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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 458983, 17 pages
http://dx.doi.org/10.1155/2012/458983
Research Article

On a Nonsmooth Vector Optimization Problem with Generalized Cone Invexity

1Department of Mathematics, Xidian University, Xi’an 710071, China
2College of Mathematics and Computer, Yangtze Normal University, Fuling 408100, China

Received 4 September 2012; Accepted 12 October 2012

Academic Editor: Ferenc Hartung

Copyright © 2012 Hehua Jiao and Sanyang Liu. 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

By using Clarke’s generalized gradients we consider a nonsmooth vector optimization problem with cone constraints and introduce some generalized cone-invex functions called K-α-generalized invex, K-α-nonsmooth invex, and other related functions. Several sufficient optimality conditions and Mond-Weir type weak and converse duality results are obtained for this problem under the assumptions of the generalized cone invexity. The results presented in this paper generalize and extend the previously known results in this area.

1. Introduction

In optimization theory, convexity plays a key role in many aspects of mathematical programming including sufficient optimality conditions and duality theorems; see [1, 2]. Many attempts have been made during the past several decades to relax convexity requirement; see [37]. In this endeavor, Hanson [8] introduced invex functions and studied some applications to optimization problem. Subsequently, many authors further weakened invexity hypotheses to establish optimality conditions and duality results for various mathematical programming problems; see, for example, [911] and the references cited therein.

Above all, Yen and Sach [12] introduced cone-generalized invex and cone-nonsmooth invex functions. Giorgi and Guerraggio [13] presented the notions of - -invex, - pseudo-invex, and -K quasi-invex functions in the differentiable case and derived optimality and duality results for a vector optimization problem over cones. Khurana[14] extended pseudoinvex functions to differentiable cone-pseudoinvex and strongly cone-pseudoinvex functions. Based on this, Suneja et al. [15] defined cone-nonsmooth quasi-invex, cone-nonsmooth pseudoinvex, and other related functions in terms of Clarke’s [16] generalized directional derivatives and established optimality and duality results for a nonsmooth vector optimization problem.

On the other hand, Noor [17] proposed several classes of -invex functions and investigated some properties of the -preinvex functions and their differentials. Mishra et al. [18] defined strict pseudo- -invex and quasi- -invex functions. Mishra et al. [19] further introduced the concepts of nonsmooth pseudo- -invex functions and established a relationship between vector variational-like inequality and nonsmooth vector optimization problems by using the nonsmooth -invexity.

In the present paper, by using Clarke's generalized gradients of locally Lipschitz functions we are concerned with a nonsmooth vector optimization problem with cone constraints and introduce several generalized invex functions over cones namely - -generalized invex, - -nonsmooth invex, and other related functions, which, respectively, extend some corresponding concepts of [12, 13, 15, 17]. Some sufficient optimality conditions for this problem are obtained by using the above defined concepts. Furthermore, a Mond-Weir type dual is formulated and a few weak and converse duality results are established. We generalize and extend some results presented in the literatures on this topic.

2. Preliminaries and Definitions

Throughout this paper, let and be two fixed mappings. and denote the interior and closure of , respectively. We always assume that is a closed convex cone with .

The positive dual cone of is defined as The strict positive dual cone of is given by The following property is from [20], which will be used in the sequel.

Lemma 2.1 (see [20]). Let be a convex cone with . Then,(a) ; (b) .

A function is called locally Lipschitz at , if there exists such that for all in a neighbourhood of .

A function is called locally Lipschitz on , if it is locally Lipschitz at each point of .

Definition 2.2 (see [16]). Let be a locally Lipschitz function, then denotes Clarke's generalized directional derivative of at in the direction and is defined as Clarke's generalized gradient of at is denoted by and is defined as Let be a vector-valued function given by , where . Then is said to be locally Lipschitz on if each is locally Lipschitz on . The generalized directional derivative of a locally Lipschitz function at in the direction is given by The generalized gradient of at is the set where is the generalized gradient of at .
Every is a continuous linear operator from to and

Lemma 2.3 (see [16]). (a) If is locally Lipschitz then, for each ,
(b) Let be a finite family of locally Lipschitz functions on , then is also locally Lipschitz and

Definition 2.4 (see [17]). A function is said to be -invex function at with respect to and , if there exist functions and such that, for every , we have In this paper, we consider the following vector optimization problem with cone constraints: where are locally Lipschitz functions on and are closed convex cones with nonempty interiors in and , respectively.
Denote the feasible set of problem .
For each and , we suppose that and are locally Lipschitz.
Now, we present the concepts of solutions for problem in the following sense.

