Necessary Optimality Conditions for a Class of Nonsmooth Vector Optimization Problems Xuan-wei ZHOU *

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1 28 Iteratioal Coferece o Modelig, Siulatio ad Otiizatio (MSO 28) ISBN: Necessary Otiality Coditios for a Class of Nosooth Vector Otiizatio Probles Xua-wei ZHOU * School of Basic Courses, Zheiag Shure Uiversity, Hagzhou 35, Chia *Corresodig author Keywords: Vector otiizatio, Necessary otiality coditios, Nosooth otiizatio, Clarke's geeralized directioal derivative. Abstract. A class of osooth vector otiizatiorobles are cosidered, where the feasible set defied by coe costrait ad the obective ad costrait fuctios are locally Lischitz. he cocet of the Clarke's geeralized directioal derivative for a locally Lischitz fuctio is itroduced. By usig this cocet, ecessary otiality coditio for the ucostraied otiizatioroble is established. Furtherore, a Slater-tye costrait qualificatio is give i such a way that they geeralize the classical oe, whe the costrait fuctios are differetiable. he, Kuh ucker ecessary otiality coditio i ters of the Clarke subdifferetials is obtaied. Itroductio Vector otiizatio is a extesio of atheatical rograig where a scaler valued obective fuctio is relaced by a vector fuctio. Necessary otiality coditios for this kid of rograig are studied by several authors i the sooth ad osooth cases [ ]. Vector otiizatiorobles with differetiable fuctios is cosidered i [2] ad the the Kuh ucker coditio for a Pareto efficiet solutio of a vector fuctio over a feasible set defied by iequality costraits is obtaied. Cosiderig quasicovex ad directioally differetiable fuctios, the result obtaied by [2] is exteded i [3]. A vector otiizatioroble with oly iequality costraits ad suose that the ivolved fuctios are locally Lischitz is studied i [4]. he K otiality coditios is give uder a geeralized Abadie tye costrait qualificatio assuig that the Clarke subdifferetials of the obective ad costrait fuctios are olytoes. he result of [4] is also exteded i [5] by cosiderig locally Lischitz fuctios, Fréchet differetiable equality costraits, ad a abstract set costrait. hese authors establish the K otiality coditios uder a exteded geeralized Guigard costrait qualificatio is established, but i this case the obective fuctio is assued to be Fréchet differetiable. I this aer, the coe efficiet solutio of a class of osooth vector otiizatiorobles is cosidered, where the feasible set defied by coe costrait ad the obective ad costrait fuctios are locally Lischitz. By usig Clarke's geeralized directioal derivative, ecessary otiality coditio for the ucostraied otiizatioroble is established. Furtherore, soe costrait qualificatios are give i such a way that they geeralize the classical oe, whe the costrait fuctios are differetiable. he, Kuh ucker ecessary otiality coditio for the coe efficiet solutio i ters of the Clarke subdifferetials is obtaied. Notatios ad Preliiaries Cosider the followig vector iiizatioroble: V i f( x) ( f( x),, f ( x)), (VMP) stg.. ( x) K, where f ( f,, f ): R R, g ( g,, g): R R satisfy local Lischitz coditios, 96

