wwwacademicpresscomHigherordergeneralizedinvexityanddualityinnondifferentiablemathematicalShashiKMishraandNormaGRuedaDepartmentofMathematicsFacultyofEngineeringandTechnologyRBSCollegeBichpu ID: 849840
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1 J.Math.Anal.Appl.272(2002)496506 www.ac
J.Math.Anal.Appl.272(2002)496506 www.academicpress.comHigher-ordergeneralizedinvexityanddualityinnondifferentiablemathematicalShashiK.MishraandNormaG.RuedaDepartmentofMathematics,FacultyofEngineeringandTechnology,R.B.S.College,Bichpuri283105,Agra,IndiaDepartmentofMathematics,MerrimackCollege,NorthAndover,MA01845,USAReceived1August2000SubmittedbyJ.P.Dauer AbstractInthispaperweconsideranumberofhigher-orderdualstoanondifferentiablepro-grammingproblemandestablishdualityunderthehigher-ordergeneralizedinvexityconditionsintroducedinanearlierworkbyMishraandRueda.2002ElsevierScience(USA).Allrightsreserved. 1.IntroductionMond[4]consideredtheclassofnondifferentiablemathematicalprogrammingproblemssubjectto wherearetwicedifferentiablefunctionsfrom,respec-tively,andisanpositivesemi-de nite(symmetric)matrix.Letsatisfy Correspondi
2 ngauthor.E-mailaddress:norma.rueda@merri
ngauthor.E-mailaddress:norma.rueda@merrimack.edu(N.G.Rueda).ResearchsupportedbytheUniversityGrantsCommissionofIndiathroughgrant#:F8-33/20010022-247X/02/$seefrontmatter2002ElsevierScience(USA).Allrightsreserved.PII:S0022-247X(02)00170-1 S.K.Mishra,N.G.Rueda/J.Math.Anal.Appl.272(2002)496506(1.1).Mond[4]de nedtheset0if0ifwhere,andestablishedthefollowingnecessaryconditionstobeanoptimalsolutionto(NDP).Proposition1.1.1If0isanoptimalsolutionofandthecorrespondingsetisempty,thenthereexist,andsuchthat WeshallmakeuseofthegeneralizedSchwarzinequality[6]Thesecond-orderMangasariantype[2]andMondWeirtype[5]dualsto(NDP)weregivenbyBectorandChandra[1]asthefollowingproblems:ND2MD subjectto whereND2D subjectto Usingthesecond-orderconvexi
3 tyconditionBectorandChandra[1]estab-lish![tyconditionBectorandChandra[1]estab-lish](849840/tyconditionbectorandchandra-1-estab-lish.jpg "tyconditionBectorandChandra[1]estab-lish")
tyconditionBectorandChandra[1]estab-lisheddualitytheoremsbetween(NDP)and(ND2MD)and(ND2D),respectively.TheMangasariantype[2]andMondWeirtype[5]higher-orderdualto(NDP)weregivenin[7]asfollows: S.K.Mishra,N.G.Rueda/J.Math.Anal.Appl.272(2002)496506 subjecttowheresubjectto Dualityresultsareestablishedunderhigher-orderinvexityandgeneralizedhigher-orderinvexityassumptionsbetween(NDP)and(NDHMD)and(NDHD),asin[7],respectively.DeÞnition1.1.Theobjectivefunctionandtheconstraintfunctions ,aresaidtobehigher-ordertypeIatwithrespecttoafunctionif,forall,thefollowinginequalitieshold: DeÞnition1.2.Theobjectivefunctionandtheconstraintfunctions ,aresaidtobehigher-orderpseudo-quasitypeIatwithrespectt
4 oafunctionif,forall,thefollowingimplicat
oafunctionif,forall,thefollowingimplicationshold: S.K.Mishra,N.G.Rueda/J.Math.Anal.Appl.272(2002)4965062.Mangasariantypehigher-orderdualityInthissectionweobtaindualityresultsbetween(NDP)and(NDHMD).ThefollowingtheoremgeneralizesTheorem4.4.1givenbyZhang[7]tohigher-ordertypeIfunctions.Theorem2.1(weakduality).befeasibleforandletfeasiblefor.If,forallfeasible,thereexistsafunctionsuchthat thenin mum(NDP)supremum(NDHMD)Proof. The rstinequalityfollowsfrom(2.1),theequalityfollowsfrom(1.3),andthesecondinequalityfollowsfrom(2.2)andSince1,bythegeneralizedSchwarzinequality(1.2),itfollowsthat Theore
