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 , 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 also. ᮊ 1997 Academic Press

1. INTRODUCTION

In a work which appeared in 1969, Tanaka, Fukushima, and Ibarakiwx 11 introduced generalized 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 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.

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0022-247Xr97 $25.00 Copyright ᮊ 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 Žٌ 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 ␦ ) 0 and c ) 0 such that 5 fyŽ.yfz Ž.5

n DEFINITION 2.2. A function f: R ª R is said to be invex if there exists n n n n a mapping ␩: R = R ª R such that, for each x, u g R ,

X fxŽ.yfu Ž.GfŽ.u;␩ 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 0gѨfŽ. u is a global minimum of f. DEFINITION 2.3. A function f is essentially pseudoconvex if there exists n n n a function ␺ : R ª R and a diffeomorphism T: R ª R such that f s ␺ (T and for any x and x

X ␺ Ž.z; z y z G 0 implies that ␺ Ž.z G ␺ 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. Pt˙˙ux 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. P˙uxŽ.0 is defined as lim Pt˙ 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 и such that

X fPŽ.uxŽ. t,Pt˙ 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 ␣ s x: fxF␣ stands for the level set of a function F и , for each ␣ 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 Tˆ such that, for any x and x , at least one of the arcs P Ž.и satisfying Ž 2.4 . is trans- 12 xx12 Ž. Ž. ÄŽŽ.. formed into the line segment from Txˆˆ12to Tx , i.e., TP ˆx,x t <0F 4ÄŽ.Ž.Ž. 4 12 tF1s1ysTˆˆ 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 ␣for each ␣ 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,␭stand for the lagrangian function given as p m

LxŽ.1,␭sÝÝ␶iifx Ž.q ␭ igx i Ž. is1 is1 ␭ m ␶ Ý p ␶ 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.␭ ␶ Ý␶ ␭ 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 ␭ and ␶ G 0, Ý␶i s 1 p m 0Ѩ␶UUfxŽ.␭Ѩ Ugx Ž. U Ž.3.2 gž/ÝÝii q i i is1 is1 U U U U U U ␭iiG0, ␶ G 0, Ý␶ is 1, gx iŽ.F0, ␭ 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 Ѩ␶Ž.ÝÝiifxŽ.q ␭iѨgx i Ž .

U U U sѨ␶Ž.ÝÝiifq␭ iigxŽ..

Since a nonnegative linear combination of regular functions is regular and its generalized is the sum of each generalized gradient, hence fromŽ. 3.2 and Ž. 3.3 ,

ppm UU UU U U ÝÝÝ␶iifxŽ.s ␶iifx Ž.q ␭iigx Ž. is1 is1 is1 p m UU FÝÝ␶iifxŽ.q ␭iigx Ž. is1 is1 p U FÝ␶iifxŽ.. 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, ␭ be essentially pseudocon¨ex for ␶iiG 0, Ý␶ s U U 1, and ␭i G 0, i s 1, 2, . . . , m. Then if there exist ␶ and ␭ G 0 such that Ž.3.1 and Ž.3.2 are satisfied for xUU, then x is an efficient solution for program Ž.3.1 . Proof. From Theorem 2.2 essential pseudoconvexity implies invexity. Hence the result follows from Theorem 3.1. The following assumption is made for the sufficient optimality condi- tions depicted in the ensuing theorem. Ž. Ž .Ž Assumption B . The functions fi1 s1, 2, . . . , p and gjj s 1, 2, . . . , m.are essentially pseudoconvex with respect to a common map- ping T. Ž. Ž . THEOREM 3.3. Consider problem 3.1 where fii i s 1, 2, . . . , p and g Ž. Ý p␶ U 1,...,m are functions satisfying the condition that is1 iif and each g i is ␶ U Ž Ý p␶ U . essentially pseudocon¨ex for some G 0 with is1 i s 1. Also the Kuhn᎐Tucker conditions Ž.3.2 and Ž.3.3 are satisfied for xU in the feasibility domain of Ž.3.1 . Then xU is an efficient solution for program Ž.3.1 . 54 R. N. MUKHERJEE

Ž. Ý p␶ U Proof. Let x be an arbitrary feasible solution of 3.1 . Since is1 iif q Ým␭U is1 iigis regular we have p m ␶UUUUf␭gxŽ.;xx ž/ÝÝiiq ii y is1 is1 p max ␰ TŽ.x xUU␰ Ѩ␶fx Ž.U s½ y g ž/Ýii is1 m ␭ѨUUgxŽ..3.4 Ž. qÝii 5 is1 ѨŽÝ p␶ UŽU.. However, since 0 g is1iifx m UU qÝ␭ѨiigxŽ. is1 Ž. U ŽUŽU.. U Ž U . it follows from 3.4 that there exists ␰ 0g Ѩ Ý␶iifx and ␰ ig Ѩ gx i , is1,...,m, such that

m ␰UUU␭␰ Ž.x x 0.Ž. 3.5 ž/0qÝii y G is1

On the other hand since gi is regular for each i,

X UUT U U gxiiŽ.Ž.Ž.Ž.;xyx smaxÄ4␰ x y x ␰ig Ѩ gxi.3.6 ŽU.Ž.ŽU. If gxiis0, then the feasibility of x implies gxFgxis0. Since gi is essentially pseudoconvex, then it can be seen that

