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含参向量优化问题真有效解映射的下半连续性.docx

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含参向量优化问题真有效解映射的下半连续性 Introduction Vectoroptimizationproblemswithconstraintsonthesolutionsethavebecomeawidelystudiedareaofresearchinrecentyears.Theseproblemsaimtomaximizeorminimizeavector-valuedfunctionsubjecttoconstraints,andoftenariseinpracticaldecision-makingproblemsinfieldssuchasengineering,finance,andlogistics.Oneimportantaspectofsolvingtheseproblemsistoobtainthesetofefficientsolutions,whicharepointsintheobjectivespacethatcannotbeimprovedinanycomponentwithoutworseninganother.Inthispaper,wefocusonakeypropertyoftheefficientset,namely,itslowersemicontinuity(LSC). Definitions LetRbeaEuclideanspace.Avector-valuedfunctionF:R->R^missaidtobecontinuousatapointx0inRifforeveryε>0,thereexistsδ>0suchthat||F(x)-F(x0)||<εforallxinRwith||x-x0||<δ.ThefunctionFissaidtobelowersemicontinuousatx0ifforeverysequence{xn}inRconvergingtox0,liminfF(xn)>=F(x0),whereliminfdenotesthelimitinferior. ConsideravectoroptimizationprobleminwhichweseektominimizeF(x)subjecttoconstraintsoftheformh(x)<=0,whereh:R->R^kisavector-valuedfunctionrepresentingtheconstraints.Apointx0issaidtobeefficientifthereexistsnoxinthefeasiblesetsuchthatF(x)isstrictlysmallerthanF(x0)andh(x)<=h(x0).Thesetofallefficientpointsiscalledtheefficientset,denotedbyE. MainResult ThemainresultofthispaperisthattheefficientsetisLSC.Thatis,if{xn}isasequenceinEconvergingtosomex0,thenx0isalsoinE.Toprovethis,weusethefollowingcharacterizationofefficientpoints:apointx0isefficientifandonlyifthereexistnonnegativevectorsλandμsuchthatF(x0)=λF(x)+μh(x)forallxinthefeasiblesetandλandμsatisfythecomplementarityconditionsλ_i>0ifF_i(x)<F_i(x0)andμ_j=0ifh_j(x0)>0. Suppose{xn}isasequenceinEconvergingtox0.LetλnandμnbethenonnegativevectorssuchthatF(xn)=λnF(x)+μnh(x)forallxinthefeasibleset.SinceFandharecontinuous,wehaveλnF(x0)+μnh(x0)->F(x0)asn->∞.Nowsuppose,forcontradiction,thatx0isnotefficient.ThenthereexistsxinthefeasiblesetsuchthatF(x)<F(x0)andh(x)<=h(x0),andsoλnF(x)+μnh(x)<λnF(x0)+μnh(x0)forallsufficientlylargen.Thiscontradictsthecomplementarityconditi