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Abstract and Applied Analysis
Volume 2016, Article ID 5367190, 6 pages
http://dx.doi.org/10.1155/2016/5367190
Research Article

Optimality Conditions for Nondifferentiable Multiobjective Semi-Infinite Programming Problems

Department of Economics, University of Messina, Via dei Verdi 75, Messina, Italy

Received 2 May 2016; Accepted 18 August 2016

Academic Editor: Jozef Banas

Copyright © 2016 D. Barilla et al. 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

We have considered a multiobjective semi-infinite programming problem with a feasible set defined by inequality constraints. First we studied a Fritz-John type necessary condition. Then, we introduced two constraint qualifications and derive the weak and strong Karush-Kuhn-Tucker (KKT in brief) types necessary conditions for an efficient solution of the considered problem. Finally an extension of a Caristi-Ferrara-Stefanescu result for the ()-invexity is proved, and some sufficient conditions are presented under this weak assumption. All results are given in terms of Clark subdifferential.

1. Preliminaries and Introduction

First, we briefly overview some notions of convex analysis and nonsmooth analysis widely used in the formulations and proofs of the main results of the paper. For more details, discussion, and applications, see [13].

Given a nonempty set , we denote with , , , and the closure of , the relative interior of , convex hull, and convex cone (containing the origin) generated by , respectively. The polar cone and strict polar cone of are defined, respectively, by

It is easy to show that if then . The bipolar theorem states thatThe cone of feasible direction of at is the cone defined by

It is worth observing that if is a minimizer of convex function on a convex set , thenwhere and denote, respectively, the normal cone of at and the convex subdifferential of at ; that is,

We observe that if is an arbitrary set, and , then

If is a collection of convex sets in , and , then it is easy to see that

Let and let be a locally Lipschitz function. The Clarke directional derivative of at in the direction and the Clarke subdifferential of at are, respectively, given byThe Clarke subdifferential is a natural generalization of the classical derivative, since it is known that when function is continuously differentiable at , . Moreover when a function is convex, the Clarke subdifferential coincides with the subdifferential in the sense of convex analysis.

In the following theorem we summarize some important properties of the Clarke directional derivative and the Clarke subdifferential from [1] which are widely used in what follows.

Theorem 1. Let and be functions from to which are Lipschitz near . Then,(i)the function is finite, positively homogeneous, and subadditive on , and(ii) is a nonempty, convex, and compact subset of ,(iii) is upper semicontinuous as a function of .

In this paper, we have considered the following multiobjective semi-infinite programming problem: where , and , , are locally Lipschitz functions from to and the index set is arbitrary, not necessarily finite (but nonempty).

For differentiable MOSIP where is finite, necessary conditions of KKT type have been established under various constraint qualifications in [4]. The Abadie constraint qualification and related constraint qualification for semi-infinite systems of convex inequalities and linear inequalities are also studied in [5]. There, the characterizations of various constraint qualifications in terms of upper semicontinuity of certain multifunctions are given.

There are only a few works available that deal with optimality conditions for MOSIP. For instance, for differentiable MOSIPs, some optimality conditions have been presented by Caristi et al. in [6]. Glover et al. in [7] considered a nondifferentiable convex MOSIP and presented optimality theorems for it. For a nonsmooth MOSIP, the “basic constraint qualification” has been studied by Chuong and Kim in [8], who have given optimality and duality conditions of Karush-Kuhn-Tucker (KKT, briefly) type for the problem which involves the notion of Mordukhovich subdifferential. Also, Gao presented some sufficient and duality results for MOSIPs under the various generalized convexity assumptions in [9, 10].

This paper is structured as follows: In Section 2 we propose a Fritz-John type necessary condition after we derive a KKT type necessary condition for optimality of the considered problem under a suitable qualification condition, and we establish the strong KKT necessary conditions for an efficient solution of the considered problem. In Section 3 we obtain an extension of a Caristi-Ferrara-Stefanescu result for the ()-invexity.

2. Necessary Conditions

As a starting point of this section, we denote with the feasible region of ; that is,For a given , let denote the index set of all active constraints at :A feasible point is said to be an efficient solution [resp., weakly efficient solution] to problem (MOSIP) iff there is no satisfying , , and [resp., , ].

For each , set

For each , setRecall the following definition from [11].

Definition 2. We say that MOSIP has the Pshenichnyi-Levin-Valadire (PLV) property at , if is Lipschitz around , and

The following condition is well known, even in differentiable cases (see, e.g., [5, 6]).

Assumption A. The index set is a nonempty compact subset of , the function is upper semicontinuous on , and is an upper semicontinuous mapping in for each .

The following lemma from [5, Theorem ] will be used in sequel.

Lemma 3. Suppose that Assumption A holds. Then, (1) is a compact set for each ,(2)the PLV property holds at each .

The following result is an extension of [6, Theorem ].

Theorem 4 (FJ necessary condition). Let be a weakly efficient solution of MOSIP. If condition A holds at , then there exist (for ) and , for , with for finitely many indexes, such that

Proof. We know from Lemma 3 that is a compact set. Thus is also a compact set, and hence, is closed.
If , by strict separation theorem we find such that for all . This implies thatSince and the PLV property is satisfied at (by Lemma 3), we haveThus, there exists such that for all . The last inequality and the fact that (since ) conclude that , and henceMoreover, we haveFor each we find such thatTake . By (21) and (23) for each we havewhich contradicts the weak efficiency of . This contradiction implies thatNow, (7) proves the result.