5 m2.2(strongduality).bealocalorglobalopti
m2.2(strongduality).bealocalorglobaloptimalsolutionofwithcorrespondingsetemptyand Thenthereexistsuchthatisfeasibleforandthecorrespondingvaluesofareequal.Ifalsoaresatis edatforallfeasible,then S.K.Mishra,N.G.Rueda/J.Math.Anal.Appl.272(2002)496506areglobaloptimalsolutionsforrespectively.Proof.Sinceisanoptimalsolutionto(NDP)andthecorrespondingsetempty,thenfromProposition1.1,thereexistsuchthat Then,using(2.3),wehavethatisfeasiblefor(NDHMD)andthecorrespondingvaluesof(NDP)and(NDHMD)areequal.If(2.1)and(2.2)aresatis edthenfromTheorem2.1mustbeanoptimalsolutionfor(NDHMD).Wenowshowthatweakdualitybetween(NDP)and(NDHMD)holdsunderweakerhigher-ordertypeIconditionsthanthosegiveninTheorem2.1.ThefollowingtheoremisageneralizationofTheorem4.4.3givenbyZhang[7].Theorem2.3(weakduali
6 ty).befeasibleforandletfeasible
ty).befeasibleforandletfeasiblefor.If,forallfeasible,thereexistsafunctionsuchthat Thenin mumsupremumProof.From,wehaveHence,by(2.4),wehave Sinceisfeasiblefor(NDHMD),weget Then,by1andthegeneralizedSchwarzinequality(1.2)itfollowsthat S.K.Mishra,N.G.Rueda/J.Math.Anal.Appl.272(2002)496506Remark2.1. thentheconditions(2.1)and(2.2)aresuf cientfortobesecond-ordertypeI.Remark2.2.Thefollowingexample(see[3])showsthatcondition(2.4)isweakerthan(2.1)and(2.2).Consider0,andde nedasinRemark2.1.Condition(2.4)issatis edatforany,whileconditions(2.1)and(2.2)aresatis edonlywhenthecomponentsofarenonnegative.
7 3.MondÐWeirtypehigher-orderdualityWenowc
3.MondÐWeirtypehigher-orderdualityWenowconsideraMondWeirtypehigher-orderdualto(NDP)asin[7]:subjectto ThefollowingtheoremisageneralizationofTheorem4.4.4givenbyZhang[7]tohigher-ordertypeIfunctions.Theorem3.1(weakduality).befeasibleforandletfeasiblefor.If,forallfeasible satisfytheconditionsofTheorem,respectively,thenin mumsupremumProof.by(2.1)by(3.1) S.K.Mishra,N.G.Rueda/J.Math.Anal.Appl.272(2002)496506 by(2.2)andby(3.2)Since1,bythegeneralizedSchwarzinequality(1.2),itfollowsthatThefollowingstrongdualitytheoremfollowsalongthelinesofTheorem2.2.Theorem3.2(strongduality).bealocalorglobaloptimalsolutionofwithcorrespondingsetemptyandletconditionbesatis ed.Thenthereexistsuchthat
8 isfeasibleforandthecorrespondingvalueso
isfeasibleforandthecorrespondingvaluesofareequal.Ifalsoaresatis edatforallfeasible,thenareglobaloptimalsolutionsfor,respec-tively.Weakerconditionsunderwhich(NDHD)isadualto(NDP)canalsobeobtained.ThefollowingisageneralizationofTheorem4.4.6givenbyZhang[7]tohigher-orderpseudo-quasitypeIfunctions.Theorem3.3(weakduality).befeasibleforandletfeasiblefor.If,forallfeasible,thereexistsafunctionsuchthat0(3.3) thenin mumsupremumProof.Sinceisfeasiblefor(NDHD),thenby(3.2),wehave by(3.4)by(3.1)by(3.3) S.K.Mishra,N.G.Rueda/J.Math.Anal.Appl.272(2002)496506Since1,bythegeneralizedSchwarzinequality(1.2),itfollowsthatRemark3.1.
