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Generalized Pseudoconvex Functions and Multiobjective Programming JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 208, 49]57Ž. 1997 ARTICLE NO. AY975281 Generalized Pseudoconvex Functions and Multiobjective Programming R. N. Mukherjee Department of Applied Mathematics, Institute of Technology, Banaras Hindu Uni¨ersity, View metadata, citation and similar papers atVaranasi, core.ac.uk 221 005, India brought to you by CORE provided by Elsevier - Publisher Connector Submitted by E. Stanley Lee Received June 1, 1995 In a recent work, cited in the Introduction, a concept of generalized pseudocon- vexity was used to obtain optimality results in nonlinear programming. In the present work we give sufficient optimality conditions, in the context of the multiobjective programming problem under the assumptions of generalized pseu- doconvexity on objective and constraint functions. An application of such a result is given for fractional programming also. Q 1997 Academic Press 1. INTRODUCTION In a work which appeared in 1969, Tanaka, Fukushima, and Ibarakiwx 11 introduced generalized pseudoconvexity and connected that with the con- cept of invexity and arcwise pseudoconvexity. In fact the concept of invexity existed earlier in the literature, e.g., Hansonwx 6 used it to give Kuhn]Tucker sufficient optimality conditions in nonlinear programming. The concept of arcwise convexity was introduced by Avriel and Zhangwx 1 , and later Singhwx 10 derived some analytical properties of arcwise convex functions and used them to characterize local global minimum properties for problems in nonlinear programming. Inwx 9 as a further extension, the concept of an arcwise pseudoconvex function was introduced. Subse- quently the unification of the three concepts like generalized pseudocon- vexity invexity and arcwise pseudoconvexity was done and under certain assumptions their equivalence was shown. In the same work optimality conditions for the nonlinear programming problem were given under the assumption of generalized pseudoconvexity. 49 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved. 50 R. N. MUKHERJEE In the present work our aim is to derive sufficient optimality conditions for a multiobjective programming problem under such generalized pseudo- convexity assumptions. Applications are given in the context of fractional programs. In the organizational set up of the paper we introduce the concepts as cited in the previous paragraph in Section 2. The unification results are mentioned thereafter. A theorem is quoted fromwx 9 which gives equiva- lence of the concepts and the underlying assumption in such an equiva- lence result. In Section 3 a multiobjective programming problem is intro- duced and sufficient optimality results are given. In Section 4 such results are used to consider the multiobjective fractional programming problem. Here a standard reference is made for ``efficient solutions'' to replace ``minimal solutions''Ž. for the single objective problem for the case of the multiobjective programming problem. 2. PSEUDOCONVEX FUNCTIONS AND THEIR GENERALIZATIONS n Let f: R ª R be a real valued function of n-variables. For a differen- tiable function f the following definition is standard for pseudoconvex functions. n DEFINITION 2.1. Let S ; R be a convex open set. Then f: S ª R is pseudoconvex on S, if for all x, y g S T =fyŽ.Ž xyy .G0 implies fxŽ.Gfy Ž.. In case the second inequality is replaced by fxŽ.)fy Ž., for all x / y, x, y g S then f is said to be strictly pseudoconvex on S. To introduce invexity we give the following preliminaries. n Let f: R ª R be a locally Lipschitz continuous function, i.e., for each n xgRthere exists d ) 0 and c ) 0 such that 5 fyŽ.yfz Ž.5<Fyyz <, when ever 55x s y - d and 55z y x F d. f is said to be regular in the sense of Clarke ifŽ. i for all d, there exists the one-sided directional derivative fxXŽ.;d lim fyŽ.Ž.td fx t;Ž. ii for all d, fxXŽ.;d s tx0wq y xr s fx0Ž.;d, where fx0Ž.;dis the generalized directional derivative defined by fyŽ.Ž.td fy 0 q y fxŽ.;dslim sup . yªx t tx0 It then follows that X T fxŽ.;dsmaxÄ4j d< j g ­ fxŽ., for any x and d,2.1Ž. where ­ fŽ.? denotes Clarke's generalized gradientwx 9 . GENERALIZED PSEUDOCONVEX FUNCTIONS 51 n DEFINITION 2.2. A function f: R ª R is said to be invex if there exists n n n n a mapping h: R = R ª R such that, for each x, u g R , X fxŽ.yfu Ž.GfŽ.u;h Žx,u ..2Ž..2 Note that if f is differentiable thenŽ. 2.2 reduces to the definition of invexity given inwx 5 . The result which connects global optimality and invexity is the following. THEOREM 2.1. The function f is in¨ex if and only if e¨ery point u such that 0g­fŽ. u is a global minimum of f. DEFINITION 2.3. A function f is essentially pseudoconvex if there exists n n n a function c : R ª R and a diffeomorphism T: R ª R such that f s c (T and for any x and x X c Ž.z; z y z G 0 implies that c Ž.z G c Ž.z ,2.3 Ž . where z s TxŽ.and z s Tx Ž.. For T s I Ž.identity mapping the above definition reduces to the ordi- nary definition of pseudoconvexity for directionally differentiable func- tions. Further if a function is differentiable, then finally the above defini- tion will lead to the classical definition of a pseudoconvex function. n n DEFINITION 2.4. For u, x g R , a continuous mapping Pux:0,1wxªR is called an arc from u to x if PuxŽ.0 s u, PuxŽ.1 s x. Ž. Ž . Ž. PtÇÇux exists and is continuous for any t g 0, 1 , and Ptux /0 for any t such that PtuxŽ. is not a global minimum of f. PÇuxŽ.0 is defined as lim PtÇ Ž.. qtx0 ux DEFINITION 2.5. The function f is called arcwise pseudoconvex if for n Ž. each u, x g R , there exists an arc Pux ? such that X fPŽ.uxŽ. t,PtÇ ux Ž. G0 implies fPŽ.uxŽ. t GfPŽ.ux Ž. t for any 0 F t F t - 1.Ž. 2.4 Ž. Ä Ž. 4 Ž. Lf a s x: fxFa stands for the level set of a function F ? , for each a g R. The following assumption is made in some of the equivalence criteria in the sequel which follows. 52 R. N. MUKHERJEE Assumption Ž.A . There exists a diffeomorphic mapping TÃ such that, for any x and x , at least one of the arcs P Ž.? satisfying Ž 2.4 . is trans- 12 xx12 Ž. Ž. ÄŽŽ.. formed into the line segment from TxÃÃ12to Tx , i.e., TP Ãx,x t <0F 4ÄŽ.Ž.Ž. 4 12 tF1s1ysTÃÃ x12qsT x <0 F s F 1. We have the equivalent characterization of various generalizations given as follows: THEOREM 2.2. Consider the following three statements. Ž.a f is essentially pseudocon¨ex Ž.b f is arcwise pseudocon¨ex Ž.c fisin¨ex. Then Ž.a implies Ž.b,and Ž.b implies Ž.c.If Assumption ŽA .holds, then Ž.a is equi¨alent to Ž.b.If f is continuously differentiable and has a compact le¨el set Ž. Ž. Ž. Lf afor each a and admits a unique minimum then b is equi¨alent to c. In the next section we introduce a multiobjective program and study the sufficient optimality conditions for such a program to have an efficient solution. 3. MULTIOBJECTIVE OPTIMIZATION PROBLEM Consider V y minŽ.fx12Ž.,fx Ž.,..., fxp Ž. Ž.3.1 Ž. n Ž. subject to gxjiF0, j s 1, 2, . , m, where f : R ª Ris1, 2, . , p n and gi: R ª R, i s 1, 2, . , m, are locally Lipschitz and regular. For Ž. Ä Ž. 4 each feasible x, we define the index set IxJi:gxi s0 . By the optimal solution ofŽ. 3.1 we mean the efficient solution. Let LxŽ.1,lstand for the lagrangian function given as p m LxŽ.1,lsÝÝtiifx Ž.q l igx i Ž. is1 is1 l m t Ý p t for g Rq and iiG 0 and s1is 1. We have the following sufficiency condition for the efficient solution of programŽ. 3.1 . m HEOREM Ž.l t Ýt l T 3.1. Let L x1, be in¨ex for iiG 0, s 1, and g Rq. UUUŽU. Let x be such that there exists l and t G 0, Ýti s 1 p m 0­tUUfxŽ.l­ Ugx Ž. U Ž.3.2 gž/ÝÝii q i i is1 is1 U U U U U U liiG0, t G 0, Ýt is 1, gx iŽ.F0, l iigx Ž.s0, Ž. 3.3 U is1, 2, . , m. Then x is an efficient solution for program Ž.3.1 . GENERALIZED PSEUDOCONVEX FUNCTIONS 53 Proof. We have U U U ­tŽ.ÝÝiifxŽ.q li­gx i Ž . U U U s­tŽ.ÝÝiifql iigxŽ.. Since a nonnegative linear combination of regular functions is regular and its generalized gradient is the sum of each generalized gradient, hence fromŽ. 3.2 and Ž. 3.3 , ppm UU UU U U ÝÝÝtiifxŽ.s tiifx Ž.q liigx Ž. is1 is1 is1 p m UU FÝÝtiifxŽ.q liigx Ž. is1 is1 p U FÝtiifxŽ.. is1 Hence fromwx 8, Theorem 2, p. 310 it follows that xUis an efficient solution of programŽ. 3.1 . Ž. THEOREM 3.2. Let L f, l be essentially pseudocon¨ex for tiiG 0, Ýt s U U 1, and li G 0, i s 1, 2, .
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