointsofthesetD2and r1i,...,rJii,(4)betheextremeraysoftheset.Letusnotethatinourspeciccase,D2iwillalwayshaveaniteoptionalsolutionbecausetheprimalwillhaveasolutionforanyvalidnetworkowxchosenonarststage.Therefore,thesolutionofD2iwillbeoneoftheextremepointspiandtheoptimalobjectivefunctionvalueziwillsatisfy: zi(x)=(pi)T(di)]TJ /F3 11.955 Tf 11.96 0 Td[(Bix)=maxk(pki)T(di)]TJ /F3 11.955 Tf 11.96 0 Td[(Bix).(4) 68

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Thus,wecanre-writeD2iD2i:zi(x)=minzizi,s.t.pji(di)]TJ /F3 11.955 Tf 11.96 0 Td[(Dix)zi,j=1,...Ij.Thelatterproblemassuresthatoptimalvaluezi(x)isadualobjectiveevaluatedatthebestextremepointsofthedualfeasibleset.NowwecanreformulatetheinitialMasterProblemandwriteitasMP:V=minx,z1,...,zkcTx+PKi=1zi(x),s.t.Ax=b,x0,(pji)T(di)]TJ /F3 11.955 Tf 11.95 0 Td[(Bix)zi,j=1,...,Ji,i=1,...,k.ThisproblemiscalledFullMasterProblem(FMP).Inthisproblemahugenumberofsecondstagevariablesisreplacedwithkextravariableszi.Besides,wehaveaddedconstraintswhosenumbercanbeexponentiallylargeinageneralcase.Letusnotethatwecanconsiderasmallsubsetofthefullsetofconstraints.WewillobtainareducedMasterProblemwhichcontainsonlymextremepointsoftheFullMasterProblem'sconstraints:RMPm:Vm=minx,z1,...,zkcTx+PKi=1zi(x),s.t.Ax=b,x0,(pji)T(di)]TJ /F3 11.955 Tf 11.96 0 Td[(Bix)zi,forsomeiandj.AfterthisproblemissolvedweobtainanoptimalobjectivevalueVm,whichisalowerboundoftheoptimalsolutionV,i.e.:VMV. 69

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Tocheckiftheoptimalitysolutionpointx,z1,...,zkforthereducedproblem,weneedtoverifywhethertheabovepointviolatesanyoftheconstraintsthatarenotincludedinthereducedproblems.Thus,wesolvethefollowingKproblem:Qi(x)=maxpipi(di)]TJ /F3 11.955 Tf 11.96 0 Td[(Bix),s.t.DTipifi,andcheckwhetherx,z1,...,zkisoptimalforthefullmasterproblem.ForeachoftheKproblemsthealgorithmwillreturnanoptimaldoubleandprimalextremesolutionspiandyioftheoptimizationproblemQi(x).Ifitappearsthatthenewsolutionviolatestheconstraint(pi)T(di)]TJ /F3 11.955 Tf 11.95 0 Td[(Bix)zi,weaddthisconstrainttothereducedmasterproblem.Afteryiarecomputedforallsecondstageproblemswecanupdateanupperbound:UB minfcTx+KXi=1fTiyig.IfithappensthatnoneoftheconstraintsofKproblemsisviolated,i.e.maxl=1,...Ii(pli)T(di=Bix)zi,8i=1,...K,thesolutionofreducedmasterproblemisthesolutionofthefullmasterproblemandwecanterminatethealgorithm.Also,wecanstopthealgorithmwhenthedifferencebetweentheupperandlowerboundsissmallerthanpredenedtolerancelevel,i.e.UB)]TJ /F3 11.955 Tf 11.95 0 Td[(Vk<.Theformulationpresentedabove( 4 )-( 4 )mightnotttheRAMofthecomputerduetoalargenumberofpossiblescenarios,andtheproblemseemstobeaperfectcandidatefortheBendersdecompositionalgorithm.However,theBenders'decompositiontechniquecannotbedirectlyappliedto( 4 )-( 4 )formulation, 70

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becauseoneoftheCVaRrelatedconstraints,namely( 4 ),violatestheblock-ladderstructureoftheconstraintmatrix.Inordertoapplythedecompositionalgorithm,weneedtomodifythestructureoftheproblem( 4 )-( 4 )bymovingriskconstraintoobjectiveusingusingthemathematicaltechniqueoftheLagrangianrelaxation.TheLagrangianrelaxationconsistsinremovingoneconstraintandinsertingitinsidetheobjectivefunctionwithsomepenaltyforitsviolation.Inthiscase,theconstraintismultipliedbyapenaltycoefcientwhichrepresentstheLagrangianmultiplier(dualvalueorsimplexmultiplier)oftheconstraint.Ifthisconstraintwereleftinitsinitialformulation,intheoptimalsolution,theLagrangianmultiplierwouldgetaspecicvalueinherenttothesolution.ThissolutionfoundforsomepenaltycoefcientcorrespondstothelevelofCVaRforthespecicquantilethatweintendtoachieve.Therefore,themagnitudeofthepenaltycoefcientallowsustoregulateourriskconstraintseverity.Thegreateritis,themoreincentivewehavetorelaxourCVaRleveland/orpercentileandexpectsmallercosts.Inthecaseofrelaxation,weproceedbyexternaliterationsonthemultipliervaluesinordertondthepenaltycoefcientthatcorrespondstothedesiredsolution.Thedesiredmultiplierlevelcanbedeterminedusingnumericalapproximationtechniques,forexampleabisectionmethod.Thus,theinitialproblemMinimizeX(i,j)2Eci,jxi,j,s.t.constratints( 4{2 )-( 4{3 ),( 4{12 )-( 4{14 );CVaRL(x,y)C,canbetransformedinto 71

