Journal of Applied MathematicsVolume 2011 (2011), Article ID 647489, 16 pagesdoi:10.1155/2011/647489

Research Article

Nonemptiness and Compactness of Solutions Set for Nondifferentiable Multiobjective Optimization Problems

Xin-kun Wu,¹ Jia-wei Chen,² and Yun-zhi Zou¹

¹College of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China²School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei 430072, China

Received 13 May 2011; Accepted 10 July 2011

Academic Editor: Yongkun Li

Copyright © 2011 Xin-kun Wu 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.

A nondifferentiable multiobjective optimization problem with nonempty set constraints is considered, and the equivalence of weakly efficient solutions, the critical points for the nondifferentiable multiobjective optimization problems, and solutions for vector variational-like inequalities is established under some suitable conditions. Nonemptiness and compactness of the solutions set for the nondifferentiable multiobjective optimization problems are proved by using the FKKM theorem and a fixed-point theorem.

1. Introduction

The weak minimum (weakly efficient, weak Pareto) solution is an essential concept in mathematical models, economics, decision theory, optimal control, and game theory. For readers’ reference, we refer to [1–11] and the references therein.

In [5], Garzón et al. studied some relationships among the weakly efficient solutions, the critical points of optimization problems, and the solutions of vector variational-like inequalities with differentiable functions. In [12], Mishra and Wang extended the work of Garzón et al. [5] to nonsmooth case. In [9], Lee et al. investigated the existence of solutions of vector optimization problems with differentiable functions. In [7], Kazmi considered the relationship between the weakly efficient solutions of a vector optimization problem and the solutions of a vector variational-like inequality with preinvex and Frechet differentiable functions. For more related work in this interesting area, we refer to [4, 10].

Motivated and inspired by the works mentioned above, we consider nondifferentiable multiobjective optimization problems (MOPs) with nonempty set constraints. The relationship among weakly efficient solutions, critical points of (MOP), and solutions of the vector variational-like inequalities (for short, (VVLI)) is presented under subinvexity, strictly pseudosubinvexity, and pseudosubinvexity conditions. By using the FKKM theorem and a fixed-point theorem, we prove the nonemptiness and compactness of solutions set for (MOP). The results presented in this paper extend the corresponding results of [5, 7, 9, 12, 13].

2. Preliminaries

Throughout this paper, without other specifications, let be the -dimensional Euclidean space, and . Let be a nonempty convex subset of , let be a subset of , and let be the relative interior of to . Let , , and let such that, for each , is a closed convex cone, , , and . The multiobjective optimization problem (for short, (MOP)) is defined as follows:

We first recall some definitions and lemmas which are needed in the main results of this paper.

Definition 2.1. A point is said to be a weakly efficient (weak minimum) solution of (MOP) if .

Definition 2.2. A real-valued function is said to be locally Lipschitz with respect to if, for each , there exist a neighborhood of and a constant such that

Remark 2.3. If , then the above definition reduces to that of local Lipschitz.

Definition 2.4. A set-valued function is said to be locally bounded at if there exist a neighborhood of and a constant such that

Definition 2.5. Let be a set-valued function. The graph of is defined as The inverse of is defined by if and only if .

Definition 2.6. Let be a nonempty subset of topological vector space . A set-valued mapping is called a KKM mapping if, for every finite subset of , where co denotes the convex hull.

Definition 2.7 (see [13]). A real-valued function is said to be subinvex at with respect to if there exists , such that , for any , where is called the -subgradient of at . The subdifferential of at , denoted , is the set of all such that

Remark 2.8. If is locally Lipschitz, then the subinvexity of with respect to collapses to the invexity of with respect to in the sense of Clarke's generalized directional derivative with respect to [1]. If for any , then the subdifferential reduces to the subdifferential in the sense of convex analysis, where .

Definition 2.9. A vector-valued function is called strictly pseudosubinvex with respect to if, for any and , where .

Definition 2.10. A vector-valued function is called pseudosubinvex with respect to if, for any and , where .

Remark 2.11. Since , it is clear that (1)if is strictly pseudosubinvex with respect to , then it is pseudosubinvex with respect to ;(2)if , are subinvex with respect to , then is pseudosubinvex with respect to .

Definition 2.12. A point is called a critical point of (MOP) if there exists , with for some , such that

Lemma 2.13 (see [14](FKKM theorem)). Let be a nonempty subset of Hausdorff topological vector space . Let be a KKM mapping such that for any is closed and is compact for some , then there exists such that for all , that is, .

