On Weak Pareto Optimality for Pseudoconvex Nonsmooth Multiobjective Optimization Problems

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1 Int. Journal of Math. Analysis, Vol. 7, 2013, no. 60, HIKARI Ltd, On Weak Pareto Optimality for Pseudoconvex Nonsmooth Multiobjective Optimization Problems A. Jaddar 1 and K. El Moutaouakil 2 Copyright c 2013 A. Jaddar and K. El Moutaouakil. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The purpose of this paper is to characterize the weak Pareto optimality for multiobjective pseudoconvex problem. In fact, it is a first order optimality characterization that generalize the Karush-Kuhn-Tucker condition. Moreover, this work is an extension of the single-objective case [6] to the multiobjective one with pseudoconvex continuous functions. Mathematics Subject Classification: 46N10, 26A51, 26A27 Keywords: Nonsmooth analysis; upper Dini subdifferential; Multiobjective problem; Pseudoconvexity; Karush-Kuhn-Tucker conditions; Weak Pareto optimal solutions 1 Introduction Generally, the different functions of a multiobjective problem dont reach there optimum at the same point. So, it si necassary to search for a compromise solution. Based on this idea, the concepts of Pareto and weak Pareto optimality are used in many contribution that characterize and give the Pareto optimality conditions (see, for instances Chankong and Haimes [2], Miettinen [14], Jahn [7], Zhang and Zuo [9], Oliveira, Beato Moreno and Moretti [18] Mohamed First University, MATSI Laboratory, Oujda, Morocco, ajaddar@gmail.com 2 Mohamed First University, Oujda, Morocco, karimmoutaouakil@yahoo.fr

2 2996 A. Jaddar and K. El Moutaouakil One of the most important is the first-order necessary optimality conditions that generalizes the Karush-Khun-Tucker condition. When constructing otimality conditions some kinds of generalized convexities have proved to be the main tool during the last decades. In this paper, we deal with a particular case of multiobjective problem, whith the notion of pseudoconvexity [13], in an extended sense using upper Dini subdifferential, for the objective functions and the constraint set. There are many contributions dealing with a smooth and nonsmooth singleobjective case (see for example Pini and Singh [19], Jaddar [6], Maeada Takashi [10], Minami [15], Diewert [3] and the referenses therein. These result were extended for nonsmooth multiobjective problems with locally lipschitz continuous functions (see for instance Huu, Myung and Sang [5], Nobakhtian [17], Staib [20] and recently in [12] by Mäkelä, Karmitsa and Eronen. The aim of this paper is to extend recent results of single-objective case of [6] to multiobjective one. In fact we characterize weak pareto minima of continuous (not necessarely locally lipschitz pseudoconvex functions in terms of upper Dini subdifferentials of Kuhn-Tucker type. The paper is organized as follows, after recalling some definitions of upper Dini subdifferential and pseudoconvexity. In section 2, we present first-order optimality conditions of weak Pareto optimality for nonsmooth multiobjective pseudoconvex optimization problems in the general case when the set of constraint is abstract. Then, in section 3, we refine this caracterization for a set of constraints including abstract set and inequality constraints. Let us recall some definitions and properties that will be used in the sequel. Let f : R n R {+ } be a function. The upper Dini derivative of f at a in the direction d is defined by Df(a; d = lim sup t 0 The upper Dini subdifferential is f(a + td f(a. t { {a R n ; a ; d Df(a; d, d R n } if a dom (f f(a = if a dom (f A function f : R n R {+ } is pseudoconvex for the upper Dini subdifferential if for all x, y R n, the following implication holds : ( x f(x : x,y x 0 = f(x f(y. The pseudoconvexity has also been defined using generalized derivatives. Nevertheless, we will adopt in the sequel this definition. Using the definition of pseudoconvexity, we can see easily that for a pseudoconvex function f, we have 0 f(x x is a global minimum of f.