2 K K R ad R are closed covex coes with oety iterior. Let D { x R g( x) K } be the feasible set, ad deote by it K the iterior of K. Defiitio he weakly local efficiet solutio is with resect to the coe K, thus x D is a weakly local efficiet solutio of (VMP) if, for soe, f ( x) f( x) itk wheever x D ad x x. he fuctios f ad g are ot assued differetiable; the Lischitz hyothesis, with Radeacher's theore, esures that they are differetiable excet o a set of zero easure. For a set S R ad oit x S, defie Sx { ( x x ), x S}. he dual coe of S is defied by S* { R s, s S}. Note that S * is a closed covex coe, eve though S eed ot be. It is well kow that, sice it K, the dual coe K * has a coact covex base B, Lea If K is a closed covex coe with oety iterior, B is a coact covex base of the dual coe K *, ad y R satisfies y ( B), the y it K. Proof If y ( B), the y K by the covex sets searatio theore; if y it K, the y is a geerator of the coe K, thus ( y) for soe ozero for which y ( y K), therefore K *; cosequetly y it K. For a locally Lischitz fuctio h at the oit x, Clarke's geeralized directioal derivative i the directio h is deoted by hx ( tv) hx ( ) h ( x, v) lisu, x x t t ad Clarke's geeralized gradiet is deoted by hx ( ) { R h( x, v) v, v R}. For a Lischitz vector fuctio f : R R, f ( x) is the covex hull of all liits of sequeces f ( x ), where x x ad the gradiet f ( x ) exists at x. Fro [6,heore 2.6.6], ( f )( x) ( f)( x) for each liear fuctioal. he followig lea will be required. Lea 2 Let A R be oety; let B R be coact, with B ; let P coeb ; let f : A R R be cotiuous, with ( v R ) f(, v) covex, ad ( u A) f( u, ) cocave ad ositively hoogeeous. he exactly oe of the followig systes is cosistet: () ( u A)( v P\{}) f( u, v) ; (2) ( v P\{})( u A) f( u, v). Proof It is a iediate secial case of [7], heore 2.. Now the cotraositive of () is(3) ( u A)( v P\{}) f( u, v). It follows that (3) holds if ad oly if (2) holds. Lea 3 Let K R be a closed covex coe with oety iterior, the dual coe K *, B a coact set with B ad K* coeb. If, for soe directio v R, ad all B, the geeralized gradiet ( f) ( x, v), the there is ( B) such that ( K*\{})( ' t (, ( B))) ( f( x tv) f( x)), where the sybol ( 't ) deotes "for all t excet a set of zero easure". Proof Suose that ( v R \{})( K*)( f) ( x, v), Usig Fubii's theore, there is soe ( ) such that ( t [, ( )]) f( x tv) exists excet a set of zero easure. Sice ( f )( x ) f( x ), 97

3 ( B)( M f( x )) Mv. Sice f ( x ) cosists of all covex cobiatios of liits of f at oits x x, t 2 f x tv v,where 2 ( ' [, ( )]) ( ) ( ) ( ). Hece ( B)( ' t [, 2( )]) ( f ( x tv) f( x)) f ( x tv) vdt. Sice B is coact, ( ) 2 ay be relaced by ( B ), ideedet of B. he ( K*\{})( ' t (, ( B))) ( f( x tv) f( x )), Otiality Coditios I the followig ecessary otiality coditio for the ucostraied otiizatioroble is established, ad Fritz Joh ad Kuh ucker ecessary otiality coditios i ters of the Clarke subdifferetials are obtaied. heore (Ucostraied ecessary otiality coditio) If f : R R satisfies local Lischitz coditio, ad x is a local weakly efficiet solutio with resect to the coe K, the, there exists B such that f ( x). Proof Suose that, for soe directio v R \{}, ad all B, there holds ( f) ( x, v). Fro Lea3, there is a sequece { t }, ad all B, ( f( x tv) f( x)). Hece, fro Lea, ( f ( x tv) f( x)) it K, for, 2,, cotradictig x the local weakly efficiet solutio with resect to the coe K. Cosequetly, ( v R \{})( K*\{})( f) ( x, v). Now the hyotheses of the Lea2 hold here with f (, v) ( f) ( x, v), sice this fuctio is covex i v, ad also cocave (ad ositively hoogeeous ) i. herefore ( *\{})( K v R \{})( f) ( x, v). Cosequetly Clarke's geeralized gradiet satisfies ( f )( x) f( x) for soe K *\{}, ad hece for soe B fro K* coeb. heore2 (Fritz Joh otiality coditio) If f : R R, g : R R satisfy local Lischitz coditios, K R ad K R are closed covex coes with oety iterior, ad x is a local weakly efficiet solutio with resect to the coe K ad the costraied coditio is g( x) K, the, there exist Lagrage u tiliers K * ad K *, ot all zero, such that f ( x) g( x), gx ( ). Proof Let F ( f, g) ; let Q K ( K ) g ( x ). Suose that ( v R \{})( Q*\{})( F) ( x, v). Fro Lea 3, we have ( Q* \{})( ' t [, ( )]) ( F( x tv) F( x)). Iarticular, fixig B, ad settig (, ) with, the ( ' t [, ( )]) ( g( x tv) g( x)), where ( ) ay be take ideedet of i the coact set A. he, fro Lea, g( x tv) g( x) itk. So that g( x tv) it K g( x ) itk K K 98

