ConicLinearOptimizationandApplMSE314LectureNote012 MathematicalProgrammingMP TheclassofmathematicalprogrammingproblemsconsideredinthiscoursecanallbeexpressedintheformPminimizesubjecttousuallysp ID: 330245
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ConicLinearOptimizationandAppl.MS&E314LectureNote#011ConicLinearProgrammingandApplicationsYinyuYeDepartmentofManagementScienceandEngineeringStanfordUniversityStanford,CA94305,U.S.A. ConicLinearOptimizationandAppl.MS&E314LectureNote#012 MathematicalProgramming(MP) Theclassofmathematicalprogrammingproblemsconsideredinthiscoursecanallbeexpressedintheform(P)minimizesubjecttousuallyspeciÞedbyconstraints:)=0 ConicLinearOptimizationandAppl.MS&E314LectureNote#013 ImportantTerms decisionvariable/activity,data/parameterobjective/goal/target,coefÞcientvector/coefÞcientmatrixconstraint/limitation/requirement,satisÞed/violatedequality/inequalityconstraint,directionofinequalityconstraintfunction/theright-handsidenonnegativityconstraint,integralityconstraintfeasible/infeasiblesolutionglobalandlocaloptimizers,globalandlocalminimumvalues ConicLinearOptimizationandAppl.MS&E314LectureNote#014 ASpecialClassofMPExamples minimizesubjecttominimizesubjectto =(2;1;1)=(1;1;1) ConicLinearOptimizationandAppl.MS&E314LectureNote#015minimizesubjectto ConicLinearOptimizationandAppl.MS&E314LectureNote#016 ConicLinearProgramming/Optimization CLPminimizesubjectto,...,m,isapointedandclosedconvexcone.LinearProgramming(LP)Second-OrderConeProgramming(SOCP)SemideÞniteProgramming(SDP)MixedConicLinearProgrammingcanbeaproductofclosedconvexcones,andaretwodifferentcones. ConicLinearOptimizationandAppl.MS&E314LectureNote#017 ConicLinearProgrammingCharacteristics linearobjectivefunction/constraintsvariablesinapointedandclosedconvexconeconicinterior,boundary,extremepoint(corner)everylocaloptimizerisglobal,andtheyformaconvex(optimizer)setmostofthempossessefÞcientalgorithmsinbothpracticeandtheory ConicLinearOptimizationandAppl.MS&E314LectureNote#018 ConicLinearOptimizationModelandFormulation SortoutdataandparametersfromtheverbaldescriptionDeÞnethesetofdecisionvariablesFormulatethelinearobjectivefunctionofdataanddecisionvariablesSetuplinearequalityandconicconstraints ConicLinearOptimizationandAppl.MS&E314LectureNote#019 Example1:ParimutuelCallAuctionMechanismI Giventhataremutuallyexclusiveandexactlyoneofthemwillberealizedatthematurity.isabetononeoraofstates,withapricelimitmaximumpricetheparticipantiswillingtopayforoneunitoftheorder)andaquantitylimit(themaximumnumberofunitsorsharestheparticipantiswillingtocontractonanorderisapaperagreementsothatonmaturityitisworthanotional$dollariftheorderincludesthewinningstateandworth$otherwise.Therearesubmittednow. ConicLinearOptimizationandAppl.MS&E314LectureNote#0110 ParimutuelCallAuctionMechanismII:orderdata thorderisgivenasisthebettingindicationrowvectorwhereeachcomponentiseither,...,aiswinningstateandisnon-winningstate;isthepricelimitforoneunitofsuchacontract,andisthemaximumnumberofcontractunitsthebetterliketobuy. ConicLinearOptimizationandAppl.MS&E314LectureNote#0111 ParimutuelCallAuctionMechanismIII:orderÞlls bethenumberofunitsorsharesawardedtothethorder.Then,thebidderwillpaytheamountandthetotalamountcollectedwouldbeIfthethstateisthewinningstate,thentheauctionorganizerneedtopaythewinningbidderswherecolumnvectorThequestionis,howtodecide,thatis,howtoÞlltheorders ConicLinearOptimizationandAppl.MS&E314LectureNote#0112 ParimutuelCallAuctionMechanismIV:worst-caseproÞtmaximization s.t.s.t.ThisisNOTalinearprogram. ConicLinearOptimizationandAppl.MS&E314LectureNote#0113 ParimutuelCallAuctionMechanismV:linearprogramming However,theproblemcanberewrittenass.t.isthevectorofallones.Thisisalinearprograms.t.,y,s