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MinimizeX(i,j)2Eci,jxi,j+(CVaRL(x,y))]TJ /F3 11.955 Tf 11.96 0 Td[(C),s.t.constratints( 4{2 )-( 4{3 ),( 4{12 )-( 4{14 ),wheretheparameteristhementionedabovemultiplier.Notethattheconstantterm)]TJ /F5 11.955 Tf 9.3 0 Td[(Ccanbeneglectedandthereforeweomititlaterinthetext.AfterperformingthestandardtransformationtheLPproblemcanbewrittenasfollowsMinimizeX(i,j)2Eci,jxi,j+ +1 S(1)]TJ /F5 11.955 Tf 11.96 0 Td[()SXs=1ts!, (4)s.t.Xfj:(i,j)2Egxi,j)]TJ /F14 11.955 Tf 23.67 11.36 Td[(Xfj:(j,i)2Egxj,i=di,8i2V, (4)0xi,jui,j,8i,j2V. (4)0xsi,jxi,j(1)]TJ /F3 11.955 Tf 11.96 0 Td[(ysi,j),8i,j,s; (4)rsijdij,8i2V,s=1,...S; (4)Xfj:(i,j)2Egxsi,j)]TJ /F14 11.955 Tf 23.67 11.36 Td[(Xfj:(j,i)2Egxsj,i=sign(di)rsi,8s,i, (4)tsXfi:di<0gli()]TJ /F3 11.955 Tf 9.3 0 Td[(di)]TJ /F3 11.955 Tf 11.95 0 Td[(rsi))]TJ /F5 11.955 Tf 11.95 0 Td[(,ts0,8s=1,2,...,S; (4),ts,xij,xsi,j,rsi2IR,8(i,j)2E,8i2V,8s=1,2,...,S. (4) Proposition4.1. Let( 4 )-( 4 )hasoptimalsolution.Then,thereexistssuchvalue=thatoptimalsolutionxofproblem( 4 )-( 4 )isalsooptimalfor( 4 )-( 4 )Weprovideaformaldescriptionofthealgorithmappliedtothenetworkowformulation 72

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Input ,,di,cij,ysij Initialize SetlowerboundLB=0,Setupperboundtothesumofmaximalpossiblecostandloss:UB=X(i,j)2Eci,jui,j+Xi:di<0li()]TJ /F3 11.955 Tf 9.3 0 Td[(di).Writeinitialreducedmasterproblem: RMP:V0=Minimizex,,z+Xfj:(i,j)2Egci,jxi,j+ S(1)]TJ /F5 11.955 Tf 11.96 0 Td[()SXs=1zs, (4) s.t.Xfj:(i,j)2Egxi,j)]TJ /F14 11.955 Tf 23.66 11.36 Td[(Xfj:(j,i)2Egxj,i=di,8i2V, (4) 0xi,jui,j,8i,j2V. (4) zs0 (4) xi,j,,zs2IR (4) Setk 0 Updatelowerbound Solvereducedmasterproblem.LetVkbetheoptimalvalueandx,zetaoptimalpointatk-thiteration.ThenLB max(LB,Vk). 73

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Updateupperbound For8s=1,...,Ssolvesecondstageproblemsforx=xand=: zs(x,)=Minimizexs,rs,tsts (4) s.t.xsij(1)]TJ /F3 11.955 Tf 11.95 0 Td[(ysij)xij, (4) Xfj:(i,j)2Egxsi,j)]TJ /F14 11.955 Tf 23.66 11.36 Td[(Xfj:(j,i)2Egxsj,i)]TJ /F4 11.955 Tf 11.96 0 Td[(sign(di)ri=0,8i2V, (4) rijdij,8i2V; (4) ts+Xi:di<0liri)]TJ /F14 11.955 Tf 26.8 11.36 Td[(Xi:di<0lidi)]TJ /F5 11.955 Tf 11.95 0 Td[(, (4) giventhatx=x,=Ifthesolutionzs(x,)>zs,addaconstraintTsRHS(x,)zstothereducedmasterproblem( 4 )-( 4 ).Here,sisthedualsolutionandRHS(x,)istherighthandsidevectorofconstraints( 4 )-( 4 )thatisalinearfunctionofrststagevariablesxand.Iftheoptimalsolutionofallzs(x,)isnolargerthanzs,thereducedmasterproblemsolvesthefullmasterproblemandweterminate.Otherwise,weupdatetheupperboundasfollowsUB min8<:UB,+Xfj:(i,j)2Egci,jxi,j+ S(1)]TJ /F5 11.955 Tf 11.95 0 Td[()SXs=1zs(x,)9=;, CheckIfUB)]TJ /F3 11.955 Tf 11.95 0 Td[(LB 74

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thenterminateotherwiseincrementk:k k+1andrepeatUpdatelowerboundandUpdateupperboundsteps. Finally Theoptimalnetworkowisrepresentedbyx.OptimalcostvalueisX(i,j)2Ecijxi,j,andthe-CVaRvalueatoptimalityis +1 S(1)]TJ /F5 11.955 Tf 11.96 0 Td[()SXs=1zs!.Thevalueofcorresponding-VaRisstoredin. 4.3ComputationalExperimentsThissectionexperimentswiththeoldandnewlossfunctionsforrobustnetworkowproblems.Firstofall,itisinterestingtoseehowthethenetworkbehaveswhenwetightentheCVaRconstraintontheloss.Inordertotestthenewlossfunctionweconsideredtheexamplefrom[ 16 ].Forthe6nodesnetworkshowninFigure 4-2 ,A,Ihavegenerated100randomscenarios.Deterministicsetupwasconsideredandthen70%CVaRwasplacedintotheobjectivefunctionwiththepenaltycoefcient=10.Wecannoticethattheowpictureslightlychangesastheriskisconsidered.Itisalsointerestingtonoticethatunlikethecaseoftheoldlossfunction,thereisnofractionowinthenetworkwiththeCVaRpenaltyintheobjective.Thealgorithmconvergedinthreeiterations: Iteration1 Upperbound=440,LowerBound=526.667. Iteration2 Upperbound=501.667,LowerBound=526.667. Iteration3 Upperbound=506.667,LowerBound=506.667. 75

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ObjectivefunctioniftheCVaRisincludedis425,CVaRvalueis8.17.Inthedeterministiccasetheobjectiveis295andtheCVaRoflossinthiscaseis23.17. AInitialNetwork B10CVaR0.7(L)inobjectuve CDeterministicsolutionFigure4-2. Newlossfunctionina6verticenetwork.A-initialnetwork,B-minimalcostowwithoutwithCVaRinobjective,CdeterministiccasewithoutCVaR 76