Definition 2.5. Let , then (a) is said to be a minimum of if for all , (b) is said to be a weak minimum of if for all , (c) is said to be a strong minimum of if for all , Based on the lines of Yen and Sach [12] and Noor [17], we define the notions as follows.

Definition 2.6. Let be a locally Lipschitz function. is said to be - -generalized invex at , if there exist functions and such that for every and ,

Definition 2.7. Let be a locally Lipschitz function. is said to be - -nonsmooth invex at , if there exist functions and such that for every , where

Remark 2.8. If , and is differentiable, then - -generalized invex and - -nonsmooth invex functions become -invex function [17]; if for all , then - -generalized invex and - -nonsmooth invex functions reduce to -generalized invex and -nonsmooth invex functions defined by Yen and Sach [12].

Lemma 2.9. If is K- -generalized invex at with respect to and , then is K- -nonsmooth invex at with respect to the same and .

Proof. Since is - -generalized invex at , then there exist and such that for every and By Lemma 2.3, for each , we choose such that Then and equivalently, Hence, is - -nonsmooth invex at with respect to the same and .

The following example shows that converse of the above lemma is not true.

Example 2.10. Let be a cone in . Assume that , where Let and be defined as and , respectively. Then at , Hence, is - -nonsmooth invex at .
It is easy to verify .
Taking and , we have Therefore, is not - -generalized invex at .
Next, we introduce several related functions of - -nonsmooth invex.

Definition 2.11. is said to be - -nonsmooth quasi-invex at , if there exist functions and such that for every ,

Definition 2.12. is said to be - -nonsmooth pseudo-invex at , if there exist functions and such that for every ,

Definition 2.13. is said to be strict - -nonsmooth pseudo-invex at , if there exist functions and such that for every ,

Definition 2.14. is said to be strong - -nonsmooth pseudo-invex at , if there exist functions and such that for every ,

Remark 2.15. If for all and is differentiable, then - -nonsmooth pseudo-invex and strong - -nonsmooth pseudo-invex functions reduce to -pseudo-invex and strong -pseudo-invex functions, defined by Khurana [14].

Remark 2.16. If for all , and is differentiable, then - -nonsmooth quasi-invex functions reduce to quasi-invex functions and - -nonsmooth pseudo-invex and strong - -nonsmooth pseudo-invex functions reduce to pseudo-invex functions [8].

Remark 2.17. If for all , then the above definitions reduce to the corresponding definitions [15]. If is differentiable, then - -generalized invex and - -nonsmooth pseudo-invex functions reduce to - -invex and - pseudo-invex functions [13], respectively.

3. Optimality Criteria

In this section, we establish a few sufficient optimality conditions for problem by using the above defined functions.

Theorem 3.1. Let be - -generalized invex and be - -generalized invex at with respect to the same and . We assume that there exist such that Then is a weak minimum of .

Proof. By contradiction, we assume that is not a weak minimum of . Then there exists a feasible solution of such that From (3.1), it follows that there exist and such that Since is - -generalized invex and is - -generalized invex at , we get Summing (3.3) and (3.5), we have As ,  from Lemma 2.1, we obtain which yields Considering positivity of and (3.4), one has From , we deduce Hence, By , relation (3.6) gives By virtue of (3.2) and , the above inequality implies that is, which is a contradiction to (3.12).
Therefore, is a weak minimum of .

Theorem 3.2. Let be K- -generalized invex and be - -generalized invex at with respect to the same and . We assume that there exist such that (3.1) and (3.2) hold. Then is a minimum of .

Proof. Assume contrary to the result that is not a minimum of . Then there exists such that From (3.1), it follows that there exist and such that Since is - -generalized invex at , we get Utilizing (3.16), we deduce According to , we obtain Next proceeding on the same lines as in the proof of Theorem 3.1, we obtain a contradiction.
Thus, is a minimum of .

Theorem 3.3. Let be - -nonsmooth pseudo-invex and be - -nonsmooth quasi-invex at with respect to the same and . We assume that there exist such that (3.1) and (3.2) hold. Then is a weak minimum of .

