向量变分不等式的正则间隙函数和误差界的一些注释李丽丽，陈纯荣重庆大学数学与统计学院，重庆401331摘要：本文主要给出文献（SunandChai,Optim.Lett.,2013）中关于（广义）向量变分不等式的误差界结果的修正版本。此外，建立了一个构造（广义）向量变分不等式的正则间隙函数的框架。文中得到的主要结果是新的。关键词：向量优化;向量变分不等式;间隙函数;误差界;标量化中图分类号：O221RemarksonRegularizedGapFunctionsandErrorBoundsforVectorVariationalInequalitiesLILi-Li,CHENChun-RongCollegeofMathematicsandStatistics,ChongqingUniversity,Chongqing401331Abstract:Inthispaper,modifiedversionsofcorrespondingresultsonerrorboundsfor(generalized)vectorvariationalinequalitiesobtainedbySunandChai(Optim.Lett.,2013)aregiven.Moreover,aframeworktoconstructregularizedgapfunctionsfor(generalized)vectorvariationalinequalitiesisestablished.Themainresultsobtainedarenewintheliterature.Keywords:Vectoroptimization;Vectorvariationalinequalities;Gapfunctions;Errorbounds;Scalarization0IntroductionThevectorvariationalinequalityproblem,asanimportantmodelinvectoroptimization(e.g.,[1,2,3,4]),hasreceivedextensivestudiesinthelastthreedecades.Manyimportantresultstovariouskindsofvectorvariationalinequalitieshavebeenestablished;forexample,see[3,4,5]andthereferencestherein.Fromthecomputationalpointofviewlikescalarcases,oneimportantresearchdirectioninvectorvariationalinequalitiesisthestudyofgapfunctionsormeritfunctions(e.g.,see基金项目：SpecializedResearchFundfortheDoctoralProgramofHigherEducation(20100191120043)andNationalNaturalScienceFoundationofChina(11301567).作者简介：LILi-Li（1988-），female，graduatestudent，majorresearchdirection：vectoroptimization.Cor-respondingauthor:CHENChun-Rong（1981-），male，associateprofessor，majorresearchdirection：vectoropti-mization.-2-[3,5,6,7,8,9]).Especially,gapfunctionsplayafundamentalroleondevelopingerrorboundsforvectorvariationalinequalities.Theerrorboundsprovidemeasureofthedistancebetweenasolutionsetandanarbitraryfeasiblepoint.Acomprehensivesurveyoftheoryandrichapplicationsabouterrorboundscanbefoundin[10,11].Recently,severalresultsoferrorboundsintermsof(regularized)gapfunctionsforsomekindsofvectorvariationalinequalitieshavebeenestablishedin[12,13,14,15,16].Amongthem,SunandChai[14]introducedgapfunctionsandtheirregularizedversionsfor(generalized)vectorvariationalinequalitieswithoutanyscalarizationparameters,andthenusedtheseregularizedgapfunctionstoobtainsomeerrorboundresultsforvectorvariationalinequalities.Unfortunately,wefindthattherearesomemistakesintheproofof[14,Theorem3.3]abouterrorbounds.Theaimofthispaperistogivemodifiedversionsofcorrespondingresultsonerrorboundsfor(generalized)vectorvariationalinequalitiesobtainedbySunandChai[14].Wenewlyprove[14,Theorem3.3]withoutimposingLipschitzpropertiesof𝑔𝑖,𝑖=1,⋅⋅⋅,𝑚(comparingLemma3andTheorem1).Inaddition,weshowthatunderassumptionsofSunandChai,thesolutionsetsol(GVVI)isasingletonsetbutnotageneralset.Weshallmentionthattherevisedresultsonerrorbounds(Theorem1andCorollary1)arestillnewintheliterature,becauseofthedifferentregularizedgapfunctionwithoutinvolvingscalarizationparametersconstructedbySunandChai.Furthermore,wealsoestablishaframeworktoconstructregularizedgapfunctionsfor(generalized)vectorvariationalinequalities(seeTheorem2),whichcontains[14,Theorem3.2andCorollary3.2]asspecialcases.So,inthispaper,weeithermodifyorextendmainresultsofSunandChai...