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AInitialNetwork BDeterministicSolution CCVaR10 DCVaR5Figure4-3. FlowscheduleinthenetworkwithandwithoutconsideringCVaRrobustconstraint.A-initialnetwork,B-minimalcostowwithoutwithoutconsideringreliabilityissues,CandD-CVaRoflossisboundedby5and10respectively. 77

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CHAPTER5ROBUSTSYSTEMIDENTIFICATIONFORSPACEWEATHERFORECASTINGItisknownthatspace,biologicalandtechnicalsystemscanbesignicantlyaffectedbytheprocessesoftransferringsolarenergytotheEarthmagnetosphereorionosphere[ 5 37 39 ].Majorfactorsofthisinuencearestudiedinthecontextofspaceweatherproblems.Solarinuenceonnear-earthspaceischaracterizedwithgeomagneticindices.TheseindicesarecomputedbasedonmeasurementsandrepresentonlyasmallpartofthecomplexphenomenonofinteractionbetweentheSunandtheEarth.Theproblemofselectingtheproperindexandrelatingittoacertainphysicalprocessisacomplextaskthatisnecessarilyprecededbyextensiveresearch[ 5 ].Therefore,itisessentialtobeabletopredictsuchcharacteristicsofgeomagneticactivityasKp-index,Ap-index,Dst-index,etc.ThepurposeofthepresentchapteristoprovideanadequatepredictionofDst-indexpredictionindetails.Accordingtothemodernviewsinscience,thisindexrepresentsglobalbehaviorofmagnetosphereplasmundertheinuenceofsolarwind.TheDstisageomagneticindexisconstructedbyaveragingthehorizontalcomponentofthegeomagneticeldfrommid-latitudeandequatorialmagnetogramsfromallovertheworld.PracticallyDstindexisdeterminedbasedonthemeasurementsfromsatellitesonhourlybasis.NegativeDstvaluesindicateamagneticstormisinprogress,themorenegativeDstisthemoreintensethemagneticstorm.ItisextremelyimportanttoselectanadequatemathematicalmodelforDst-indexprediction.Thereareseveralmodelsdescribedintheliterature,suchasneuralnetwork,regression,principalcomponentbasedmodels,andgroupmethodofdatahandlingmodels.Anon-linearblack-boxmodelischosenforthisproblem.Theprimaryfeatureofnonlinearblackboxisusingtimeseries,composedoftwosolarwindparameters,asaninputandDst-indexasanoutput[ 5 ].Obviously,thecomplexityofthisproblemwillincreasealongwiththenumberofmodelvariables,whenthemulti-dimensiontimeseriesisusedforcharacterizingsolarwind. 78

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5.1ModelingMagnetosphereasaBlackBoxPresently,identicationandpredictionmodelsarewidelyusedforstudyinglinearandnon-linearprocessesinthespace[ 7 12 14 31 34 37 ].ItispossibletopredicttheEarthmagnetospherebybuildingblackboxmodelwhichrelatedsolarwindparameters(suchasmagneticeldcharacteristics,solarwindspeed,etc)tomeasurablegeophysicsindices(forexampleKporDstindices).Tobuildsuchamodelweshouldkeepinmindthatthemagnetospherestaysinweakturbulentstate.Itisknownthatlinearprocessesinplasmaleadtotheenergyexchangebetweenwavesandparticles,whilenonlinearprocessesleadtotheexchangeviathreeorfourwaveinteractions.Typically,thehigherthenthirdorderdegreeofwaveinteractionsareneglectedinplasmatheory[ 5 7 56 ]thatsignicantlysimpliesthemodeling.SolarwindparametersarerecordedbysatellitesonaregularbasiswhilegeomagneticindicesaremeasuredontheEarthsurface.Therefore,thetimeseriescorrespondingtosolarwindcharacteristicsandgeomagneticindicescanbeusedasinputandoutputfornon-linearmodel.Inordertobuildthemodelitself,theknownmethodsofdynamicsystemidenticationcanbeapplied[ 34 37 ].Themostdirectapproachtomagnetospheremodelingconsistsinaccountingfortheentirechaininteractionsinmagnetosphereinthemodel.Suchanapproach,however,doesnotlookimplementablebecauseofinsuperablemathematicalandphysicalcomplexityofthemagnetosphereprocesses.Theimpossibilityofdirectmodelingstimulatedresearcherstolookforalternativeapproaches.Oneofthemisbasedonusingqualitativeconsiderations.Forexample,letusconsiderthereasoningbehindthefunctionalinterdependenceofDst-indexandproductofsolarwindvandsoutherncomponentofinterplanetmagneticeldBz Dst=f(vBz).(5) 79

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Itisknownthattheinducedbysolarwindelectriceld~Einuencesonplasmaparticles[ 5 ].Theelectriceld~EisadominatingfactorofmasstransitioninmagnetosphereundertheunperturbedconditionsanddeterminesthekineticenergyoftheparticlesW=W(~E).Achargedparticle,whichhaskineticenergyW=W?+Wkandchargevalueq,driftswiththeaveragespeedof ~VD=W?+2Wk qB3h~BrBi.(5)ThetotaldriftoftheparticlesinducescircleelectriccurrentneartheEarth.Thus,theeld~Edeterminedtheamountofelectriccurrent.Theelectriceldthateffectivelypenetratesthemagnetosphereisdeterminedbythesoutherncomponentofelectriceld j~Ej=1 cvBz(5)andthecirclecurrent~IdenesEarthmagneticelddepression(i.e.Dstindex).Apparently,theremustexistnonlinearinterdependenceofDstindexandthementionedelectriceld.Basingonthedetectedfunctionaldependency,wecanreconstructDstindexfromthemeasuredvaluesofsolarwindandsoutherncomponentofmagneticeld.ThisapproachwasutilizedintheseriesofworksonidenticationofdiscreteDstindexpredictionmodels[ 7 12 ]andexplanationonthephysicalprocessesthatdrivetheindexdynamic.OneofthesimplestwaysofsuchidenticationistoconstructanonlineartransferfunctionusingFourierrepresentationforinputandoutputdata.Asimilarapproachisusedinthischapter.Thenotionofblack-boxiswellknownincomputerscience,electricalengineeringandotherengineeringelds.Itisbasedontheassumptionthatinnercomponentsofasystemarenotimportantfordescribingthesystem.Itissufcienttoknowtherelationbetweeninputandoutputtocharacterizethesystem.Theinternalstateofsuchasystemissetbytheexternalfactoru(t),whichisusuallyamultidimensiontimeseries. 80

