Abstract and Applied AnalysisVolume 2011 (2011), Article ID 432963, 16 pagesdoi:10.1155/2011/432963

Research Article

Second-Order Optimality Conditions for Set-Valued Optimization Problems Under Benson Proper Efficiency

Qilin Wang¹ and Guolin Yu²

¹College of Sciences, Chongqing Jiaotong University, Chongqing 400074, China²Research Institute of Information and System Computation Science, Beifang University of Nationalities, Yinchuan 750021, China

Received 4 October 2011; Accepted 18 November 2011

Academic Editor: D. Anderson

Copyright © 2011 Qilin Wang and Guolin Yu. 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.

Some new properties are obtained for generalized second-order contingent (adjacent) epiderivatives of set-valued maps. By employing the generalized second-order adjacent epiderivatives, necessary and sufficient conditions of Benson proper efficient solutions are given for set-valued optimization problems. The results obtained improve the corresponding results in the literature.

1. Introduction

The investigation of the optimality conditions is one of the most attractive topics of optimization theory since many optimization problems encountered in economics, engineering, and other fields involve vector-valued maps (or set-valued maps) as constraints and objectives (see [1–3]). With the concept of contingent derivative for a set-valued map (see [3]), Corley [4] investigated optimality conditions for general set-valued optimization problems. But it turns out that the optimality conditions do not coincide under standard assumptions. Jahn and Rauh [5] introduced the contingent epiderivative of a set-valued map and then obtained unified necessary and sufficient optimality conditions. The essential differences between the definitions of the contingent derivative and the contingent epiderivative are that the graph is replaced by the epigraph and the derivative is single-valued. But the existence of the contingent epiderivative of a set-valued map in a general setting is an open question. To overcome the difficulty, Chen and Jahn [6] introduced a generalized contingent epiderivative of a set-valued map which is a set-valued map. They proved that the generalized contingent epiderivative exists under standard assumptions and obtained a unified necessary and sufficient condition in terms of the generalized contingent epiderivative. As to other concepts of epiderivatives for set-valued maps and applications to optimality conditions, one can refer to [7–11].

Since higher-order tangent sets introduced in [3], in general, are not cones and convex sets, there are some difficulties in studying higher-order optimality conditions and duality for general set-valued optimization problems. