ThepositivenessoflowerlimitsoftheHoffmanconstant - PDF document

JGlobOptim DOI10.1007/s10898-011-9729-7 ThepositivenessoflowerlimitsoftheHoffmanconstant inparametricpolyhedralprograms A.Jourani · D.Zagrodny ©TheAuthor(s)2011.ThisarticleispublishedwithopenaccessatSpringerlink.com Abstract If K ( t ) aresetsofadmissiblesolutionsinparametricprogramsthenitisnatural AnswerstothisquestionarerelatedtotheproblemofthecontinuityorLipschitzcontinuity ofthevaluefunction,namelyhavingthelowersemi-continuityof K ( · ) wegettheupper semi-continuityofthefunctioneasilyandtheLipschitz-likepropertyof ( · ) leadstothe Lipschitz-continuityofit.HereinsufÞcientconditionstogetthesepropertiesofthepolyhe- dralmultifunctionofadmissiblesolutionsaregivenintermsofthelowerlimitoftheHoffman lowerlimitoftheHoffmanconstantarepositive. Keywords Parametricprogramming · Hoffmanconstant · Errorbound · Moscoconvergence · AttouchÕstheorem Convexfunctions · Subdifferentials · Polyhedrals MathematicsSubjectClassiÞcation(2000) 49J40 ThispaperwaswrittenwhilethesecondauthorwasvisitingProfessorattheUniversityofBurgundy.The A.Jourani UniversitŽdeBourgogne,UFRSciencesetTechniques,InstitutdeMathŽmatiquesdeBourgogne,UMR 5584CNRS,BP47870,21078DijonCedex,France e-mail:jourani@u-bourgogne.fr D.Zagrodny FacultyofMathematicsandNaturalScience,CollegeofScience,CardinalStefanWyszy« Dewajtis5,01-815Warsaw,Poland D.Zagrodny( B ) SystemResearchInstitute,PolishAcademyofSciences,ul.Newelska6,01-447Warsaw,Poland e-mail:d.zagrodny@uksw.edu.pl 123 JGlobOptim1IntroductionbearealBanachspaceandbeitstopologicaldual.Forasetofindices,ametricspaceandmappings,westudythefollowinginequalitysystem0forallreferstothepairingbetween.Thissystemisviewedasdependingontheparameter,soforeach,letbethe(possiblyempty)setofsolutionsto)withrespectto.WeareinterestedinalocalbehaviorofthemultifunctionaroundaÞxedelement.OurattentionismainlyfocusedonconditionsunderwhichLipschitz-likeatt,inthesensethatthereexistaneighborhoodUoftsuch(2)orKislowersemi-continuousatt,i.e.liminfdenotestheballat0withradiusandliminfstandsforthelowerlimitofsets.Thesepropertiesplayacentralroleinparametricprogramming,werefertoo5]forseveralfactsonthecontinuityof.Theyallowtoinvestigatethebehaviorandthepropertiesofsolutionsetsofoptimizationproblemsundervariationsofthedescribingparameters.Tobemoreprecise,letusconsiderthefollowingproblemisagivenfunction,whichisassumedtobeconvexinthesecondvariable.Changingover,wewillgetafamilyofproblemswhosevaluesandsetsofsolutionsaregiven,respectively,by:={.Theobtainedfunctioniscalledvalueormarginalorcostfunction.Thebehaviorofitisrelatedtothatofthesolutionsets.Inordertoobserveitletusindicatesomelinksamong.ForthisreasonÞxandsupposethatarecontinuous.Itiseasytoobservethatthefollowingimplicationlowersemi-continuityofKatholdstrue,seealso[,Theorems4.2.1and4.2.2].Additionally,imposingauniformcom-pactnessassumptiononandsomecontinuitypropertieson,wehavelowersemi-continuityofaswellasthefollowingequivalenceuppersemicontinuityofuppersemicontinuityofholdstrue.Thisfactiscommonlyknownsee[,Theorem5]or[,Proposition12],wereferalsoto[].Ofcoursetogetmoresmoothwehavetoassumemoreoninvolvedfunctions,wereferto[]andthereferencesthereinforseveralfactsonthat. JGlobOptimWeseethatthelowersemi-continuityofisessentialtogettheuppersemi-continuity.HereinweprovideitusingtheHoffmanconstant.Thisconstantisgivenby