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Thisfactoriscalledtheinputofdynamicsystem.Theinnerstateoftheblackboxattimetcouldbesetbyvector (t)ifweknewit.Sincewedonotactuallyknowthisvector,wecanusesomeexperimentalmeasurementsy(t)instead.Weassumethaty(t)issomehowfunctionallydependenton (t).Letuscally(t)anoutputofthenonlinearblackbox.Then,theproblemofsystemidenticationconsistsinsearchinganalyticaldescriptionofthemodel(i.e. (t))basedontheinputandoutput.Ifthesystemislinear,theoutputy(t)canberepresentedastheconvolutionoftheu(t)andimpulsefrequencycharacteristicsh() y(t)=Z10h()y(t)]TJ /F5 11.955 Tf 11.96 0 Td[()d.(5)Ifweswitchfromtimevariablettofrequencyf,Fouriertransformationh()functioniscalledlinearamplitude-frequencycharacteristicsHlin(f).Inthiscaseitbindsspectralcomponentsofinputandoutputwiththeequation yf=Hlin(f)u(f).(5)TheabsolutevalueofcomplexvalueHlin(f)characterizedthedegreeofamplifyingthesystem'sinputspectrumcomponent.Thephasecorrespondstothedelaybetweentheinputandoutputatfrequencyf.Thef()andHlin(f)functionsareequallyvaluablefordescribingtheblackbox.Thus,theoutputofthelinearsystemsiscompletelydenedbytheinputandpulse-frequencycharacteristics.Theintegralequation( 5 )canbeeasilygeneralizedfornonlinearcaseusingVolterraseries[ 48 ].Inthiscase,theinputsignalisdenedbyequation y(t)=Z10h1()y(t)]TJ /F5 11.955 Tf 11.95 0 Td[()d+Z10Z10h2(1,2)u(t)]TJ /F5 11.955 Tf 11.95 0 Td[(1)u(t)]TJ /F5 11.955 Tf 11.96 0 Td[(2)d1d2++Z10...Z10hi(1,...,i)u(t)]TJ /F5 11.955 Tf 11.96 0 Td[(1)...u(t)]TJ /F5 11.955 Tf 11.95 0 Td[(i)d1...di+...,(5)wherehi(1,...,i)isthei-thVolterrakernel.Ifallthekernelsareknown,itispossibletocomputetheoutputandtostudythepropertiesofthesystem. 81

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Inacaseofdiscreteinput,theintegralsin( 5 )arereplacedwithsums yk=Xn1h1(n1)uk)]TJ /F6 7.97 Tf 6.59 0 Td[(n1+Xn1,n2h1(n1)h2(n1,n2)uk)]TJ /F6 7.97 Tf 6.59 0 Td[(n1uk)]TJ /F6 7.97 Tf 6.58 0 Td[(n2+...+Xn1,...,nih1(n1)...hi(n1,...,ni)uk)]TJ /F6 7.97 Tf 6.59 0 Td[(n1...uk)]TJ /F6 7.97 Tf 6.59 0 Td[(ni,(5)wherekrepresentsdiscretizedtime.Fouriertransformationappliedto( 5 )leadstothefollowingequationfortheoutput: yf=H1(f)U(f)+Xf1,f2:f1+f2=fH2(f1,f2)uf1uf2+....(5)Hi(f1,,fi)isageneralizedfrequencytransferfunctionwhichaccountsnonlinearpropertiesofthesystem.Aswepreviouslymentioned,suchrepresentationcanbelimitedwiththreewaveinteractionforweakturbulentspaceplasma.Therefore,thenonlinearprocesseshavingwaveorderfourandmorecanbeneglected.Apparently,H1(f)hasthephysicalmeaningequivalenttoHlin(f).FunctionH2(f1,f2)dependsontwofrequenciesandrepresentsnonlinearquadraticinteractionbetweenthecomponentsf1andf2(ficantakenegativevalue.Italsorepresentsenergytransfertothesystemoutputoffrequencyf=f1+f2.Letusnotethat( 5 )canbestrictlyjustiedusingthetheoryofturbulentplasma[ 49 56 ].ThecorrespondencebetweenmathematicalformulationoftheturbulentplasmatheoryandVolterraseriesisimportantfortheanalysisofanexperimentalmeasurement.Inthischapter,wewillusediscreteblack-boxmodelthatallowsrepresentingtheoutputwithVolterraseries: y(k)=F[y(k)]TJ /F4 11.955 Tf 11.95 0 Td[(1),...,y(k)]TJ /F3 11.955 Tf 11.96 0 Td[(ny),...,u(k)]TJ /F4 11.955 Tf 11.95 0 Td[(1),...,u(k)]TJ /F3 11.955 Tf 11.96 0 Td[(nu),(k),...,(k)]TJ /F3 11.955 Tf 11.95 0 Td[(n)],(5)whereF[]isapolynomofvariablesu(k),y(k),and(k).Theleastsquaresmethodistraditionallyusedfordeterminingunknowncoefcientsoflinearblackboxmodelfromexperimentaldata.However,thisapproachisnotapplicabletodiscretepolynomialmodelbecausethenumberofpolynommembers 82