ConicLinearOptimizationandAppl.MS&E314LectureNote#0114 Example2:PortfolioManagement denotetheexpectedreturnvectordenotetheco-variancematrixaninvestmentportfolio,andletbetheinvestmentproportionvector.Then,onemanagementmodelis:minimizesubjecttoisthevectorofallones.Thisisaquadraticprogram ConicLinearOptimizationandAppl.MS&E314LectureNote#0115.Thentheproblemcanbewrittenasminimizesubjecttowhichisamixedlinearandsecond-orderconeprogram ConicLinearOptimizationandAppl.MS&E314LectureNote#0116 RobustPortfolioManagement Inrealapplications,maybeestimatedundervariousscenarios,sayfor,...,mminimizesubjecttominimizesubjecttoThiscanbefurtherreducedtoamixedlinearandsecond-orderconeprogram ConicLinearOptimizationandAppl.MS&E314LectureNote#0117 Example3:GraphRealization GivenagraphV,EandsetsofnonÐnegativeweights,sayi,j,thegoalistocomputeaintheforagivenlowdimension,i.e.toplacetheverticesofsuchthatEuclideandistancebetweeneverypairofadjacentverticesi,j(orbounded)bytheprescribedweight ConicLinearOptimizationandAppl.MS&E314LectureNote#0118 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0. 5 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 Figure1:50-node2-DGraphRealization ConicLinearOptimizationandAppl.MS&E314LectureNote#0119 Figure2:A3-DTensegrityGraphRealization;pictureprovidedbyAnstreicher ConicLinearOptimizationandAppl.MS&E314LectureNote#0120 Figure3:TensegrityGraph:ANeedleTower;pictureprovidedbyAnstreicher ConicLinearOptimizationandAppl.MS&E314LectureNote#0121 Figure4:MolecularConformation:1F39(1534atoms)with85%ofdistancesbelowAand10%noiseonupperandlowerbounds ConicLinearOptimizationandAppl.MS&E314LectureNote#0122 GraphRealizationII:DistanceGeometryModel Systemofnonlinearequationsfori,j,ij,k,jarepossiblepointswhoselocationsareknown. ConicLinearOptimizationandAppl.MS&E314LectureNote#0123 GraphRealizationIII:NonlinearOptimization Nonlinearleast-squaresk,jk,jEitheroneisanon-convexoptimizationproblem. ConicLinearOptimizationandAppl.MS&E314LectureNote#0124 GraphRealizationIV:MatrixRepresentation x1x2...xn]bethematrixthatneedstobedetermined.ThenThenIX]T[IX](ak;ej),whereeiisthevectorwithatthethpositionandzeroeverywhereelse.i,j,ij,k,jdenotestheGrammatrix ConicLinearOptimizationandAppl.MS&E314LectureNote#0125 GraphRealizationV:SDPRelaxation matrixinequalityisequivalenttoThisisthesemideÞnitematrixcone ConicLinearOptimizationandAppl.MS&E314LectureNote#0126 GraphRealizationVI:SDPstandardform Findasymmetricmatrixsuchthati,j,ij,k,jThisissemideÞniteprogrammingwithanullobjective.Iftherankofisconstrainedtobe,thenitisequivalenttotheoriginalgraphrelaxationproblem. ConicLinearOptimizationandAppl.MS&E314LectureNote#0127 Example4:TheKissingProblem Givenaunitcentersphere,themaximumnumberofunitspheres,indimensions,cantouchorthecentersphere?GeneralSolutionsdoesnotexist.DelsarteMethoduseslinearprogrammingtoprovideanupperboundonthenumberofspheres.K(1)=2,K(2)=6,K(3)=12,K(8)=240,K(24)=196650.K(4)=24:provedusingDelsarteMethodbyOlegMusinonly3yearsago.Forotherdimensions,lowerboundshavebeenprovidedbyconstructingalatticestructure.TherealsoexistsaboundusingtheRiemannzetabutisnon-constructive ConicLinearOptimizationandAppl.MS&E314LectureNote#0128 TheKissingProblemasaGraphRealization ballsallkissthecenterballatthesametimeDoesthefollowingproblemhasasolution?solution?x1x2...xn]isthematrixthatneedstobedetermined.Thenitcanbeformulatedasarank-constrainedfeasibilityproblem;rank ConicLinearOptimizationandAppl.MS&E314LectureNote#0129TheSDPfeasibilityrelaxationisButtheSDPsolutionmaynotprovideaproperranksothatweconstructanonzeroSDPobjectivefortheSDPrelaxations.t.For12spheres,SDPmethodprovidesthefollowingrealization: ConicLinearOptimizationandAppl.MS&E314LectureNote#0130Figure5:12Spheresin3-D