ai(t),x+bi(t)],K(t):={see[],and:=.ManyauthorshavepresentedandstudiedexplicitrepresentationsofHoffmanconstants,wereferto[]andreferencestherein,seealso[].InIn7,Theorem5.1](undertheassumption)itisshownthatistheFenchelsubdifferentialoftheconvexfunctionisthedistancebetween0andwithrespecttothenormofRelation()isequivalenttothefollowingoneobtainedin[].ThisrepresentationoftheHoffmanconstantallowsustouseasubddifferentialcalculustoshowthattheinequalityliminfentailsthelowersemi-continuityof,seeTheorem,ortheLipschitzcontinuity,see,thus()and()hold.Unfortunately,thefunctionisnotlowersemi-continuouseveninsimplecases,asitisshowninSect..Itmeansthatitisnotenoughtoimposeconditionspreservingthat0togetthepositivenessofthelowerlimit.Theproblemismuchmorecomplicated.InSect.wepresentconditionsimplyingthepositivenesswheneverisÞniteordenumerable.ThecaseisdenumerableinvolvestheAttouchtechniqueofapproximationofsubgradientsbyÒbetterÓonesingettingtheinequal-ity,seeTheorems.ThistechniquecanbeusedonlyinreßexiveBanachspacesormoregenerallyinweaklycompactlygeneratedBanachspaces.WedonotknowhowtogetthisresultsingeneralBanachspaces,thisisanopenproblem.WheneverisÞniteorcanbeexpressedasthemaximumofaÞnitenumberofafÞnefunctions,see[]forsomeinformationonthistechnique,itiseasiertoevaluatethesubdifferential,so()canbeappliedtogettheinequalityliminf0,seeExampleandPropositionLetusalsomentionthatwheneverisaÞniteset,,functionalsdonotdependonandatleastoneofthemisdifferentfromzero,thenliminf0,seeandCorollaryfordetails.WhenwecompareProposition,Theoremsitturnsoutthattheyareofdifferentnature.Wepresenttheminthosemiscellaneousformsinordertopointoutthatthereareseveralpossibilitiestopreservethepositivenessofthelimitliminfbyexamples.OfcoursetherearepossibilitiestoproducetheoremslikePropositionthereßexiveorweaklycompactlygeneratedBanachspacesetupwithdenumerablefamiliesofafÞnemappings,andthereverseisalsopossible.Finallywewouldliketothanktherefereeforhisremarks,whicheliminatedsomegapsinthepresentation. JGlobOptim2PropertiesofsubgradientsofconvexfunctionsInthissectionseveralpropertiesoflowersemi-continuousfunctionsdeÞnedonarealBanachspacearerecalled,wereferto[]forthedefinitionoflowersemi-continuousproperconvexfunctionandtheirproperties.WhenisaBanachspacethentheweaktopologyisdenotedby,theweaktopologyby,wereferto[]forthedefinitionsoftheweaktopologies,weakconvergence,weakconvergenceandforthedefinitionofthereßexiveBanachspace.TheclosedandtheopenunitballsofaredenotedbyThe(Fenchel)subdifferentialofaconvexfunction{}atapointisthesubsetofthedualspacegivenbyisapointofthedomainof,wheredom:={,andthesubdifferentialistheemptysetotherwise.Itfollowsfromthisdefinitionthatforevery0thefollowingassertionsareequivalent,)(10)Either()or()ensuresthatisanisolatedminimumofBelowwerecalltworesultsallowingustoapproximateasubgradientofaconvexfunc-tionbysubgradientsofconvexfunctions,whichsubgradientsareeasiertocalculate.ForthisreasonletusrecallthenotionoftheMoscoconvergence.Indoingthiswefollow[],wereferalsoto[]formoreinformationontheMoscoconvergence.DeÞnition2.1beaBanachspaceand{}forevery.WesaythatMoscoŠifthetwofollowingconditionsaresatisÞed:(S1)wheneverisasequenceweaklyconvergentto,thenliminfŠ(S2)foreachthereexistsasequenceconverginginnormtoforwhichŠItisnotdifÞculttonoticethatifisanondecreasingsequenceoflowersemicon-tinuousconvexfunctionsandforeveryŠthen(S1)and(S2)aresatisÞed.FirstresultconcerningtheapproximationsofsubgradientsbybetteronesisaconsequenceoftheAttouchtheorem(thenecessitypart),see[Theorem2.2Letf{+}beconvexlowersemi-continuosfunctionsonareßexiveBanachspaceXandfMoscoŠ.Foranyxtherearesequencessuchthata:limŠŠforeverynandxŠWerecallthataBanachspaceisWCG(weaklycompactlygenerated)ifthereexistsaweaklycompactsubsetthatspansadenselinearspacein,onecanalwaysassumethatisconvex,wereferto[]fordetailedinformationonWCGspaces.Below JGlobOptimwerecall,see[],thatifisaweaklycompactlygeneratedBanachspaceandMoscoŠ,thenforeverythereisasequenceandlimŠforevery,wereferto[]formore.Theorem2.3LetXbeaWCGBanachspace,beÞxedandf{+}bealowersemi-continuousconvexfunctionsuchthatfAssumethatf{+}arelowersemi-continuousconvexfunctionssuchthat:MoscoŠthereisanopennonemptysubsetUofXandaconstantcsuchthatforeveryuandnwehavefThentherearesequencesEandsuchthat:iii:limŠŠiv:ŠFinallyletusrecalltheEkelandvariationalprinciple,see[]formore.Theorem2.4(EkelandVariationalPrinciple)Assumethatf{}isalowersemi-continuousfunctiononaBanachspaceX,boundedfrombelow.ForanyEsuchthatfthereisapointzXsatisfyingforeveryx3Examplesoflowersemi-continuityofadmissiblesetswiththelackoflowersemi-continuityoftheerrorboundsInthissectionthreesimpleexamplesofparametricconvexprogramsarepresented,wherethesetsoftheadmissiblesolutionsarelowersemi-continuouswithrespecttoparameterbuttheHoffmanconstantsarenotlowersemi-continuous.Thusthelowersemi-continuityofthemultifunctionofadmissiblesolutionsmaynotbelinkedtothelowersemi-continuityoftheerrorboundfunction.Example3.1Letusput:=[andforeveryandforeveryLetusobservethatisLipschitzcontinuousonandforeverythefunctionisconvex,moreoverif0 JGlobOptimForeverystandsforthesubdifferentialofwithrespecttothesecondvariableat:={Foreveryywehave2andliminf(12)isnotlowersemi-continuousat0but=]Šforeveryislowersemi-continuousonBelowweprovideanotherexample,wherethesamephenomenaoccursbutthelowerlimitin()isequalto0,henceweinferthatthepositivenessofthelowerlimitoftheHoffmanconstantsisnotnecessaryforthelowersemi-continuityoftheadmissiblesetsofsolutions.ThesecondexampleisaslightmodiÞcationoftheÞrstone,namelyExample3.2if0Forevery:={Foreveryywehave1andliminfisnotlowersemi-continuousat0but=]Šforevery,soislowersemi-continuousonThethirdexampleisjusttoshowthatevenassumingthatbdconvforeverysubsetwedonothavethelowersemi-continuityoftheerrorbounds,seealso()inRemarkExample3.3 if0Forevery JGlobOptim:={Foreveryywehave),2andliminfisnotlowersemi-continuousat0butbdconvforeverysubsetOfcourse=]Šforevery,soislowersemi-continuousonLetusobservethatalltheexamplescanberearrangedtohavecoercive,sotheadmis-siblesetswouldbebounded.Forthisreasonitisenoughtoadd1andinthedefinitionoftakethemaximumfromthefourafÞnefunctionsinsteadofthethree,see4ThepositivenessofthelowerlimitsoftheHoffmanconstantsThroughoutthepaperletbearealBanachspace,beametricspace,beanonemptysetofindicesandthefamilyofmappingsbegiven,whereLetusdeÞneandforevery,):={:={wheretheinfimumovertheemptysetisWestartwithasimpleobservationthathaving0insidetheinteriorofthesubdifferentialwegetthatthelowerlimitoftheerrorboundsispositive,namelyProposition4.1FixXandletusassumethatf{+}suchthatfandforsomeandeverytTinsomeneighborhoodoftsayforeveryt,wehave,)andK,thenforeverytwehave,thusliminf