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dramaticallyincreasesalongwithpolynomdegreeandtimeseriessize.Therefore,wewillapplythestatisticalstructureidenticationtechniquesknownasleaps-and-boundsintheliterature[ 40 ].Theideaoftheprocedureistoaddonlysuchnonlinearcomponentsthatsignicantlyimprovespredictionqualityandpreventmodelovertting.Weaddonlythecomponentwhosecombinationsignicantlycontributestothepredictionqualitymeasuredusingcertaincriteria.Letusassumethemagnetospheretobeasystemwithoneinputandoneoutput.Letusnotethatthemodelcanalsobeimprovedbyaddingextratermsandnoiseparameters.Wedidnotstudythispossibilityatthepresentstageofthework.ThecomputationalexperimentsdemonstratedreliablepredictionefciencywhenBzvischosenasaninputparameter. 5.2RobustModelReconstructionFromthephisycalconsiderationsstatedintheprevioussectionitfollowsthatthemagnetospherecanbemodeledasalinearofbilinearsystemwherestatevectorx(t)representsthemagnetosphereindexwhilecontrolvectoru(t)correspondstoasolaractivity.Sincethemagnetosphereisdrivenbysolarparameteraswellasinternalstate,thedynamicsystemcanbemodeledasbilinearsystemwithpositivefeedback.Abilinearsystemissetbythefollowingsetofdifferentialequations: _x(t)=(qXi=1Aiui(t))x(t)+Bx(t)+f,(5)whereAi,Baremnmatrices,fisn-dimensionalvector.u():IR7!IRqisaninputorcontrolfunctionandx():IR7!IRnisastateofthesystem.ByidenticationweunderstandrestorationofparametersAi,Bandfifinputu()andoutputx(t)areknownatsometimemoments.TosolvethisproblemusuallythesystemisdiscretizedusingRunge-Kuttaorothermethods xk+1=(qXi=1Aiuk)xk+Bxk+f.(5) 83

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Thus,wecanconsiderthesystemindiscretetimeperiods,forexampleeveryhour,andmodelthemagnetosphereusingadiscretebilinearsystemThen,basedonknownxkandukwedetermineAi,Bandftominimizethedifferencebetweenpredictedstateonthenextstepxk+1andobservablestatexk+1.Thatiswearriveatregression-typeoptimizationproblem: minAi,B,fD(e1,e2,...,eN), (5) s.t.ek=xk)]TJ /F4 11.955 Tf 12.14 0 Td[(xk,k=1,...,N, (5) xk+1=(qXi=1Aiuk)xk+Bxk+f,k=0,...,N)]TJ /F4 11.955 Tf 11.95 0 Td[(1. (5) [ 19 ]consideredleastsquaresdeviationmeasureD()=kek22thathasanalyticalsolution.Laterin[Yatsenko2009]Yatsenkoetalclaimedthatifdeviationofstateislimiteditleadstomorestablesolutions.ThenaturaldeviationmeasureofsuchkindisaCVaRbaseddeviationmeasures(i.e.D(e)=CVaR(e))]TJ /F4 11.955 Tf 12.38 0 Td[(CVaR()]TJ /F3 11.955 Tf 9.3 0 Td[(e)).Usingstandardlinearizationtechniqueswecanreducesystemreconstructiontolinearproblemtoobtainstablesolutions. 5.3NonlinearStructureReconstruction Table5-1. Leaps-and-boundbasedvariableselectionforxednumberofregressorsk=1,...,8foronestepaheadforecasting(linearmodel).+incolumnskindicatesthatthecorrespondingvariableisaddedtothemodel.MaximalnumberofregressorsFactor12345678 Dst++++++++V()]TJ /F4 11.955 Tf 9.3 0 Td[(1)Bz()]TJ /F4 11.955 Tf 9.3 0 Td[(1)+++++++V()]TJ /F4 11.955 Tf 9.3 0 Td[(1)++++++V()]TJ /F4 11.955 Tf 9.3 0 Td[(1)By()]TJ /F4 11.955 Tf 9.3 0 Td[(1)++++By()]TJ /F4 11.955 Tf 9.3 0 Td[(1)Bz()]TJ /F4 11.955 Tf 9.3 0 Td[(1)+++V()]TJ /F4 11.955 Tf 9.3 0 Td[(1)Bx()]TJ /F4 11.955 Tf 9.3 0 Td[(1)++Bz()]TJ /F4 11.955 Tf 9.3 0 Td[(1)++Bx()]TJ /F4 11.955 Tf 9.3 0 Td[(1)Bz()]TJ /F4 11.955 Tf 9.3 0 Td[(1)+ Itisimportanttoknowwhichcomponentsandtheircombinationofsolaractivityarethebestpredictorofthesystem.Itisalsoessentialtodeterminehowmanystepsintime 84

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Table5-2. Leaps-and-boundbasedvariableselectionforxednumberofregressorsk=1,...,10foronestepaheadforecasting(bilinear).+incolumnskindicatesthatthecorrespondingvariableisaddedtothemodel.MaximalnumberofregressorsFactor12345678 Dst()]TJ /F4 11.955 Tf 9.3 0 Td[(1)++++++++V()]TJ /F4 11.955 Tf 9.3 0 Td[(1)Bz()]TJ /F4 11.955 Tf 9.3 0 Td[(1)+++++++V()]TJ /F4 11.955 Tf 9.3 0 Td[(1)++++++Dst()]TJ /F4 11.955 Tf 9.3 0 Td[(1)Bz()]TJ /F4 11.955 Tf 9.3 0 Td[(1)++++Dst()]TJ /F4 11.955 Tf 9.3 0 Td[(1)V1()]TJ /F4 11.955 Tf 9.3 0 Td[(1)Bz()]TJ /F4 11.955 Tf 9.3 0 Td[(1)++++Dst()]TJ /F4 11.955 Tf 9.3 0 Td[(1)V1()]TJ /F4 11.955 Tf 9.3 0 Td[(1)+++By()]TJ /F4 11.955 Tf 9.3 0 Td[(1)++V()]TJ /F4 11.955 Tf 9.3 0 Td[(1)By()]TJ /F4 11.955 Tf 9.3 0 Td[(1)+Dst()]TJ /F4 11.955 Tf 9.3 0 Td[(1)By()]TJ /F4 11.955 Tf 9.3 0 Td[(1)Bz()]TJ /F4 11.955 Tf 9.3 0 Td[(1)+ areenoughforreliableprediction.WehavetakenrealdataavailableontheNASAwebsite[ 41 ]collectedfortenyearsperiodfromJanuary1,2000untilDecember31,2010andappliedvariableselectiontechniques.Inordertodeterminethebestpredictorsthevariableselectionstatisticaltechniqueknownasleapsandboundswasappliedtoallpossiblesolverwindcomponentsasmagneticeld(Bx,By,Bz)andsolarwindspeed(V)aswellastheirproducts.Theideabehindleapsandboundstechniqueistoxthenumberofregressorsandconsiderallpossiblesubsetsofthexedsize.Ifthenumberofregressorishighttheenumerationbecomescomputationallychallengingandthereforevariousheuristicsareusedthatdescribedin[ 40 ].TheexperimenthasshownthatthebestpredictorsareDstindex,Bzmagneticeldcomponentandtheirproducts(refertotables 5-1 and 5-2 .Thisfactagreeswiththepreviousresearch.Ithasalsobeenexperimentallyestablishedthataddingmorethantwotimeperiodintothemodelhasalmostnoeffectonpredictionquality.Thus,thereisnonecessitytoconsidermanystepsasithasbeendoneinpreviousworks[ 19 61 ]. 85