Lemma 2.14 (see [13]). Let be a nonempty and convex subset of Hausdorff topological vector space , and let be two set-valued maps such that, for each . If there exist a nonempty compact convex set and a nonempty compact set such that, for each , there exists , such that , then there exists such that .

We consider the following vector variational-like inequality (for short, ): find such that for any , there exists such that

3. Relationships between (MOP) and (VVLI)

In this section, we will investigate the properties of -subdifferential of the function , and the relationships among weakly efficient solutions, critical points of (MOP), and the solutions of (VVILP).

Theorem 3.1. Let be subinvex with respect to , then the following statements are true: (i)for each , is a nonempty closed-convex subset of , (ii)if for each , then is -monotone, that is, for any , (iii)if is locally Lipschitz with respect to , is continuous in the second argument, and , for any , then is closed, and is upper semicontinuous, (iv)if is an open map, for any , and is locally Lipschitz with respect to , then is locally bounded on .

Proof. Assertions (i), (ii), and (iv) are shown in [13]. We only need to prove assertion (iii). Let , with and . Since and is locally Lipschitz with respect to , there exist a neighborhood of and a constant such that Then there exists such that for all , and so Consequently, we have as . It follows from that that is, . Hence, . In view of (i), we have that is upper semicontinuous. This completes the proof.

Theorem 3.2. Let be pseudosubinvex with respect to . If is a solution of , then is a weakly efficient solution of (MOP).

Proof. Let be a solution of (VVLI). If is not a weakly efficient solution of (MOP), then there exists such that Since is pseudosubinvex with respect to , we have which contradicts the assumption. This completes the proof.

Corollary 3.3. Let be subinvex with respect to . If is a solution of , then is a weakly efficient solution of (MOP).

Theorem 3.4. Let be an open map, and continuous and affine in the first argument, let , for any . Let be subinvex and locally Lipschitz with respect to . If is a weakly efficient solution of (MOP), then is a solution of .

Proof.. Let be a weakly efficient solution of (MOP). If is not a solution of (VVLI), then We assert that Let Suppose to the contrary that then there exist and such that Since is -monotone, we have that is, As a consequence, and it follows that which contradicts the assumption.On the other hand, let and let such that . For any given , set , for any . Then as . Since is affine in the first argument, and , we have that is, Since is locally bounded, there exist a neighborhood of , and such that for any , and , we have Then there exists , such that for any , and so Consequently, has a convergent subsequence. Without loss of generality, let . By Theorem 3.1 (iii), we have . Since is closed, it follows that Thus, it follows from (3.9) that that is, there exist and such that Since we have Thus, that is, which contradicts the assumption. This completes the proof.

Theorem 3.5. Let be subinvex with respect to , and let be strictly pseudosubinvex with respect to . If is a critical point of (MOP), then is a weakly efficient solution of (MOP).

Proof. Let be a critical point of (MOP). If is not the weakly efficient solution of (MOP), then there exists such that By the strict pseudosubinvexity of with respect to , one has that is, Thus, we have Since is a critical point of (MOP), there is , with for some , such that Set , then and so which is a contradiction. This completes the proof.

Remark 3.6. If for each , then the following statements are true in the sense of Clarke’s generalized directional derivative [12]: (1)all critical points of (MOP) are weakly efficient solutions of (MOP) if and only if is strictly pseudoinvex with respect to ;(2)if is strictly pseudoinvex with respect to , and locally Lipschitz, then the critical points, the weakly efficient solutions of (MOP), and the solutions of are equivalent.

4. Existence of Weakly Efficient Solutions for (MOP)

In this section, we present several existence theorems for (MOP), by using the FKKM theorem and a fixed-point theorem.

Theorem 4.1. Let be nonempty convex. Suppose that the following conditions are satisfied: (i) for any , (ii) is affine, is continuous, (iii)the set-valued function is given by for any such that is closed, (iv) are subinvex and locally Lipschitz with respect to , (v)there exists a nonempty closed bounded set such that, for each , there exists , and for any , such that then the solutions set of (MOP) is nonempty compact.