X U giŽ.x ; x y x F 0.Ž. 3.7 ŽU.Ž.Ž. Thus, if gxi s0 then it follows from 3.6 and 3.7 that

UT ␰iŽ.x y x F 0.Ž. 3.8a UUUŽU. However, ␭iiG 0 for all i, and in particular ␭ s 0 whenever gxi-0. Therefore byŽ. 3.5 and Ž 3.8a . we must have

UT U ␰ 0Ž.x y x G 0.Ž. 3.8b Ý p␶ XŽU . Now from the definition of is1 iifx;xyxit follows that p X UU Ý␶iifxŽ.,xyx G0. is1 GENERALIZED PSEUDOCONVEX FUNCTIONS 55

Also from Definition 2.3, it follows that

pp U ÝÝ␶iifxŽ.G ␶iifx Ž .. is1 is1

Since x was arbitrary it follows that xUis an efficient solution ofŽ. 3.1 . For the general case when T / I, we proceed as follows. The following lemma is proved inwx 9 .

LEMMA 3.4. Under Assumption Ž.B,i.e., fii and g are essentially pseudo- con¨ex with respect to a common mapping T, the feasible region of Ž.3.1 is connected. We have the following theorem. U THEOREM 3.5. Suppose Assumption Ž.B is satisfied. Then if for x in the feasible domain of Ž.3.1 the Kuhn᎐Tucker conditions are satisfied then xU is an efficient solution of Ž.3.1 . i Ž.i Proof. Let ␺0 , ␺jii s 1,..., p; js1,...,m be such that f s ␺0(T Ž. Ž . is1,..., p and giis ␺ (Tjs1,...,m . Then consider the program

p Ui minimize Ý␶␺i 0Ž.z is1

subject to ␺iŽ.z F 0, i s 1,2,..., m,3.9Ž.

U U and let z s TxŽ .Ž.. Since ٌTx is nonsingular, conditionsŽ. 3.2 hold if and only if

p m 0Ѩ␶␺UUiŽ.z ␭␺ UU Ž.z gž/ÝÝi0q ii is1 js1 U U U U U ␶iiiiG0, Ý␶ s 1, ␺ Ž.z F 0, ␭ ␺i Ž.z s 0; i s 1,2,...,m.

Then by the proof of Theorem 3.4 it follows that zU is efficient for the corresponding multiobjective program with ␺-functions. From which it easily follows that xU is an efficient solution of multiobjective program Ž.3.1 . Finally observe that the minimum set of Ž. 3.9 is convex from the i Ž. pseudoconvexity of ␺0 and ␺j j s 1, 2, . . . , m . In the same manner in which Lemma 3.4 is derived it can now be shown that the efficiency set of Ž.3.1 is connected Ž via the connectedness of the minimum set of the scalarized program. . 56 R. N. MUKHERJEE

4. APPLICATIONS: FRACTIONAL PROGRAMMING

We consider a multiobjective program with fractional objectives of the type

fx12Ž. fx Ž. fxp Ž. V-minimize , , . . . ,Ž. 4.1 ž/gx12Ž. gx Ž. gxp Ž. Ž. n Ž.n subject to hxF0, where fii: R ª Ris1, 2, . . . , p and g : R ª R nm Ž.is1, 2, . . . , p are real valued locally Lipschitz functions and h: R ª R n is a vector valued locally Lipschitz function over R . Also fii, g and h are Ž. regular. The usual assumption is that gi ) 0 i s 1, 2, . . . , p . One can see thatŽ. 4.1 is equivalent to a parametric program of the type

U U V-minimize Ž.fx111Ž.y¨gx Ž.... fxppp Ž.y¨gx Ž. Ž4.2 . Ž. ŽUU U. subject to hxF0, where ¨ 12, ¨ ,...,¨pis some preassigned parameter vectorŽ seewx 1. . Under the assumption as in Section 3 for a Lagrangian

␭ ␶ U ␭T ␭ m␶ ␶ LxŽ1 .sÝÝiiŽ.fx Ž.y¨ iigx Ž.q h, for g RqiiG 0, s 1, we can give sufficient optimality conditions for efficiency points of pro- gramŽ. 4.2 . Further, Assumption Ž. B will lead to sufficient optimality conditions like Theorem 3.4 and optimality conditions for the parametric programŽ. 4.2 , which in turn gives appropriate optimality conditions for programŽ. 4.1 via equivalence of Ž. 4.1 and Ž. 4.2 .

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