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CHAPTER6CONCLUSION 6.1ThesisContributionFindingasolutionwhichisrobusttovariousuncertaintiesisachallengingtaskthathasbeenresearchedextensivelyoverthelastdecade.Motivatedbytheriskmanagingtechniquesinnancialengineering,theauthorutilizedandextendednancialframeworksfornonnancialapplications.ThedissertationdevelopedmathematicalprogrammingmodelsutilizingacoherentquantitativeriskmeasureConditionalValue-at-Risk(CVaR).Thesemodelsallowachievingrobustandefcientperformanceofsensorsscheduling,networkowandsolarwindpredictioninthepresenceofuncertainfactorswhichoftenneedtobeconsideredinpractice.Idevelopedamathematicalframeworkforsolvingaclassofsensorsschedulingproblems.BasedonthemodelingconsideredbyJavuzandJeffcoatin[ 62 63 ],Iintroducedandformulatedthreerobustoptimizationproblems:twofordeterministicandoneforstochasticcase.Theobtained0-1problemsarealsoreformulatedintermsofcardinalityfunctions.NumericalexperimentsareconductedusingtwocommercialsolversILOGCPLEXandAOrDaPSG.CPLEXgivesexactsolutionsforsmallproblems.Bothsolversgiveanapproximatesolutioninreasonabletimeforlargeproblems.Forlargeproblemswithmanystochasticscenarios,PSGisfasterthanCPLEXinndinggoodqualityapproximatesolutions.Also,PSGhasanintuitiveuserfriendlyinterface.ThesefeaturesofPSGmakeitanefcientandconvenienttoolforsolvingrobustsensorsschedulingproblemsinuncertainenvironments.LPbasedmodelfornetworkowproblemssubjecttouncertainarcfailureshasbeendeveloped.Ihaveintroducedalossfunctionthatcharacterizesactuallossofowinthenetworkwhencertainarcsaredestroyed.TherobustnetworkdesignproblemunderconsiderationextendsapproachdevelopedbyBoginskietal[ 16 ].Iproposedanewmorerealisticlossfunctionthatmoreadequatelydescribesactualharmcaused 86

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bynetworkcomponentfailures.Theproposedlossfunctionmakestheoptimizationproblemsmoredifculttosolvecomparedtotheformulationwithpreviouslossfunction.Inparticular,theresultinglinearproblemincludessecondstagevariablesthatmakeastraightforwardwayofsolvingLPimpossibleforalargesizerealworldproblem.Inordertohandlethecomplexity,IhaveproposedthetechniqueofLagrangianRelaxation.TheCVaRconstraintisrelaxedandinsertedintheobjectivefunctionwithpenaltycoefcients.Then,IutilizedaspecialstructureoftheconstraintmatrixandusedBenders'decompositiontechniqueswhichallowsolvinglargescaletwostagestochasticproblemsefciently.Ihavealsosuggestedarobustapproachtosystemidenticationthatextendstheframeworkforsystemidenticationdevelopedat[ 19 ].TheCVaR-baseddeviationmeasureintheobjectivefunctionservesasaregularizationcomponentandprovidesmorestablerestoreddynamicsystemsusedforspaceweatherprediction. 6.2FutureWorkRobustnetworkowmodelsneedtoaddresstheoreticalaspectsofcomplexity.Namely,theoreticalestimationshouldbeestablishedthatprovidesboundsonhowmanysamplingscenariosarerequiredfortheaccurateapproximationofriskmeasures.Suchapproximationwasdevelopedin[ 16 ]whereBoginskietalinvestigatedhowmanyscenarioswererequiredtoensurethatthetruevalueoftheCVaRfunctionF(x,)wascloseenoughtothesample(scenario-based)valuethatisusedintheLPformulationwithahighcondencelevel.Inparticular,apolynomialupperboundonthenumberofscenarioswasfoundforaparticularproblem.Duetocomplexityoftheconsideredformulation,itisnotpossibletomapthepreviouslyusedtechniquesdirectlytothenewframework.Therefore,ndinggoodestimationsisamatteroffutureresearchendeavors.Theapproachconsideredforspaceweatherpredictionprovidesreliablepredictionformagnetosphereindices.Furthertheoreticalconditionofsystemstabilityshouldbeobtainedforlinearandespeciallynonlinearcases.Thestudyofstabilityfornonlinear 87

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systemisaverychallengingtaskinitselfandcouldbeagreattopicoffuturestudies.Furthercomputationalexperimentsarerequiredtocalibratemodelparameters,suchasCVaRpercentilelevel. 88