JGlobOptimProofForeveryandeverybytheequivalence()wehaveThusif,thenwhichimpliesforevery.Inordertocompletetheproofletusobservethat0forevery.Infact,by()wehaveforevery,so0andbythedefinitionofcanbeconsideredwhenthevaluesofthefunctionarecalculated,butthen,thus(keepinmindthattheinfimumovertheemptysetisBelowweprovideanexampleshowingthatwhenever0isintheinteriorofthepolyhedrongeneratedbyaÞnitefamily,then()issatisÞedwithExample4.2Assumethatforgiven0and,wehave0,thesetsarenonemptyforeverycloseto,themappingsequi-continuousat,themappingsdonotdependon,i.e.forevery,anddclconv,)thentheassumptionsoftheabovepropositionaresatisÞed,whereisdeÞnedin()andstandsfortheclosurewithrespecttotheweaktopology.Indeed,becauseoftheequi-continuityof,thereexistsaneighborhoodsuchthatsuchthatandletetbearbitrary.Foreachconv,),thereexistaÞnitesubset,)andnon-negative1,suchthatthatai(t),x+bi]Š iJi[ai(t0),0+bi]Š= iJ[iai(t)Šai(t0),x]+Henceforeachconv,)Thelastinequalityisalsotrueforallandhenceisarbitraryinn,wehave JGlobOptimUsingtheassumptionthatthemappingsdonotdependon,wehaveforall,andthenorequivalently, )andthisisexactlyrelation(Remark4.3Theexampleaboveholdstrueifwereplaceandcondi-tion()bythefollowingoneconvRemark4.4IfthespaceisaÞnitedimensional,isÞnite,themappingsarecontinuousat,then()implies(Itisalsoeasytoobservethatassuming,similarlyto[],thatisÞniteandforeverysubsetbdconveither()holdstrueoritdoesnotholdbutthen()ensures, )convforsome0andaneighborhoodof,sayItisnaturaltoaskwhathappensif()doesnothold.BelowwegivepartialanswerstothisquestionwheneverthesetiseitherÞniteordenumerable.InthePropositionbelowweassumeonlythatsetsofalmostactiveconstraintsareÞnite.Proposition4.5LetusÞxtTandassumethatforsomeandaneighborhoodoftsayUT,thesets,)(20)arenonemptyandÞniteforeveryt.Ifforsome, )domfconv,)domfforeveryt,thusliminfwhereNdomfisthenormalconetodomfaty,i.e.domf:={domf JGlobOptimProofLetusÞxsuchthatthatandxf(t,y)=.Take.Bythelowersemi-continuityandtheconvexityofthereis0suchhforeveryry0,µ],hencethesetet0,µ]I(t,y+s(uŠy),)isÞnite.BecauseoftheÞnitenessof,)andthecontinuityofonthesegmentmenty,u]wehave),),),for0smallenough.Indeed,if),),)foreveryandsome0,thenthereisasuchthat),),)forsomeandevery(keepinmindthat),)isÞnite).,),acontradiction.Thus),),),for0smallenough.Takeanyandobservethatforforsmallenoughweobtain,)Nowapplyingthestandardprocedure,seeforexample[,Theoremp.87],wegetconv,)conv,)Hencebytheassumptionsforevery0weget(keepinmindthatthedistancefromtheemptysetisObservethatifthesetdeÞnedin()isemptyforsome,so(issatisÞed.InthepropositionbelowweassumethatisaÞniteset,atleastoneofisnotequaltozeroandalldonotdependon.Beforestatingthisresult,letusset:={arelinearlyindependentforallForeverysuchthat0Farkaslemma(forcones)tellsusthatthat0,+[conv(22)Proposition4.6SupposethatXisaHilbertspace,IisaÞniteset,atleastoneofaisnotequaltozeroandalladonotdependont.ThenforeachtTsuchthatK,we