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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.

    ;Receivedf199801l4.Revised:l9980427.

    ;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>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>0.r1

    ;Theorem3.2.Letfandg.,i1….,m,bequasidifferentiablefunctionsonRandprein— ;vexwithrespecttothesame,let(,)beageneralizedKTpairofproblem(P).Then ;thereexistsJD>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‟>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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