Proof. Define a set-valued mapping by For any finite set , let , then is a compact convex set. Define another set-valued mapping by, Obviously, for any , that is, is nonempty. Let be a net such that From Theorem 3.1(iii), we conclude that is closed. Therefore, there exists , such that . It follows from the closedness of that Thus, , that is, is closed, and so is compact, since is compact.It is easy to prove that, for any finite set . In fact, if there exists , with such that then for any , By the convexity of , Since is affine in the first argument, we have Now from the assumption that for any , we get , which is a contradiction. Therefore, is a KKM mapping. By Lemma 2.13, there exists such that that is, From assumptions, we have , and moreover, , that is, has the finite intersection property. Consequently, , that is, there exists such that for any , there exists , such that From Theorem 3.1, is a weakly efficient solution of (MOP).Denote the solutions set of (MOP) by . Let , such that , then for any , there exists , such that From Theorem 3.1, we have that is locally bounded on and is closed. Thus, there exists such that , where . Since is continuous in the second argument and is closed, we have Thus, , that is, is closed. From assumptions, we get , and so is compact. This completes the proof.

Example 4.2 .. Let for any , and for any , where . Let . For each , then and so . Since we have . Therefore, Similarly, one has Consequently, for any such that Then it is easy to check that all assumptions in Theorem 4.1 hold and .

Corollary 4.3 (see [9]). Let be invex in the sense of Clarke's generalized directional derivative with respect to and locally Lipschitz. Suppose that other conditions are the same as in Theorem 4.1, then the solutions set of (MOP) is nonempty compact.

Corollary 4.4. Let be nonempty convex. Suppose that the following conditions are satisfied: (i) for any , (ii) is affine and is continuous, (iii)the set-valued function is given by for any such that is closed, (iv) are subinvex and locally Lipschitz with respect to , (v)there exists , such that , is closed and bounded, then the solutions set of (MOP) is nonempty compact.

Proof. Let Obviously, , and moreover, is nonempty closed bounded. Therefore, for any , there exists such that for any , By Theorem 4.1, the solutions set of (MOP) is nonempty compact. This completes the proof.

Theorem 4.5. Let be a nonempty convex set. Suppose that the following conditions are satisfied: (i) for any , (ii) is affine and is continuous, (iii)the set-valued function is given by for any such that is closed, (iv) are subinvex and locally Lipschitz with respect to , (v)there exists such that for any and , such that then the solutions set of (MOP) is nonempty compact.

Proof. Since there exists , such that for any , and such that there exists , such that for each , with , such that Taking , where , and let . Clearly, is closed bounded. Then for any , there exists , and for any , such that By Theorem 4.1, the solutions set of (MOP) is nonempty compact. This completes the proof.

Theorem 4.6. Let be nonempty convex. Suppose that the following conditions are satisfied: (i) for any , (ii) is an open mapping, is affine, and is continuous, (iii)the set-valued function is given by for any such that is closed, (iv) are subinvex and locally Lipschitz with respect to , (v)there exist a nonempty closed bounded set and a nonempty bounded closed set such that, for each , there exist and such that then the solutions set of (MOP) is nonempty compact.

Proof. Define set-valued mapping by Obviously, for any , that is, .For any net , with , we have, for any , that is, In view of the continuity of with respect to the second argument and closedness of , we obtain and so Consequently, , that is, is closed. By Theorem 3.4, we only need to prove that (VVLI) has a solution.Suppose to the contrary that has no solution, then From the proof of Theorem 3.4, we have Therefore, for each .Define set-valued mappings by, respectively, Obviously, for each .For any finite set , there exist such that By the -monotonicity of , we have, for each , It follows that Since , for any , we get that is, Therefore, for any , and thus Since is affine in the first argument, it follows that We have , that is, for any . In view of the closedness of , one has as an open set in . Therefore, . From (4.34), we have that is, Furthermore, From assumption , one has By Lemma 2.14, there exists such that . Since for any , we have . Thus, that is, , which is a contradiction. Therefore, (VVILP) has a solution. From Theorems 3.2 and 3.4, (MOP) has a weakly efficient solution.Similarly, we can show that is compact.This completes the proof.

Remark 4.7. As previously mentioned, the results presented in this paper extend some corresponding results in [5, 7, 9, 12, 13]. For instance, Theorems 4.1 and 4.6 extended the corresponding results given by Kazmi [7], and Lee et al. [13], Lee et al. [9] from preinvex functions to subinvex functions. Corollaries 4.3 and 4.4 generalized the results given by Lee et al. [13] from differentiable functions to nondifferentiable functions.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 60804065); academic award for excellent PhD candidates funded by Ministry of Education of China; the Fundamental Research Fund for the Central Universities (no. 201120102020004).

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