JGlobOptimMoreover,ifKfortneart,thenliminfProofWeusethesameideasasintheproofof[,Theorem4.1].Forthesakeofthereaderconvenience,wegiveadetailedproofoftheabovepropositionherein.LetusobservethatthefollowinginclusionholdstrueconvLetusÞxandsetconvThen,sincearelinearlyindependent,wehaveObservethat,whereisdefiniteasin().Sousingtheequivalence),wegetTheproofisthenterminatedifweshowthatforeachthereexistssuchthatIndeed,let,thenthereexistssuchthatorequivalentlyBy((0,+[conv.Sothereexistnotallequaltozero(because),suchthatarelinearlyindependentthenputandobservethatforevery0andthentheresultfollowsfrom().Sosupposethereexist,notallequaltozerosuchthatHenceforall JGlobOptimOurproblemistoÞnd0andsuchthat0and.Setandsupposethat.Forall0iff .Soletbesuchthatmin µi=Ši0 andput .Then0andByinductionweshowthatisapositivecombinationoflinearlyindependentfamilyof,with,orequivalently,(keepinmindthat0foreveryTakingintoaccountthatequalitycanbeexpressedastwoinequalitiesweobtainthefollowingcorollaryofPropositionCorollary4.7beaÞnitefamilyofvectorsofaHilbertspaceX,withatleastoneofanotbeingequaltozero,andlet,foreachibeafunction.Considertheset:={andthefunctionfdeÞnedbyIfSfortneart,thenliminfInordertodealwiththesetofindexesbeingdenumerable,infactitisenoughtohavethatthesetdeÞnedin()isatmostcountable,andtoget()weneedamoresophisticatedtool.Namely,inwhatwedohereinisemploymentofanapproximatetechnique.WeapproximatesubgradientsoftheconvexfunctiondeÞnedin()bysubgradientsoffunctions·+whichsubdifferentialsarepossibletocalculateexplicitly.ForthisreasoninthenexttheoremweusetheAttouchtheorem.Thepricefortheuseofthetoolistheneedtoassumethatthespaceisreßexive.Theorem4.8LetXbeareßexiveBanachspaceandtTbeÞxed.Assumethatforaandaneighborhoodoft,sayUT,foreverytweareabletochooseanonemptydenumerablesubsetIIsuchthatifKifdomfwherep,and, )convforeveryt,soliminf JGlobOptimProofWeshowthatforeverysuchthatinfwehaveclconvForthispurposeletusÞxsuchthatinf.Bythelowersemi-continuityofthereis0suchthatforevery.LetusassumethatanddeÞneasequenceofconvexasfollows(v)),vForeverywehave(v)(v),(v),v),istheMoscolimitofthesequenceandTheorembeappliedtothesequence,whereisequalto0ontheballandoutsidetheball.HenceanysubgradientisthestronglimitofsubgradientsofbutforlargeenoughwegetclconvclconvThuswehaveclconvwhichbytheassumptionsimpliesLetusÞx.Forany0suchthatforevery.Since,soforeveryandhence,whichby()impliesConsiderthecaseandÞx(ifdomandwearedone).Thesetisconvexandclosed,sobythereßexivitythereissuchthat.Foreveryywehave (z+s(x(z)Šzf(tf(t,z) d(z,Kf(t123 JGlobOptimwhichby()implies Letusassumethat.TakeanyyandÞndsuchthat,v)1and,v)(v,.Choose,v) (v,,.ByTheorem,appliedto,thereissuchthat(v,,w),v),,w),),w))(thusf+(t,w),w),and,w),w))andforsome,w),whichcontradictsthechoiceof.Weconcludethatinthecaseby()weget0forevery,soeitherdomordom(inthelattercaseinf0).Weexcludethelattercase.ForthisaimÞx .ByTheoremthereissuchthat,w),w),),w))andw forsome,w).Henceby)wegetbutitcontradictsthechoiceof,soitisimpossiblethat.ThusforeverywheneverwhichimpliesthestatementsLetusobservethat()isfulÞlledwhenever, )convseealsoExample.Ofcoursehavingdenumerableweseethatitiseasytocheckthatimplication()issatisÞed(forexampleputtingforall,seeTheoremsoassumptionsofTheoremarenotdifÞculttobeveriÞedinthiscase.Wheneverassumedtobeseparableandthefamilyisboundedforevery,thenimplication()isalsovalid,evenwhenisnotdenumerable,itisdiscussedintheRemarkbelow.Remark4.9beaseparableBanachspaceandthefamilybeboundedforevery.Foreveryforwhichdomletuschoseadenumerablesubsetsuchthatdomandforeverythereisasequence(thechoiceofbecarriedoutasfollows:takeacountabledensesubsetof,),andput:={,)).Foreveryletuschooseadenumerablesubsetsuchthat