The necessary conditions of Fritz-John type can be viewed as being degenerate when the multiplier corresponding to the objective function vanishes, because the function being minimized is not involved. In the next theorem we derive a Karush-Kuhn-Tuker type necessary condition for optimality of MOSIP under a suitable qualification condition.

Definition 5. Let . We say that MOSIP satisfies the Zangwill CQ (ZCQ briefly) at , if

Theorem 6. Let be a weakly efficient solution of MOSIP, ZCQ hold at , and be a closed cone. Then there exist (for ) and , for , with for finitely many indexes, such that

Proof. We first claim thatOn the contrary, suppose that there exists such that for all . Thus, there exist positive scalars such thatTake . By above inequalities, for each we havewhich contradicts the weak efficiency of . Thus, (28) is true.
If , there exists a sequence in converging to . Owing to (28) and continuity of function , we deduce thatWe thus proved that (by assumption of ZCQ at )Since and , the last relation implies that the following convex problem has a minimum at :Hence, by (4), (6), and (11) we obtain thatNow, the closeness of , (2), (7), and (8) prove the results.

In almost all examples, we were not able to obtain positive KKT multipliers associated with the vector-valued objective function; that is to say, some of the multipliers may be equal to zero. This means that the components of the vector-valued objective function do not have a role in the necessary conditions for weak efficiency. In order to avoid the case where some of the KKT multipliers associated with the objective function vanish for the MOSIP, an approach has been developed in [5], and strong KKT necessary optimality conditions have been obtained. We say that strong KKT condition holds for a multiobjective optimization problem, when the KKT multipliers are positive for all the components of the objective function. Below, we establish the necessary strong KKT conditions for an efficient solution (not a weak efficient solution) of MOSIP under a suitable qualification condition.

Let . On the lines of [4], for each , define the set For the sake of simplicity, we denote by in this paper.

Definition 7. Let . We say that MOSIP satisfies the strong Zangwill CQ (SZCQ briefly) at , if

Theorem 8 (strong KKT necessary condition). Let be an efficient solution of MOSIP. If in addition, SZCQ and the conditionhold at , then exist (for ) and , for , with for finitely many indexes, such that

Proof. We present the proof in four steps.
Step 1. We claim thatIt suffices only to prove thatOn the contrary, suppose that for some there is a vector such thatBy the definition of , there exists such that for each . Thus, owing to the definition of we obtain thatOn the other hand, (41) leads to . This means that exists, satisfyingThe above inequality with (42) implies that, for each with , we haveThis contradicts the efficiency of . Therefore, our claim holds.
Step 2. Let for some . Then, there exists a sequence in converging to . By (40) and continuity of we concluded thatTherefore,and henceStep 3. We claim thatOn the contrary we suppose that (48) does not hold. ThenThus, by the strong convex separation theorem [3, Theorem ] and noting that is a convex cone, it follows that there is a hyperplane separating and properly. Hence, there exists a vector satisfyingThus, owing to SZCQ and we conclude thatwhich contradicts (47).
Step 4. The result is immediate from (48), (8), and the fact that (see, [3, Theorem ])

3. Sufficient Conditions

Similar to [6], let and be given functions satisfyingObserve that an element of is represented as the order pair with and .

In [6] a differentiable function was named -invex at with respect to if, for each , We extend this result as follows.

Definition 9. The locally Lipschitz function , at , is called generalized -invex at with respect to if, for each , it satisfies In the rest of this paper, we will always assume to be equal to the set of the feasible solution of MOSIP.

Theorem 10 (sufficient KKT condition). Suppose that there exist a feasible solution and scalars (for ) with and a finite set and scalars (for ) such thatMoreover if the functions and the functions (for ) are -invex at , and for all , then is a weak efficient solution for MOSIP.

Proof. Inclusion (57) implies that some (for ) and (for ) exist, satisfying Taking and , we conclude that Owing to these equalities, (54), , and convexity of , we obtain thatNow, if is not a weak efficient of MOSIP, there exists a point such that for all . Hence, by generalized -invexity of functions we haveSimilarly, for each , we haveFrom (61), (62), and , we conclude thatwhich contradicts (60). This contradiction shows that is an weak efficient for MOSIP.

Strengthening the assumptions concerning s we obtain sufficient conditions for efficiency.

Theorem 11 (strong sufficient KKT condition). Suppose that there exist a feasible solution and scalars (for ) and a finite set and scalars (for ) such thatMoreover if the functions and the functions (for ) are -invex at , and for all , then is an efficient solution for MOSIP.

Proof. Similar to (61) and (62), if is not efficient, we find and such thatFrom these and (for all ) and (for all ), we obtainwhich contradicts (60).

Remark 12. Similar to [6], we can define some weaker -invexiy assumption for the function , and then, we can prove some weaker sufficient conditions for optimality of MOSIP. Since the proof of these extensions is similar to previous theorems, we omit them.

Competing Interests

The authors declare that they have no competing interests.

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