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1、MetaheuristicsforMultiobjectiveOptimizationCarlosA.CoelloCoelloCINVESTAV-IPNDepto.deIngenier´ıaEl´ectricaSecci´ondeComputaci´onAv.InstitutoPolit´ecnicoNacionalNo.2508Col.SanPedroZacatencoM´exico,D.F.07300,MEXICOccoello@cs.cinvestav.mx1MotivationMostproblemsinnaturehaveseveral(possiblyconfl
2、icting)objectivestobesatisfied.Manyoftheseproblemsarefrequentlytreatedassingle-objectiveoptimizationproblemsbytransformingallbutoneobjectiveintoconstraints.2Whatisamultiobjectiveoptimizationproblem?TheMultiobjectiveOptimizationProblem(MOP)(alsocalledmulticriteriaoptimization,multiperforman
3、ceorvectoroptimizationproblem)canbedefined(inwords)astheproblemoffinding(Osyczka,1985):avectorofdecisionvariableswhichsatisfiesconstraintsandoptimizesavectorfunctionwhoseelementsrepresenttheobjectivefunctions.Thesefunctionsformamathematicaldescriptionofperformancecriteriawhichareusuallyincon
4、flictwitheachother.Hence,theterm“optimize”meansfindingsuchasolutionwhichwouldgivethevaluesofalltheobjectivefunctionsacceptabletothedecisionmaker.3AFormalDefinitionThegeneralMultiobjectiveOptimizationProblem(MOP)canbeformallydefinedas:Findthevector~x∗=[x∗,x∗,...,x∗]Twhichwillsatisfythem12nineq
5、ualityconstraints:gi(~x)≥0i=1,2,...,m(1)thepequalityconstraintshi(~x)=0i=1,2,...,p(2)andwilloptimizethevectorfunctionf~(~x)=[f(~x),f(~x),...,f(~x)]T(3)12k4Whatisthenotionofoptimuminmultiobjectiveoptimization?Havingseveralobjectivefunctions,thenotionof“optimum”changes,becauseinMOPs,wearere
6、allytryingtofindgoodcompromises(or“trade-offs”)ratherthanasinglesolutionasinglobaloptimization.Thenotionof“optimum”thatismostcommonlyadoptedisthatoriginallyproposedbyFrancisYsidroEdgeworthin1881.5Whatisthenotionofoptimuminmultiobjectiveoptimization?ThisnotionwaslatergeneralizedbyVilfredoPar
7、eto(in1896).AlthoughsomeauthorscallEdgeworth-Paretooptimumtothisnotion,wewillusethemostcommonlyacceptedterm:Paretooptimum.6DefinitionofParetoOptimalityWesaythatavectorofdecisionvariables~x∗∈FisParetooptimaliftheredoesnotexistanother~x∈Fsuchthatf(~x)≤f(~x∗)foriialli=1