JGlobOptimandputLetusobservethatforthesetimplication()holdstruewheneverdomInfact,wehaveforevery.Takesuchthat.ObservethatŠŠ,thus.Letusalsoobservethattheboundednessofthefamilyandnon-emptinessofthedomaindomfdomfTheargumentsusedintheproofofTheoremallowsustoextendPropositiontheÞnitecasetothedenumerableoneinthereßexiveBanachsetting.Theorem4.10LetXbeareßexiveBanachspaceandtTbeÞxed.Assumeforsomeandaneighborhoodoft,sayUT,theset,)isnonemptyanddenumerableandforeveryt,domfisclosedandforsomeconv,)domfProofandÞxsuchthat.Bythelowersemi-continuityofthereis0suchthat0forevery.Lett.LetusassumethatanddeÞneasequenceofconvexfunctionsasfollows(v)),vForeverywehave(v)(v).convergestosomefunctionsuchthat.SoistheMoscolimitofthesequence(thefunctionsareconvexandlowersemi-continuous)andTheoremcanbeappliedtothe,whereisequalto0onthesetoutsidethisset.Henceforany,anysubgradientsubgradientf +D(t)}](u )isastronglimitofsubgradients,withforall,butclconv JGlobOptimTheassumptionsofthetheoremandthelastinclusionensurethatstronglyconvergesto,wehaveandhenceencef +D(t)}](u )) .(28)Now,pickorequivalentlySinceforall,weobtainnf +D(t)](y).Combiningthisrelationwith(),itfollowsthatandhenceItfollowsthatInthediscretecase,i.e.,theassumptionsofTheoremcanberelaxed.Theorem4.11LetXbeareßexiveBanachspaceandtTbeÞxed.AssumeIandforsomegivenneighborhoodoft,sayUT,andforsomethefollowinginequalityconvProofTheproofissimilartothepreviousonebyconsideringthesequence,where(v)),vwhichconvergesto JGlobOptimLetusalsonoticethatthecondition, )convisequivalentto, )clconvwhichinthereßexiveBanachspacesisthesameas, )conv(30)InTheoremweuse()inordertoguaranteethepositivenessoflowerlimitoferrorbounds.InanonreßexiveBanachspace()implies()butthereverseisnotalwaystrue.Ofcoursewecoulduse()inTheoreminsteadof()andtheresultwouldbethesame.HoweverinnonreßexiveBanachspacesitisnotpossible.Inthenexttheoremweproposearesultwhere()isusedinsteadof()andthespaceisassumedtobeweaklycompactlygenerated.Howeverasapriceforthatwehavetoassumethefamilyoffunctionsconsistsofcontinuousfunctions.Thecontinuityassumptionscanberelaxedusingatechnicalconditionfrom[],forthesakeofsimplicitywedonotdoitÑtheinter-estedreadercandoitrepeatingtheideas.Letusstartwithanexampleilluminating(Example4.12beaBanachspaceandbeaconvexweakclosedsubsetsuchthat0,forexample,where.Assumethatforeverywehavethen()and()aresatisÞedfor0sufÞcientlysmall.Intheproofofthetheorembelowweneedthepropertythatwhenevertheinteriorofthedomainisnonempty,i.eintdomthentheHoffmanconstantcanbecalculatedintheinteriorofthedomain,namelywehaveProposition4.13ForeverytTsuchthatintdomfwehaveintdomfProofAssumethat()doesnothold,i.e.intdomFixany),and�0suchthat d(z,Kf(tf(t,z) intdom(32)NowletusapplyTheoremforthefunction.Thereissuchthat,).Hence0and,so,),whichcontradictsthelastinequalityin