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REFERENCES [1] 2009.Americanoptimaldecision,portfoliosafeguard.URL http://www.aorda.com [2] 2010.Ilogcplex.URL http://www.ilog.com/products/cplex/ [3] Abbasi,AmeerAhmed,MohamedYounis.2007.Asurveyonclusteringalgorithmsforwirelesssensornetworks.Comput.Commun.30(14-15)2826. [4] Ahuja,R.K.,T.L.Magnanti,J.B.Orlin.1993.NetworkFlows:Theory,AlgorithmsandApplications.PrenticeHall. [5] Akasofu,S.I.,SChapman.1972.Solar-TerrestrialPhysics.ClarendonPress,Oxford,England. [6] Akyildiz,IanF.,TommasoMelodia,KaushikR.Chowdhury.2007.Asurveyonwirelessmultimediasensornetworks.Comput.Netw.51(4)921. [7] Balikhin,M.A.,I.Bates,S.Walker.2001.Identicationoflinearandnonlinearprocessesinspaceplasmaturbulencedata.AdvancesinSpaceResearch28(5)787. [8] Ben-Tal,Aharon,LaurentElGhaoui,ArkadiNemirovski.2009.RobustOptimiza-tion.PrincetonUniversityPress. [9] Benders,J.F.1962.Partitioningproceduresforsolvingmixed-variablesprogrammingproblems.NumerischeMathematik4(1)238. [10] Bertsekas,D.P.1998.NetworkOptimization-ContinuousandDiscreteModels.AthenaScientic,Belmont. [11] Bertsimas,D.,M.Sim.2003.Robustdiscreteoptimizationandnetworkows.MathematicalProgramming98(1)49. [12] Billings,S.A.1989.Identicationoflinearandnonlinearprocessesinspaceplasmaturbulencedata.InternationalJournalofControl50(5)1897. [13] Billings,S.A.,W.S.F.Voon.1983.Structuredetectionandmodelvaliditytestsintheidenticationofnonlinearsystems.ControlTheoryandApplications.IEEE,193. [14] Billings,S.A.,Q.M.Zhu.1994.Nonlinearmodelvalidationusingcorrelationtests.InternationalJournalofControl60(5)1107. [15] Biswas,Pratik,YinyuYe.2004.Semideniteprogrammingforadhocwirelesssensornetworklocalization.IPSN'04:Proceedingsofthe3rdinternationalsymposiumonInformationprocessinginsensornetworks.ACM,NewYork,NY,USA,46. 89

PAGE 90

[16] Boginski,VladimirL.,ClaytonW.Commander,TimofeyTurko.2009.Polynomial-timeidenticationofrobustnetworkowsunderuncertainarcfailures.OptimizationLetters3(3)461. [17] Buczak,AnnaL.,Henry(Hui)Wang,HoushangDarabi,MohsenA.Jafari.2001.Geneticalgorithmconvergencestudyforsensornetworkoptimization.Inf.Sci.Inf.Comput.Sci.133(3-4)267. [18] Campi,M.C.,G.Calaore.2004.Decisionmakinginanuncertainenvironment:thescenario-basedoptimizationapproach.J.Andrysek,M.Karny,J.Kracik,eds.,MultipleParticipantDecisionMaking.AndwancedKnowledeInternational,99. [19] Cheremnykh,O.,V.Yatsenko,O.Semeniv,IU.Shatokhina.2008.Nonlineardynamicsandpredictionforspaceweather.UkrainianJounalofPhysics53(5)504. [20] Chhetri,AmitS.,DarrylMorrell,AntoniaPapandreou-Suppappola.uary.Nonmyopicsensorschedulinganditsefcientimplementationfortargettrackingapplications.EURASIPJ.Appl.SignalProcess.2006(1)9. [21] Commander,C.W.,P.M.Pardalos,V.Ryabchenko,S.Sarykalin,T.Turko,S.Uryasev.2008.Robustwirelessnetworkjammingproblems.C.W.Commander,M.J.Hirsch,R.A.Murphey,P.M.Pardalos,eds.,LectureNotesinControlandInformationSciences.Springer,399. [22] Commander,C.W.,P.M.Pardalos,V.Ryabchenko,S.Uryasev.2007.Thewirelessnetworkjammingproblem.JournalofCombinatorialOptimization14:4481. [23] Corea,G.A.,V.G.Kulkarni.1990.Minimumcostroutingonstochasticnetworks.OperationsResearch38(3)527. [24] Dantzig,G.B.1963.LinearProgrammingandExtensions.PrincetonUniversityPress,NewJersey. [25] Dantzig,G.B.1963.LinearProgrammingandExtensions.PrincetonUniversityPress,NewJersey. [26] Doulliez,PierreJ.,M.R.Rao.1971.MaximalFlowinaMulti-TerminalNetworkwithAnyOneArcSubjecttoFailure.MANAGEMENTSCIENCE18(1)48. [27] Ferentinos,KonstantinosP.,TheodoreA.Tsiligiridis.2007.Adaptivedesignoptimizationofwirelesssensornetworksusinggeneticalgorithms.Comput.Netw.51(4)1031. [28] Glockner,GregoryD.,GeorgeL.Nemhauser,CraigA.Tovey.2001.Dynamicnetworkowwithuncertainarccapacities:Decompositionalgorithmandcomputationalresults.Comput.Optim.Appl.18(3)233. 90

PAGE 91

[29] Hollick,Matthias,IvanMartinovic,TronjeKrop,IvicaRimac.2004.Asurveyondependableroutinginsensornetworks,adhocnetworks,andcellularnetworks.EUROMICRO'04:Proceedingsofthe30thEUROMICROConference.IEEEComputerSociety,Washington,DC,USA,495. [30] Jeong,Jaehoon,SarahSharafkandi,DavidH.C.Du.2006.Energy-awareschedulingwithqualityofsurveillanceguaranteeinwirelesssensornetworks.DIWANS'06:Proceedingsofthe2006workshoponDependabilityissuesinwirelessadhocnetworksandsensornetworks.ACM,NewYork,NY,USA,55. [31] Johansen,T.A.1997.Constrainedandregularizedsystemidentication.In:PreprintsIFACSymposiumonSystemIdentication,Kitakyushu.1467. [32] Klappenecker,Andreas,HyunyoungLee,JenniferL.Welch.2008.Schedulingsensorsbytilinglattices.PODC'08:Proceedingsofthetwenty-seventhACMsymposiumonPrinciplesofdistributedcomputing.ACM,NewYork,NY,USA,437. [33] Koutsonikolas,Dimitrios,SaumitraM.Das,Y.CharlieHu.2007.Pathplanningofmobilelandmarksforlocalizationinwirelesssensornetworks.Comput.Commun.30(13)2577. [34] Kuntsevich,V.2006.Controlunderuncertainty:Assuredresultsincontrolandidenticationproblems.NaukovaDumka,Kiev,Ukraine. [35] Kuorilehto,Mauri,MarkoHannikainen,TimoD.Hamalainen.2005.Asurveyofapplicationdistributioninwirelesssensornetworks.EURASIPJ.Wirel.Commun.Netw.5(5)774. [36] Li,Y.,M.T.Thai,,W.Wu(eds).2007.WirelessSensorNetworksandApplications.Springer. [37] Ljung,L.1999.SystemIdentication-TheoryFortheUser.PTRPrenticeHall,UpperSaddleRiver,N.J. [38] Lobo,M.S.,M.Fazel,S.Boyd.2007.Portfoliooptimizationwithlinearandxedtransactioncosts.AnnalsofOperationsResearch152(1)376. [39] Lundstedt,Gleisner,H.Gleisner,H.Lundstedt,P.Wintoft.1996.Predictinggeomagneticstormsfromsolar-winddatausingtime-delayneuralnetworks. [40] Miller,A.2002.SubsetSelectioninRegression.Chapman&Hall/CRC. [41] NASA.2010.OMNIsatellitedatabase.URL http://nssdc.gsfc.nasa.gov/omniweb [42] Niculescu,Dragos,BadriNath.2003.Adhocpositioningsystem(aps)usingaoa.The28thConferenceonComputerCommunications.IEEE,1734. 91