3 On weak Pareto optimality 2997 Moreover, according to Aussel [1], for any continuous function f, f is pseudoconvex { f is quasiconvex, 0 f(x x is a global minimum 2 Optimality Conditions for Nonsmooth Multiobjective Problem In the present section, we present some necessary and sufficient optimality conditions for multiobjective optimization within the meaning of weak Pareto optimality and without explicit the set of feasible solutions. First, consider the following general multiobjective optimization problem : { minimize {f1 (x,f (P 2 (x,...,f q (x} subject to x C, where f k : R n R for k =1,...,q are lower semicontinuous (l.s.c. pseudoconvex functions. Let x 0 C, the upper Dini tangent cone at x 0 is defined by T C (x 0 ={ d R n ; δ >0 such that t ]0,δ[; x 0 + td C }. The polar of T C (x 0, also known as the normal cone to C at x 0, denoted N C (x 0, is given by N C (x 0 ={ u R n ; u, d 0, d T C (x 0 }. Generally, the costs functions of a multiobjective problem conflicting with each others. That is why, there is a little possibility to find an optimal solution minimizing all the costs functions simultanously. To overcome to this shortcoming, most of the authors have opted for the Pareto optimality concept. Definition 2.1. A vector x is said to be a global Pareto optimum of (P, if there does not exist x C such that f k (x f k (x 0 for all k =1,...,q and f l (x f l (x 0 for some l, and a global weak Pareto optimum of (P, if there does not exist x C such that f k (x f k (x 0 for all k =1,...,q. Vector x 0 is a local (weak Pareto optimum of (P, if there exist δ 0 such that x 0 is a global (weak Pareto optimum on B(x 0,δ C Next, we can state the necessary optimality condition of problem (P with arbitrary nonempty closed convex set C R n Proposition 2.2. If x 0 C to be a local weak Pareto optimum of of (P, then 0 f i (x 0 +N C (x 0 Proof. Let x 0 be a local weak Pareto optimum. Then, there exists ε > 0 such that for every y C B(x 0,ε there exists k {1, 2,...,q} such that

4 2998 A. Jaddar and K. El Moutaouakil f k (y f k (x 0. Let d T C (x 0. Thent t n 0 there exists an index n 0 such that x 0 + t n d C B(x 0,ε for all n n 0. Then for every n n 0 there exists k n {1, 2,...,q} such that f kn (x 0 + t n d f kn (x 0. Since the set of index k n is finite, there exists k {1, 2,...,q} and subsequence (t nm (t n such that f k(x 0 + t n d f k(x 0. For all m large enough we have t 1 (f k(x nm 0 + t nm d f k(x 0 0. It follows that Df k(x 0,d 0, d T C (x 0. Then, by Lemma 2.1 and Theorem 2.2 of [6] and their proofs, we get 0 f k(x 0 +N C (x 0. Therefore 0 f i (x 0 +N C (x 0. Before given a sufficient condition for global optimality, we will need the folowing basic result of convex anlysis (see for example [12] : Lemma 2.3. Let C i R n,, 2,...,q be nonempty convex sets, then ( { } co C i = λ i x i x i C i,λ i 0, λ i =1, where co(a denotes the convex hull of a set A. In the next, we gives a characterization of multiobjective optimization for any closed convex constraint C. Theorem 2.4. Let x 0 C such that f i (x 0 are nonempty for any i {1,...,q}. Then x 0 is a global weak Pareto minimum of (P if and only if ( 0 co f i (x 0 + N C (x 0 Proof. The necessity follows( directly from Proposition 2.2. For sufficiency let 0 co f i (x 0 + N C (x 0. Then there exist x co f i (x 0 such that x,x x 0 0, have for all x C λ k x k,x x 0 0, k=1 x C. Then by Lemma 2.3 we with λ k 0,x k f k(x 0 and λ k =1 Then there exist x k f k(x 0 with λ k 0 such that λ kx k,x x 0 0, x C. Then by pseudoconvexity of f k, f k(x f k(x 0. Thus, x 0 is a global weak Pareto optimum. Basing in these results, we study in the next section the inequality constraints case. k=1