Proof. It follows from (3.1) that there exist and such that Suppose that is not a weak minimum of . Then there exists such that Since is - -nonsmooth pseudo-invex at , we deduce By and Lemma 2.1, we obtain From and , we have which implies, By and ,   gives Taking (3.2) into account, one has Next we prove If , inequality (3.29) holds obviously.
If , from (3.28) and Lemma 2.1, we deduce Since is Q- -nonsmooth quasi-invex at , we have From and , it follows that (3.29) also holds.
Similarly, by Lemma 2.3, inequality (3.29) gives which yields, Hence, which is in contradiction with (3.26).
Therefore, is a weak minimum of .

The following example illustrates the above theorem.

Example 3.4. Consider the vector optimization problem where , , and are defined as Let and be defined as and , respectively. It is easily testified that and are - -nonsmooth pseudo-invex and - -nonsmooth quasi-invex at , respectively. The feasible set of is given by .
It is also easy to verify .
Taking and , we have which imply that (3.1) and (3.2) hold.
Therefore, by Theorem 3.3, is a weak minimum of .

Theorem 3.5. Let be strong K- -nonsmooth pseudo-invex and be Q- -nonsmooth quasi-invex at with respect to the same and . We assume that there exist such that (3.1) and (3.2) hold. Then is a strong minimum of .

Proof. From (3.1), it follows that there exist and such that Assume that is not a strong minimum of . Then there exists such that Since is strong - -nonsmooth pseudo-invex at , we deduce Next proceeding on the same lines as in the proof of Theorem 3.3, we get a contradiction.
Hence is a strong minimum of .

Theorem 3.6. Let be strict K- -nonsmooth pseudo-invex and be Q- -nonsmooth quasi-invex at with respect to the same and . We assume that there exist such that (3.1) and (3.2) hold. Then is a minimum of .

Proof. From (3.1), it follows that there exist and such that By contradiction, assume that is not a minimum of . Then there exists such that Since is strict - -nonsmooth pseudo-invex at , we have Next as in Theorem 3.3 we arrive at a contradiction.
Therefore, is a minimum of .

4. Duality

In relation to , we consider the following Mond-Weir type dual problem: Denote the feasible set of problem by , namely, , .

Now, we establish weak and converse duality results.

Theorem 4.1 (Weak duality). Let and . If is K- -nonsmooth pseudo-invex and is Q- -nonsmooth quasi-invex at with respect to the same and , then

Proof. Since , from , it follows that there exist and such that By contradiction, we assume that .
Since is - -nonsmooth pseudo-invex at , we have By and Lemma 2.1, we get From and Lemma 2.3, we deduce which yields Using (4.2), we obtain From , it follows that there exists such that Hence, From and , we find that Now we claim that
If , then (4.11) holds trivially.
If , from (4.10) and Lemma 2.1 we have As is - -nonsmooth quasi-invex at , we obtain which means that (4.11) also holds and is equivalent to which is a contradiction to (4.9). Thus

Theorem 4.2 (Weak duality). Let and . If is K- -generalized invex and is Q- -generalized invex at with respect to the same and , then

Proof. Since , from , it follows that there exist and such that We assume contrary to the result that Since is - -generalized invex and is Q- -generalized invex at , we get Summing (4.18) and (4.19), we have By and Lemma 2.1, we obtain which yields Applying positivity of and (4.17), we get By the fact that , one has Hence, From and (4.20), we obtain As and , we get Using the above relation, (4.27) yields that is, which contradicts (4.26). Therefore,

Theorem 4.3 (Converse duality). Let and . Assume that is K- -nonsmooth pseud-invex and is Q- -nonsmooth quasi-invex at with respect to the same and . Then is a weak minimum of .

Proof. Since , from , it follows that there exist and such that Assume contrary to the result that is not a weak minimum of . Then there exists such that Since is - -nonsmooth pseudo-invex at , we have By and Lemma 2.1, we get which is equivalent to which yields Using (4.32), we obtain As , thus , where .
Hence, By and , we have By the similar argument to that of Theorem 4.1, we can prove that which is equivalent to which is in contradiction with (4.39).
Therefore, is a weak minimum of .

Theorem 4.4 (Converse duality). Let and . Assume that is K- -generalized invex and is Q- -generalized invex at with respect to the same and . Then is a weak minimum of .

Proof. The proof of the above theorem is very similar to the proof of Theorem 3.1, except that for this case we use the feasibility of for instead of the relations (3.1) and (3.2).

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 60974082) and supported by Foundation Project of China Chongqing Education Commission.

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