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GENERALIZED

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GENERALIZEDGENERA

GENERALIZED

App1.Math.JCU

;14B(1999),8589

;GENERALIZEDKTCoNDITIoNSAND

;PENALTYFUNCTIoNSFORQUASIDIFFERENTIABLE

;PRoGRAMMING

;YinHongyouXuChengxian

;Almb-set.Inthispaper+aquasidifferemiableprogrmmmngproblemwithinequaUtyconstraints ;isconsidered.First,ageneralformofoptimalityconditionsforthisproblemissiven+whichcon- ;tainstheresultsofLuderer.KuntzandSchohes.Next.anewgeneralizedK-Tconditioni3de- ;rioed.Thenewoptimalityconditiondoesn~tUgeLuderer‟sregularityassumptionandjt3La-

;grangianmultipliersdon‟tdependOntheparticularelementsinthesuperdifferentialsoftheob

;jeetfunctionandconstraintfunctions.Finally,apenaltyfunctionfortheproblemisstudied.Suf- ;ficicntconditionsofthepenaltyfunctionattainingglobalminimumareobtained. ;?1Introduction

;LetCCRbeanonemptyopenset.Werecallthedefinitionofquasidifferentiablefunc- ;tionsinthesenseofDamyanovandRubinov.Thefunctionf:CRissaidtobequasidif

;ferentiableatECiffisdirectionallydifferentiableatzandthereexistthecompactcon

;vexsetsOf(x),Of(x)CRsuchthat

;,;,)=d(,lOf(x))d(?l一百,(z)),

;whered(?lA)~n3ax{(?,):?A)isthesupportfunctionofthecompactconvexsetA.

;Theorderedpairofsets(Of(x),Of(x))iscalledquasidifferentialofthefunctionfatz. ;Of(x)andOf(x)arecalledsubdifferentialandsuperdifferentialoffatzrespectively.The ;functionfissaidtobeaquasidifferentiablefunctiononCiffisquasidifferentiableateach ;pointofC.

;Considerthefollowingquasidifferentiableprogrammingproblemwithinequalitycon- ;straints:

;?

;St

;

;?0,i1…,I..(z)?,=….,m,

;wheref,,i1….,m,arequasidifferentiablefunctionsonR.

;1991MRSuhjectClassification:49K30,90C30.

;Keywords:Quaaidifferentiableprogrammingproblems,generalizedK-Tconditions,peaahyfunctions. ;

;86App1.Math.JCL~Vo1.14,Set.B

;AgeneralizedKuhnTuckernecessaryconditionofproblem(P)(inshort,generalized ;KTcondition)wasgivenbyLudererwithalocalsolutionsatisfyingaregularity~di

;tion[z3.

;DeficiencyofLudererNresultisthat,first,theregularityconditionistOOstrongand ;dependsonthechoiceofthequasidifferentialsoftheconstraintfunctionsinproblem(P)} ;s‟econd,inthegeneral~edK—TconditionLagrangianmultipliersarenotgroupofdefinite ;numbers,theyarechangedwiththechangeofelementsinsuperdifferentials.Recently‟un—

;deraweakerconstraintqualificationKuntzandSch0ltesgaveaequivalentformof ;Luderer‟sresuItc.

;Thispaperwillfirstgiveamoregeneralformofnecessaryconditionsofproblem(P), ;whichc0ntainstheresultsofLuder~r,KuntzandScholtesNext,weshallderivenew ;generalizedK—Tcondition.Thenewoptimalityconditiondoesn‟tuseLuderer~regularity

;assumptionanditsLagrangianmultipliersareindependentoftheparticularelementsinso

;perdifferentials.Finally,weshallalsostudypenaltyfunctionproblemsofproblem(P). ;?2NewGeneralizedK-TConditions

;Letg(z)max{g,(z):i1….,).Foranyfeasiblepointxofproblem(P)wede—

;noteJ)f:g.)0,i=1….,m}.

;Theorem2.1.Letx.bealocalsolutionofproblem(P),(3f(x),Of(x.))aquasidiffer

;entialoffatx,(a(z),百毋(z.))aquasidifferentialohedirectionalderivativeg(z‟;.)at

;thepoint0143.