PAGE 92

[43] Pardalos,P.M.,Y.Ye,C.W.Commander(eds)V.Boginski.toappearin2009.Sensors:Theory,Algorithms,andApplications.Springer. [44] Pemberton,JosephC.,IIIFlaviusGaliber.2001.Aconstraint-basedapproachtosatellitescheduling.DIMACSworkshopononConstraintprogrammingandlargescalediscreteoptimization.AmericanMathematicalSociety,Boston,MA,USA,101. [45] Rockafellar,R.T.,S.Uryasev.2000.Optimizationofconditionalvalue-at-risk.JournalofRisk221. [46] Rockafellar,R.T.,S.P.Uryasev.2002.Conditionalvalue-at-riskforgenerallossdistributions.JournalofBankingandFinance261443. [47] Rudafshani,Masoomeh,SuprakashDatta.2007.Localizationinwirelesssensornetworks.IPSN'07:Proceedingsofthe6thinternationalconferenceonInformationprocessinginsensornetworks.ACM,NewYork,NY,USA,51. [48] S.,Boyd,ChuaL.O.1985.Fadingmemoryandtheproblemofapproximatingnonlinearoperatorswithvolterraseries.IEEETransactionsonCircuitsandSystems32(11)1150. [49] Sagdeev,R.Z.,A.A.Galeev.1969.NonlinearPlasmaTheory.Benjamin,WhitePlains,NewYork. [50] Sarykalin,S,GSerraino,SUryasev.2008.VaRvsCVaRinriskmanagementandoptimization.INFORMSTutorial. [51] Schrijver,A.2002.Onthehistoryofthetransportationandmaximumowproblems.MathematicalProgramming91(3)437. [52] Singh,SumeetpalS.,NikolaosKantas,Ba-NguVo,ArnaudDoucet,RobinJ.Evans.2007.Simulation-basedoptimalsensorschedulingwithapplicationtoobservertrajectoryplanning.Automatica43(5)817. [53] So,AnthonyMan-Cho,YinyuYe.2007.Theoryofsemideniteprogrammingforsensornetworklocalization.Math.Program.109(2)367. [54] Uryasev,S.2000.Conditionalvalue-at-risk:Optimizationalgorithmsandapplications.FinancialEngineeringNews141. [55] Venkatesh,Swaroop,R.MichaelBuehrer.2006.Alinearprogrammingapproachtonloserrormitigationinsensornetworks.IPSN'06:Proceedingsofthe5thinternationalconferenceonInformationprocessinginsensornetworks.ACM,NewYork,NY,USA,301. [56] V.N.,Tsytovich.1972.AnIntroductiontotheTheoryofPlasmaTurbulence.Pergamon,Oxford,NewYork. 92

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[57] Wang,Chen,LiXiao.2008.Sensorlocalizationinconcaveenvironments.ACMTrans.Sen.Netw.4(1)1. [58] Wu,Kui,YongGao,FuluLi,YangXiao.2005.Lightweightdeployment-awareschedulingforwirelesssensornetworks.Mob.Netw.Appl.10(6)837. [59] Wu,Kui,ChongLiu,JianpingPan,DandanHuang.2007.Robustrange-freelocalizationinwirelesssensornetworks.Mob.Netw.Appl.12(5)392. [60] Yan,Ting,YuGu,TianHe,JohnA.Stankovic.2008.Designandoptimizationofdistributedsensingcoverageinwirelesssensornetworks.Trans.onEmbeddedComputingSys.7(3)1. [61] Yatsenko,V.A.,O.K.Cheremnykh,V.M.Kuncevich,SemenivO.V.2009.Geomagneticactivitymodelidenticationandspaceweatherforecasting.ProblemsofControlandInformatics(6)114. [62] Yavuz,M.,D.E.Jeffcoat.2007.Ananalysisandsolutionofthesensorschedulingproblem.AdvancesinCooperativeControlandOptimization,vol.369.Springer,167. [63] Yavuz,M.,D.E.Jeffcoat.2007.Singlesensorschedulingformulti-sitesurveillance.Tech.rep.,AirForceResearchLaboratory. [64] Yick,Jennifer,BiswanathMukherjee,DipakGhosal.2008.Wirelesssensornetworksurvey.Comput.Netw.52(12)2292. [65] Yuan,Yong,ZongkaiYang,MinChen,JianhuaHe.2006.Asurveyoninformationprocessingtechnologiesinwirelesssensornetworks.Int.J.AdHocUbiquitousComput.1(3)103. 93

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BIOGRAPHICALSKETCH MykytaBoykowasborninDnipropetrovsk,Ukraine.Hereceivedhisbachelor'sandmaster'sdegreesinappliedmathematicsfromDnipropetrovskNationalUniversityin2000and2001respectively.MykytaBoykoworkedasasoftwaredeveloperfrom2000until2005forlargeinternationalsoftwarecompanies,wherehehadanopportunitytoworkoncomplexenterprisebusinesssolutions.In2006,MykytaBoykojoinedthegraduateprograminIndustrialandSystemsEngineeringattheUniversityofFlorida.HereceivedhisMasterofSciencedegreeinindustrialandsystemsengineeringfromtheUniversityofFloridainAugust2007.MykytaBoykoistheauthorofseveralscienticpapersandsurveyspublishedinpeer-reviewedjournalsandbooks.HealsolecturesprogrammingclassesattheUniversityofFlorida. 94