9 then(NDHD)becomes(ND2D).4.GeneralMondÐ
then(NDHD)becomes(ND2D).4.GeneralMondÐWeirtypehigher-orderdualityInthissection,weconsiderthefollowinggeneralhigher-orderdualto(NDP):subjectto where ,withTheorem4.1(weakduality).befeasibleforandletfeasiblefor.If,forallfeasible,thereexistsafunctionsuchthat S.K.Mishra,N.G.Rueda/J.Math.Anal.Appl.272(2002)4965060(4.1) Thenin mumsupremumProof.Sinceisfeasiblefor(NDHGD),wehave,forall by(4.2)sinceisfeasiblefor(NDHGD) S.K.Mishra,N.G.Rueda/J.Math.Anal.Appl.272(2002)496506Since1,bythegeneralizedSchwarzinequality(1.2),itfollowsthatTheproofofthefo
10 llowingstrongdualitytheoremfollowsalongt
llowingstrongdualitytheoremfollowsalongthelinesofTheorem2.2,thereforewestatetheresultbutomittheproof.Theorem4.2(strongduality).bealocalorglobaloptimalsolutionofwithcorrespondingsetemptyandletconditionbesatis ed.Thenthereexistsuchthatisfeasibleforandthecorrespondingvaluesofareequal.Ifalsoaresatis edatforallfeasible,thenareglobaloptimalsolutionsforrespectively.Remark4.1.,then(NDHGD)becomes(NDHMD)andtheconditions(4.1)and(4.2)ofTheorem4.1reducetothecondition(2.4)ofTheorem2.3.forsome ,then(NDHGD)becomes(NDHD)andtheconditions(4.1)and(4.2)reducetotheconditions(3.3)and(3.4),respectively,ofTheorem3.3.Theorem4.3(strictconverseduality).beanoptimalsolutionofwithcorrespondingsetempty.Letconditionsbesatis edat,andletconditionsofTheorembesatis edforallfeasible.Ifisanoptimalsolutionofandif,forall
11 S.K.Mishra,N.G.Rueda/J.Math.Ana
S.K.Mishra,N.G.Rueda/J.Math.Anal.Appl.272(2002)496506then,i.e.,solvesReferences[1]C.R.Bector,S.Chandra,Secondorderdualitywithnondifferentiablefunctions,workingpaper[2]O.L.Mangasarian,Secondandhigher-orderdualityinnonlinearprogramming,J.Math.Anal.Appl.51(1975)607620.[3]S.K.Mishra,Secondordergeneralizedinvexityanddualityinmathematicalprogramming,Optimization42(1997)5169.[4]B.Mond,Aclassofnondifferentiablemathematicalprogrammingproblems,J.Math.Anal.Appl.46(1974)169174.[5]B.Mond,T.Weir,Generalizedconvexityandhigherorderduality,J.Math.Sci.1618(19811983)7494.[6]F.Riesz,B.Sz.-Nagy,FunctionalAnalysis,FrederickUngarPublishing,NewYork,1955,trans-latedfromthe2ndFrencheditionbyL.F.Boron.[7]J.Zhang,Generalizedconvexityandhigherorderdualityformathematicalprogrammingproblems,Ph.D.thesis,LaTrobeUniversity,Australia(1