;Proof.Sincexisalocalsolutionofproblem(P),fromaresultinIs]wehave ;0?co~(3f(x)+?.)U(g(z.,0)ncoU(Og.)+))].

;,?,‟)

;Therefore,thereexistthenonnegativenumbersr,rn,ri,i?I(x),whicharerelativeto

;,,

;i?.

;Theorem2.2.UndertheassumptionsofTheorem2.1,thereexisttheconstants?O,i?

;I(x),suchthat

;

;z+

;l(x

;.?

;U?，如)+,(?f?)‟

;forall【?,(),?,().

;1‟roof.Since,()+Of()andaag(z;0)arecompactconvexsetsinRand0

;g(z;O),thereexist??O,d&gt;0suchthat

;11catesthatthereexistl?af(x)andz?巩g({O)suchthat

;+w.+(?)0.

;I?扭

;Theab0Veequa1itytogetherwith(2.3)and(2.4)yields?)?M/d,Theref.re,

;forall.?百()and,?,(),?l(x),thenonnegativemultipliersatisfying

;(2.5)isboundedfromabove,?,().WedenotetheleastUp.),

;areindependentoftheparticularelementsinsuperdifferentials. ;?3PenaltyFunctionProblems

;Inthissection.westudypenaltyfunctionproblemsofproblem(P).Forthisaimwe ;needthefollowingdefinitionsandtheorem.

;Definition3.1.[LetxbeafeasiblepointofproexfunctionfisdirectionallydifferentiableonRthen

;,(),()?,(;(,))foral1x,?R.

;Considerthepenaltyfunction

;P(x,P),()+P?max{g.(),o),v?R,P&gt;0.r1

;Theorem3.2.Letfandg.,i1….,m,bequasidifferentiablefunctionsonRandprein— ;vexwithrespecttothesame,let(,)beageneralizedKTpairofproblem(P).Then ;thereexistsJD&gt;OsuchthatP(x,P)attainsaglobalminimumat‟forallJD?JD..

;Proof.Since(z,)isageneralizedKTpairofproblem(P),inviewofTheorem3.1we

;have

;

;No.1YinHongyou.eta1.QUPROGRAMMING89 ;,(„;(,‟))+:

ax{:(„{7/(,‟)),o}?0,V?R..

;LP‟&gt;max{:iEl(x‟)}.Notethat‟isfeasibleandthefunctj.ns,,,fl….,,

;arepreinvexonwithrespectto.ThusforallP?P‟webave

;P,P)P(x‟,P),()f(x‟)+P?_-,max{g,(x),0}?‟

;,()f(x‟)+P?.?m.,max{g()一毋(„),0}?

;,(z‟;(z,z‟))+P?(.,raax{gl(„{7/(x,X‟)),0)?

;f(x‟;(z,z‟))+?.a,max(z‟;7/(x,‟)),0}?0,

;thatis,P(x,P)attainsglobalminimumatz-. ;References

;lDemy”.,V?F?andRuhinov,A.

;M.,QuasidifferentialCalculus,Optimi?tions0flre,wYo,

;1986.

;2Ludr‟B?,Directionalderivativeestimatesfortheoptimalvaluefun

;ctionofqua3idiHenliablep廿

;grammingpioblem,Math.Programming,1991,51(3)l333~348.

;3K”“t,L,Scholtes‟S?,Constraintqualificationsinquasidifferentiableoptimiz

;ation,Math.Pmam,

;ruing,199a,60(3):339~347.

;4Clarke‟F?H?,OptimizationandNonsmoothAnalysis,W/ley—Intersc/enee,NewYork,1983.

;5M”,H?andZhang?K?C.,Quasidifferentlahleprogrammmg:optimalcondltnsaI1deneraIized ;KTpairs,J.ShaanxiNormalUniv,1995,23(\$up.):100~105.

;6Jeyk”,W?‟StrongandweakinvexityinmathematicaIpr.gramng

;,Methodsof0perionsRe-

;SchoolofScience,Xi‟anJiaotongUniv.,Xian710049.

;

;

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