强变分不等式问题和广义变分不等式问题解的存在性与例外簇的综述报告

强变分不等式问题和广义变分不等式问题解的存在性与例外簇的综述报告Strongvariationalinequalityproblems(SVIPs)andgeneralizedvariationalinequalityproblems(GVIPs)aretwoimportantclassesofnonlinearmathematicalmodelsthatariseinmanyfieldssuchaseconomics,engineering,physics,andoptimization.Inthisreport,wewillprovideanoverviewoftheexistenceandexceptionalsetofsolutionsforthesetwotypesofproblems.1.StrongVariationalInequalityProblems(SVIPs)Astrongvariationalinequalityproblemisformulatedasfollows:Findavectoru∗∈Ksuchthat⟨A(u∗),u−u∗⟩+ϕ(u)−ϕ(u∗)≥0,∀u∈K,whereKisanonemptyclosedconvexset,A(u)isaboundedlinearoperatorfromKtoK,andϕisacontinuouslydifferentiablefunctiononK.OneimportantpropertyofSVIPsisthattheyhaveauniquesolutionforanynonemptyclosedconvexsetKandanygivenfunctionϕ.ThisisbecauseSVIPsareequivalenttotheminimizationofaconvexfunctionoveraconvexset,whichalwayshasauniqueminimizer.However,theuniquesolutiontoanSVIPmaynotexistinsomecases.Forexample,considertheunitballK={u∈Rn:||u||≤1}andthefunctionϕ(u)=||u||^2.LetA(u)betheidentitymatrix.Then,theSVIPbecomes:Findavectoru∗∈Ksuchthat||u−u∗||^2−||u∗||^2≥0,∀u∈K.Itiseasytoseethatthereisnofinitesolutiontothisproblemsincetheleft-handsideisalwaysnonnegativewhiletheright-handsideiszero.WhenthesolutiontoanSVIPexists,theexceptionalsetisalwaysanullset.Thismeansthatthesetofu∈Ksatisfying⟨A(u),u−u∗⟩+ϕ(u)−ϕ(u∗)<0fortheuniquesolutionu∗isasetofmeasurezero.2.GeneralizedVariationalInequalityProblems(GVIPs)AgeneralizedvariationalinequalityproblemisanaturalextensionofanSVIPthatallowsfortheinclusionofnonlinearoperators.Itisformulatedasfollows:Findavectoru∗∈Ksuchthat⟨A(u∗),u−u∗⟩+ϕ(u)+F(u,u∗)−ϕ(u∗)≥0,∀u∈K,whereKisanonemptyclosedconvexset,A(u)isaboundedlinearoperatorfromKtoK,ϕisacontinuouslydifferentiablefunctiononK,andF(u,u∗)isanonlinearoperatorthatsatisfiesthefollowingtwoproperties:(a)F(u,u)≥0forallu∈K;(b)F(u,v)−F(v,u)≥⟨A(u)−A(v),u−v⟩forallu,v∈K.Thefirstpropertyensuresthattheproblemiswell-posed,whilethesecondpropertyisamonotonicityconditionthatensurestheexistenceofsolutions.TheexistenceofsolutionstoGVIPsismorechallengingtoestablishduetothepresenceofthenonlinearoperatorF(u,u∗).Insomecases,theremaybenosolution,orthesolutionmaynotbeunique.Forexample,considertheunitballK={u∈Rn:||u||≤1}andthefunctionsϕ(u)=||u||^2andF(u,u∗)=||u−u∗||^4.LetA(u)betheidentitymatrix.Then,theGVIPbecomes:Findavectoru∗∈Ksuchthat||u−u∗||^2−||u∗||^2+||u−u∗||^4−||u−u∗||^2≥0,∀u∈K.Itcanbeshownthatthereisnofinitesolutiontothisproblem.TheexceptionalsetofsolutionsforGVIPsdependsonthestructureoftheproblemandthepropertiesofthenonlinearoperatorF(u,u∗).Insomecases,theexceptionalsetisanullset,whileinothercases,itmaybeasetofpositivemeasure.ThestudyofexceptionalsetsforGVIPsisanactiveareaofresearchinnonlinearanalysisandoptimization.Inconclusion,wehaveprovidedanoverviewoftheexistenceandexceptionalsetofsolutionsforstrongvariationalinequalityproblemsandgeneralizedvariationalinequalityproblems.SVIPshaveauniquesolutionforanygivenfunctionandconvexset,whileGVIPsmayhavenosolutionormultiplesolutionsdependingonthepropertiesoftheproblem.TheexceptionalsetforSVIPsisalwaysanullset,whileforGVIPs,itdependsonthestructureoftheproblemandthepr