5 On weak Pareto optimality Inequality constraints The main purpose of this section is rewriting the characterization established in the previous section for the Inequality constraints case. In this regard, the fact that the C structure contains some inequality constraints enables us to refine the characterization given previously. Let S be a closed convex set, and h pseudoconvex and continuous such that {x C; h(x 0}, known as the Slater condition. In this part, we consider first the case where C = {x S R n ; h(x 0}. Let x 0 C such that h(x 0 = 0 and h(x 0, then we have : Theorem 3.1. A necessary and sufficient condition ( for a point x 0 C to be a weak Pareto optimum of (P is that 0 co f k (x 0 + Cl (R + h(x 0 + N S (x 0. where Cl(A designe the closure of a set A. Proof. According ( to Theorem 2.4, x 0 C is a weak Pareto optimum of (P if and only if 0 co f i (x 0 + N C (x 0. By the Slater condition, we can see that : k=1 N C (x 0 =N {x R n ; h(x 0}(x 0 +N S (x 0. By Lemma 3.2 and Theorem 3.4 of [6], we get N C (x 0 =Cl(R + h(x 0 + N S (x 0, Remark 3.2. If, in addition, h(x 0 is bounded, then it becomes convex and compact set, therefore R + h(x 0 is closed and then concide with it s closure and a Lagrange multiplier appears. Now we shall consider problem (P with m inequality constraints : minimize {f 1 (x,f 2 (x,...,f q (x} subject to g i (x 0, for all i =1,...,m, and x S. ( Suppose that for all i =1,...,m, g i is continuous pseudoconvex functions such that : Dg i (x 0 ; d = sup x,d, ( x g i (x 0 In order to treat the problem ( , we need the next two Lemma :

6 3000 A. Jaddar and K. El Moutaouakil Lemma 3.3. co g i (x 0 = g(x 0 ( i I(x 0 where I(x ={i; g(x =g i (x}, g(x = max hg i(x and co(a is the closed 1 i m convex hull of a set A. Proof. Let s show the first inclusion co g i (x 0 g(x 0. We know i I(x 0 that g(x 0 is closed convex subset of R n. Then it is sufficient to show the implication x i I(x0 g i (x 0 = x g(x 0. Let i I(x 0 such that x g i (x 0, then for all d one has, x,d Dg i (x 0,d. According to [4] (x,d Then, x,d Dg i (x 0,d Dg(x 0,d for all d, which means that x g(x 0. Conversely, suppose by contradiction that there exist x g(x 0 such that x / co ( i I(x0 g i (x 0. Then by the Hahn Banach separation theorem, there exist d and ε 0 such that x,d y,d + ε, y co g i (x 0 i I(x 0 According to (??, we can see that for any y g i (x 0 with i I(x 0 such that Dg i (x 0,d=Dg(x 0,d, we have y,d + ε x,d Dg i (x 0,d=Dg(x 0,d. From, ( , we get the desired contradiction. Lemma 3.4 ( [6]. The function g is pseudoconvex and satisfies the equality ( So, we have Proposition 3.5. x 0 is a weak Pareto optimum of ( if and only if ( 0 co f i (x 0 + Cl R + co g j (x 0 + N S (x 0. ( j I(x 0

7 On weak Pareto optimality 3001 Moreover, if in addition, g j (x 0 are bounded, then x 0 is a weak Pareto optimum of ( if and only if 0 m λ i f i (x 0 + μ j g j (x 0 +N S (x 0. j=1 Where λ i 0 for all i =1,...,q such that λ i =1and μ j j =1,...,m and μ j g j (x 0 =0, for all j =1,...,m. 0 for all Proof. It follows directly from Lemma 3.4, Theorem 3.1 and Lemma 3.3, that x 0 is a weak Pareto optimum of ( if and only if the inclusion ( holds. Moreover, when g j (x 0 are bounded, Remark 3.2 and Lemma 2.3 gives the classical multipliers rule 0 λ i f i (x 0 + m μ j g j (x 0 +N S (x 0, where λ i 0 for all i =1,...,q and μ j 0 for all j =1,...,m such that λ i = 1 and μ j g j (x 0 =0,j =1,...,m,(μ j = 0 whenever the constraint is not active at x 0. References [1] Aussel D., Subdifferential properties of quasiconvex and pseudoconvex functions: Unified approach. J. Optimization Theory Appl. 97, No.1, 29 45, (1998. [2] Chankong V. and Haimes Y.Y., Multiobjective Decision Making: Theory and Methodology, North-Holland, Elsevier Sciences Publishing Co., Inc., New York, (1983. [3] Diewrt W. E., Alternative Characterizations of Six Kinds of Quasiconcavity in the Nondifferentiable Case with Applications to Nonsmooth Programming, Generalized Concavity in Optimization and Economics (Eds. Schaible, S. and Ziemba, W. T., Academic Press, New York, pp , (1981. [4] Giorgi G.and Komlósi S., Dini derivatives in optimization. II. Riv. Mat. Sci. Econ. Soc. 15, No.2, 3 24, (1992.

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