向量变分不等式求解的一个邻近型方法陈纯荣重庆大学数学与统计学院，重庆401331摘要：本文基于每一迭代步对迭代矩阵的向量化处理，在有限维空间中提出了求解弱向量变分不等式(WVVI)的一个邻近型方法。在适当假设下，证明了若(WVVI)存在强解时，构造的迭代子序列收敛到(WVVI)的一个解。进一步地，如果(WVVI)的解集即为强解集，则整个迭代序列收敛于(WVVI)的一个强解。关键词：向量优化;向量变分不等式;邻近点方法;收敛性中图分类号：O221AProximalMethodforSolvingVectorVariationalInequalitiesCHENChun-RongCollegeofMathematicsandStatistics,ChongqingUniversity,Chongqing401331Abstract:Inthispaper,basedonchoosingateachiterationadifferentvectorizationtotheiteratedmatrix,aproximal-typemethodforsolvingtheweakvectorvariationalinequalityproblem(WVVI)infinite-dimensionalspacesisproposed.Underappropriateassumptions,itwasprovedthatthegeneratedsubsequenceconvergestoasolutionofproblem(WVVI),iftheproblem(WVVI)hasstrongsolutions.Moreover,ifthesolutionsetof(WVVI)coincideswithitsstrongsolutionset,thenthewholesequenceconvergestoastrongsolutionofproblem(WVVI).Keywords:Vectoroptimization;Vectorvariationalinequalities;Proximalpointmethods;Convergence0IntroductionThevectorvariationalinequality(VVI,forshort)wasfirstintroducedbyGiannessi[1].Thisproblemhasreceivedextensivestudiesinthelastthreedecades.Manyimportantresultstovariouskindsofvectorvariationalinequalities(VVIs,forshort),especially,theexistencesofsolutionshavebeenestablished,forexample,see[2,3,4]andthereferencestherein.Nowadays,VVIsappearinmanyimportantproblemsfromtheorytoapplicationssuchasvectoroptimiza-tionproblems,vectornetworkequilibriumproblemsandsoon.AmongmanydesirabletopicsofVVIs,thedesignofalgorithmsforVVIstogetherwithconvergenceanalysisisimportantin基金项目：SpecializedResearchFundfortheDoctoralProgramofHigherEducation(20100191120043)作者简介：CHENChun-Rong（1981-），male，associateprofessor，majorresearchdirection：vectoroptimization.---本文来源于网络，仅供参考，勿照抄，如有侵权请联系删除----2----本文来源于网络，仅供参考，勿照抄，如有侵权请联系删除---boththeoryandmethodology.Buttothebestofourknowledge,therewasnearlynoanyresultforthisaspectintheliterature,andhenceitisnecessarytostudythealgorithmsforVVIs.Ourmaininterestistoinvestigateso-calledproximal-typemethodsforVVIs.Itiswell-knownthattheproximalpointalgorithm(PPA,forshort)isoneofefficientalgorithmsforsolvingvariationalinequalitiesandmaximalmonotoneoperators.ThismethodwasintroducedbyMartinet,andfurtherdevelopedandstudiedbyRockafellar[5],fortheproblemoffindingzerosofamaximalmonotoneoperator.Fromthenon,manypapershavebeendevotedtoinvestigatevariousproximalpointalgorithmsforvariationalinequalitiesandtheirapplications,suchas[6,7,8,9]andthereferencestherein.Onecanalsofindcompletereviewsofthistopicinthemonographs[10,11,12].Vectoroptimizationproblemshadfoundalotofapplicationsinreallife,suchaseconomicstheory,managementscienceandengineeringdesign.Recently,someiteratedmethodsforsolv-ingvectoroptimizationproblemshavebeenproposed:steepestdescentmethodsforvectoroptimization[13,14],projectedgradientmethodstovectoroptimization[15,16],anefficientinterior-pointmethodforconvexmulticriteriaoptimizationproblems[17],Newton’smethodsforunconstrainedmultiobjective/vectoroptimization[18,19,20],etc.ItisworthnoticingthatBonneletal.[21]recentlyconstructedavector-valuedproximalpointalgorithmtoaconvexvectoro...