JGlobOptimTheorem4.14LetXbeaweaklycompactlygeneratedBanachspaceandtTbeÞxed.Assumethatforsomeandaneighborhoodoft,sayUT,thesetsintdomfarenonemptyforeverytandtheset,)isnonemptyanddenumerable,andforsome, )conv,),foreveryt,thusliminfProofItfollowsfromPropositionthatitisenoughtoshowthatforeveryintdomwehaveconv,),ForthisreasonletusÞxsuchthatintdom).Bythelowersemi-continuityofthereis0suchthatintdomisboundedfromtheaboveon.Letusassumethat,)anddeÞneasequenceofconvexfunctionsasfollows(v)),vForeverywehave(v)(v),(v),v),istheMoscolimitofthesequence,whereequalto0ontheballandoutsidetheballandTheoremcanbeapplied(keepinmindthatbythechoiceofandtheassumptionintdomwehaveintdom,whichimpliesthatthesequenceisuniformlyboundedfromtheabove).HenceanysubgradientistheweaklimitofasequenceofsubgradientsofÕsbutconvconv,),Thuswehaveconv,),whichbytheassumptionsimpliesforevery JGlobOptimhenceitfollowsthatliminfLetuspointoutthatintheproofoftheabovetheoremweneedonlytheassumptionthatforeverysuchthatthereis0suchthattheset,)isnonemptyanddenumerable.5Lipschitz-likeandlowersemi-continuitypropertiesoftheadmissiblesetsInthissectionweshowthatifliminfthenthemappingofadmissiblesetsofsolutionsislowersemi-continuousatTheorem5.1LetXbearealBanachspace,Tbeametricspace,tTanditsneighbor-hoodUTbegiven,f{+}besuchthat:={isnonemptyatt,andforeverytTthefunctionfisproperconvexlowersemi-continuousandforeveryx,andforeverysequenceTconvergingtotthereisasequence,XconvergingtoxsuchthatlimsupŠThen,thefollowinginequalityliminfentailsthelowersemi-continuityofKattProofAssumethatliminf0.LetusÞx,asequenceconvergingtoandasequence,convergingtosuchthatlimsupŠFirstletusobservethatif,thentheproperconvexlower-semi-continuousisboundedfrombelowby0.ThustheEkelandVariationlPrinciple,see,ensurestheexistenceofapairsuchthatliminfbutitwouldbeacontradictionforlargeenough,soforÕslargethesetsarenon-empty.Sincethesequenceconvergesto,thecaseforeveryimpliesthestatementimmediately.SoletusconsiderthecasewheneverinÞnitemanyof JGlobOptimÕsareoutof.Withoutlossofthegeneralitywemayassumethatforeverylarge.Usingequivalence()wegetŠlimsupŠwhichimpliestheexistenceofasequencesuchthatforeverylargeenough,thusthelowersemi-continuityisproved.Inseveralcases()isasimpleconsequenceofimposedassumptionsontheinvolvedfunctions.Forexample,letuspointoutthatifweassumethatiscontinuouson,forexampleassumingthatisÞniteandarecontinuous,weget()withforevery.Condition()isalsofulÞlledwithforevery,wheneverisuppersemi-continuousforevery.IfforeveryconvergingwehaveMoscoŠ,then()issatisÞedtoo.Thusweseethat()canbeentangledinotherassumptionsandinseveralcaseswegetitimmediately.beoftheformisadenumerableset.WeendowwithametricdenotedbyOuraimhereistoshowhowtousethepositivenessofthelowerlimitoftheHoffmancon-stanttogetakindofLipschitz-likepropertyoftheadmissiblesetdeÞnedbythefunctionTheorem5.2LettT.SupposethatthereexistandaneighbourhoodUoftsuchthatthereexistsasuchthatliminf ThenthereexistsaneighborhoodUoftsuchthatProofBy(ii),thereexistsaneighborhoodsuchthat orequivalently0bearbitrary,.If,thenwearedone.Otherwise,relation()togetherwithwithf(t,x)Šf(t ,x)] a(r+1)dT(t,t )whichcompletestheproof.Letusobservethatwheneverisuniformlyboundednear,theabovetheoremassertsthatisLipschitzcontinuousat JGlobOptimOpenAccessThisarticleisdistributedunderthetermsoftheCreativeCommonsAttributionNoncommer-cialLicensewhichpermitsanynoncommercialuse,distribution,andreproductioninanymedium,providedtheoriginalauthor(s)andsourcearecredited.References1.Attouch,H.:VariationalConvergenceforFunctionsandOperators.PitmanAdvancedPublishingPro-gram,Boston(1984)2.Attouch,H.,Beer,G.:Ontheconvergenceofsubdifferentialsofconvexfunctions.Arch.Math.389Ð400(1993)3.AzŽ,D.:AuniÞedtheoryformetricregularityofmultifunctions.J.ConvexAnal.,225Ð252(2006)4.AzŽ,D.,Corvelec,J.