约束多目标优化问题中约束处理方法综述

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2012.12*(,410205):多目标优化;约束处理;修正算法*基金项目:湖南省科技计划项目(No.S2010G2013)收稿日期:2012-11-12修稿日期:2012-12-17作者简介:王杰文(1962-),男,教授,研究方向为计算机应用技术、信息技术教育应用约束条件的处理是求解约束多目标优化问题的关键,常用方法包括拒绝不可行解法、罚函数法,以及各种修正算法。近年来,将约束转化为一个优化目标的方法受到关注,就此进行综述。::1007-1423(2012)36-0012-04DOI:10.3969/j.issn.1007-1423.2012.36.0030,。、。,,,;,;,,,,。,,。(MOP):,:miny=F(x)=(f1(x),f2(x),…,fk(x))s.t.gi(x)≤0,i=1,2,…,m:x=(x1,x2,…,xn)∈Xy=(y1,y2,…,yn)∈YX={(x1,x2,…,xn)|li≤xi≤ui,i=1,2,…,n}L=(l1,l2,…,ln)U=(u1,u2,…,un)n,km。X,LU,Y。。1,“”,H“”。A,gi(A)≤0,Hi=0;gi(A)0,Hi=gi(A)A:H(x)=ΣHi(i=1,2,…,m),,H。[2]A:HApk=1Σ{max(0,gi(A))}2,[1],。[3]A(t):趤趯2012.12H(A(t))=1ppk=1Σck(A(t))c*(A(t)),:A(t)∈[L,U],i=1,2,…,NtP(t),p。ck(A(t))=max(gk(A(t)),0),k=1,2,…,p,c*(A(t))=maxck1≤k≤p(A(t))。A(t)D,A(t)D,,。A(t),。[4]A():H(A)=Jj=1Σ[min{0,gj(A)}]2+mk=J+1Σ[hk(A)]2:gj(A)、hk(A)(),Jm。A,A,,;A,。2,(multi-ObjectiveOptimizationProblems,MOPs),。,Deb(NSGAII)[5]:AB(A刍B),:(1)AB;(2)AB,AB;(3)ABAB。Deb,Pareto。,,Coello[6],Pareto。[7]、Pareto。G(x)=H(x)。G(x)f(x)f(x):f(x)=(F(x),G(x)),n、klm-ln、k+1,:minimizey=f(x)=(F(x),G(x))wherex=(x1,x2,…,xn)∈XX={(x1,x2,…,xn)}l=(l1,l2,…,ln),u=(u1,u2,…,un),,。,,。,[8]:(1)Pareto;(2)Pareto;(3),,。[9]。,:(1)、;(2);(3)Pareto,Pareto。Wang[10]Pareto,趤趰2012.12(AdaptiveTradeoffModel)。fp():fp=0,0fp1fp=1,。[11]Agent,,。,,。3,,,,。:ui(x)=max{gi(x),0},H(x)=mi=1Σui(x)A、B,gi(A)=1,i=1,2,…,m;mi(B)=m+1,gi(B)=0,i=1,2,…,m-1H(A)=m,H(B)=m+1,[1],A刍B,ui(A)=max(0,gi(A)),ui(B)=max(0,gi(B)),i=1,2,…,mAB,Pareto,AB。[2]:H(A)=m,H(B)=(m+1)2,A刍B。[3],,H(A)=1,H(B)=1/m,B刍A,。,,,:(1)ui(A)=max(0,gi(A)),ui(B)=max(0,gi(B)),i=1,2,…,mParetoA、B;(2)gi(A),gi(B),i=1,2,…,mParetoA、B。,,,,。,,Pareto,,Pareto,。,。4,,,,,,。[1],,,,..,2007,29(11):2688~2692[2],,..,2006,(23):47~51[3],..,2007,29(2):277~280[4],,,..(),2005,45(1):103~106[5]KDeb,APratap,TMeyarivan.ConstrainedTestProblemsforMulti-ObjectiveEvolutionaryOptimization.In:EZitzler,KDeb,LotharThiele,etaleds.Procofthe1stInt'lConfonEvolutionaryMulti-CriterionOptimizationBerlin:Springer-Verlag,2001,284~298[6]CoelloCAC.TheoreticalandNumericalConstraint-Han-dlingTechniquesusewithEvolutionaryAlgorithms:aSurveyoftheStateoftheArt[J].ComputerMethodsinAppliedMec-hanicsandEngineering,2002,191(11-12):1245~1287[7],,..,2008,19(11):2943~2956[8],,,.[J].,2009,20(1):11~29[9]CAIZi-xing,WANGYong.AMultiobjeetiveOptimization-BasedEvolutionaryAlgorithmforConstrainedOptimization[J].IEEETransactionsonEvolutionaryComputation,2006,10(6):658~675[10]WANGYong,CAIZi-xing,ZHOUYu-ren,ZENGWei.AnAdaptiveTradeoffModelforConstrainedEvolutionaryOpti-mization[J].IEEETransactionsonEvolutionaryComputation,2008,12(1):80~92[11],.Agent.(),2011,41(S1):173~177(下转第19页)趤趲2012.12ResearchonInformationSystemPracticalEvaluationMethodBasedonBPNetworkZHANGChao1,2,WEILi-hao1,2,WANGTian1,2(1.GuangdongPowerScienceResearchInstitute,Guangzhou510600;2.GuangdongPowerGridCorporationInformatizationEvaluationLaboratory,Guangzhou510600)Keywords:BPNeuralNetwork;InformationSystem;PragmaticEvaluationAccordingtotheexistinginsufficiencyofthecurrentpracticalevaluationmethodofinformationsystem,proposesthepracticalevaluationmethodbasedontheBPneuralnetwork.Analysestheimpactofinformationsystemonthebasisofpracticalevaluationfactors,introducesgeneticalgo-rithmofneuralnetworkintothepracticalevaluationofinformationsystem,anditleadstothesuccessfulconstructionofpracticalevaluationmodeltotheinformationsystembasedonBPneuralnetwork.Empiricalstudyshowsthatthemodelcansupportthetraditionalinformationlevelevaluationprocesscomplexity,evaluationresultisobjectiveandreasonable.Abstract:4,,BP。,,。[1],,,,..,2010(04)[2],.APF.,2011(9)[3],,.[J].,2002(03)[4],,,.BP.,2011(12)SurveyofConstraint-HandlingMethodonMulti-ObjectiveOptimizationProblemsWANGJie-wen(DepartmentofInformationScienceandTechnology,HunanFirstNormalUniversity,Changsha410205)Keywords:Multi-ObjectiveOptimization;Constraint-Handling;AdjustingAlgorithmItisthekeydealingwithconstraintstosolvemulti-ObjectiveOptimizationproblem.Regularly,constraint-handlingmethodincludesdeletingtheinfeasiblesolutions,usingapenaltyfunctions,adjustingalgorithm,andetc.Recently,aconstraint-handlingtechniquebasedonmulti-Objec-tiveOptimizationisbeingconcerned.Abstract:!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(上接第14页)趤趽

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