強(qiáng)變分不等式問(wèn)題和廣義變分不等式問(wèn)題解的存在性與例外簇的綜述報(bào)告

強(qiáng)變分不等式問(wèn)題和廣義變分不等式問(wèn)題解的存在性與例外簇的綜述報(bào)告Strongvariationalinequalityproblems(SVIPs)andgeneralizedvariationalinequalityproblems(GVIPs)aretwoimportantclassesofnonlinearmathematicalmodelsthatariseinmanyfieldssuchaseconomics,engineering,physics,andoptimization.Inthisreport,wewillprovideanoverviewoftheexistenceandexceptionalsetofsolutionsforthesetwotypesofproblems.1.StrongVariationalInequalityProblems(SVIPs)Astrongvariationalinequalityproblemisformulatedasfollows:Findavectoru?∈Ksuchthat?A(u?),u?u??+?(u)??(u?)≥0,?u∈K,whereKisanonemptyclosedconvexset,A(u)isaboundedlinearoperatorfromKtoK,and?isacontinuouslydifferentiablefunctiononK.OneimportantpropertyofSVIPsisthattheyhaveauniquesolutionforanynonemptyclosedconvexsetKandanygivenfunction?.ThisisbecauseSVIPsareequivalenttotheminimizationofaconvexfunctionoveraconvexset,whichalwayshasauniqueminimizer.However,theuniquesolutiontoanSVIPmaynotexistinsomecases.Forexample,considertheunitballK={u∈Rn:||u||≤1}andthefunction?(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??+?(u)??(u?)<0fortheuniquesolutionu?isasetofmeasurezero.2.GeneralizedVariationalInequalityProblems(GVIPs)AgeneralizedvariationalinequalityproblemisanaturalextensionofanSVIPthatallowsfortheinclusionofnonlinearoperators.Itisformulatedasfollows:Findavectoru?∈Ksuchthat?A(u?),u?u??+?(u)+F(u,u?)??(u?)≥0,?u∈K,whereKisanonemptyclosedconvexset,A(u)isaboundedlinearoperatorfromKtoK,?isacontinuouslydifferentiablefunctiononK,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}andthefunctions?(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