4 By a siilar arguet, settig (, ) with, the ( ' t [, ( )]) ( f( x tv) f( x )), 2 where 2 ( ) ay be take ideedet of i the coact set B. he, fro Lea, f ( x tv) f( x ) itk. hus, for a sequece of t, g( x tv) K ad f ( x tv) f( x) itk, cotradictig that x is a local weakly efficiet solutio. herefore ( v R \{})( Q*\{})( F) ( x, v). A alicatio of Lae2 shows, as i the roof of heore, that there exists (, ) K * K * such that ( F) ( x, v) ( f g) ( x, v). herefore, for this (, ) K * K *, f ( x) g( x). here reais that the case gx ( ) is obvious. Now we obtai a ecessary coditio of Kuh-ucker tye. Firstly, we give a costrait quaificatio that esures that. he followig Slater-tye costrait qualificatio does this. Slater-tye costrait qualificatio : ( G g( x))( x R ) G( x) it K. heore 3 (Kuh-ucker otiality coditio) If f : R R, g : R R satisfy local Lischitz coditios, K R ad K R are closed covex coes with oety iterior, x is a local weakly efficiet solutio with resect to the coe K ad the costraied coditio is g( x) K, ad Slater-tye costrait qualificatio : ( G g( x))( x R ) G( x) it K holds, the, there exist Lagrage utiliers K * \{} ad K *, such that ( ) ( ) f x g x, gx ( ) Proof Assue that (, ), ad( G g( x))( x R ) G( x) it K. Suose, if ossible that ; the fro the hyothesis, ad G for soe G g( x ) ), fro heore 2. he Gx ( ) by Slater-tye costrait qualificatio, cotradictig the hyothesis. herefore, this is K *\{}. Coclusio his aer resets a ew tye of osooth vector otiizatiorobles. his ew vector otiizatiorobles exted Pareto efficiet solutio to coe efficiet solutio ad oly iequality costraits to coe costrait. he obective ad costrait fuctios are locally Lischitz that geeralizes the classical oe, whe the obective ad costrait fuctios are differetiable. he ew cocet of the Clarke's geeralized directioal derivative for a locally Lischitz vector fuctio is itroduced. By usig this cocet, Kuh ucker ecessary otiality coditio for the coe efficiet solutio i ters of the Clarke subdifferetials is obtaied. As oited out by a aoyous referee, we will study this tye of osooth vector otiizatiorobles by cosiderig locally Lischitz fuctios, Fréchet differetiable equality costraits, locally Lischitz iequality costraits ad a abstract set costrait. We will study ew costrait qualificatio that has a sigificat role i otiizatiorobles. By usig this costrait qualificatio, Kuh ucker ecessary otiality coditio for the coe efficiet solutio will be give. Refereces [].Q. Bao, P. Guta, B.S. Mordukhovich, Necessary coditios i ultiobective otiizatio with equilibriu costraits, J. Oti. heory Al. 35(2) (27) [2].Q. Bao, B.S. Mordukhovich, Variatioal riciles for set-valued aigs with alicatios to ultiobective otiizatio, Cotrol Cyberet. 36(3)(27)

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