-N.:OnthesensitivityanalysisofHoffmanconstantsforsystemsoflinearinequal-ities.SIAMJ.Optim.(4),913Ð927(2002)5.Bank,B.,Guddat,J.,Klatte,D.,Kummer,B.,Tammer,K.:Non-LinearParametricOptimization.Akademie-Verlag,Berlin(1982)6.Bosch,P.,Henrion,R.,Jourani,A.:Errorboundsandapplications.Appl.Math.Optim.,161Ð181(2004)7.Cornejo,O.,Jourani,A.,Zalinescu,C.:Conditioningandupper-Lipschitzinversesubdifferentialsinnonsmoothoptimizationproblems.J.Optim.TheoryAppl.(1),127Ð148(1997)8.Deville,R.,Godefroy,G.,Zizler,V.:SmoothnessandRenormingsinBanachSpaces.LongmanScientiÞcandTechnical,Essex(1993)9.Ekeland,I.:Nonconvexminimizationproblems.Bull.Am.Math.Soc.,443Ð474(1979)10.Hoffman,A.J.:Onapproximatesolutionsofsystemsoflinearinequalities.J.Res.Natl.Bureau,263Ð265(1952)11.Hogan,W.H.:Point-to-setmapsinmathematicalprogramming.SIAMRev.,591Ð603(1973)12.Holmes,R.:GeometricFunctionalAnalysisanditsApplications.Springer,NewYork(1975)13.Jourani,A.:HoffmanÕserrorbound,localcontrollabilityandsensitivityanalysis.SIAMJ.Control,947Ð970(ErratuminthesameJournal)(2000)14.Klatte,D.,Thiere,G.:Errorboundsforsolutionsoflinearequationsandinequalities.ZORMath.MethodsOper.Res.,191Ð214(1995)15.Li,W.:ThesharpLipschitzconstantsforfeasibleandoptimalsolutionsofaperturbedlinearprogram.Lin-earAlgebraAppl.,15Ð40(1993)16.Mangasarian,O.L.:Aconditionnumberoflinearinequalitiesandequalities.MethodsOper.Res.3Ð15(1981)17.Mangasarian,O.L.,Shiau,T.H.:Lipschitzcontinuityofsolutionsoflinearinequalities,programsandcomplementarityproblems.SIAMJ.ControlOptim.,583Ð595(1987)18.Meyer,R.R.:Thevalidityofafamilyofoptmizationmethods.SIAMJ.Control,41Ð54(1970)19.Mohebi,H.,Naraghirad,E.:Closedconvexsetsandtheirbestsimultaneousapproximationpropertieswithapplications.Optim.Lett.(4),313Ð328(2007)20.Mordukhovich,B.S.:VariationalAnalysisandGeneralizedDifferentiation.I:BasicTheory.GrundlehrenSeries(FundamentalPrinciplesofMathematicalSciences).Vol.330,Springer,Berlin(2006)(584p)21.Mordukhovich,B.S.:VariationalAnalysisandGeneralizedDifferentiation.II:Applications.GrundlehrenSeries(FundamentalPrinciplesofMathematicalSciences).Vol.331,Springer,Berlin(2006)(592p)22.Phelps,R.R.:ConvexFunctions,MomnotoneOperatorsandDifferentiability.2ndedn.Springer,Berlin(1993)23.Robinson,S.M.:Boundsforerrorinthesolutionsetofaperturbedlinearprogram.LinearAlgebra,69Ð81(1973)24.Tardella,F.:Existenceandsumdecompositionofvertexpolyhedralconvexenvelopes.Optim.(3),363Ð375(2008)25.Zagrodny,D.:Ontheweak*convergenceofsubdifferentialsofconvexfunctions.J.Conv.Anal.213Ð219(2005)26.Zagrodny,D.:MinimizersofthelimitofMoscoconvergingfunctions.Arch.Math.(5),440Ð445(2005)27.Zagrodny,D.:Aweak*approximationofsubgradientofconvexfunction.ControlCybern.793Ð802(2007)28.Zùalinescu,C.:SharpestimatesforHoffmanÕsconstantforsystemsoflinearinequalitiesandequali-ties.SIAMJ.Optim.,517Ð533(2003)29.Zheng,X.-Y.,Ng,K.-F.:HoffmanÕsleasterrorboundsforsystemsoflinearinequalities.J.Glob.(4),391Ð403(2004)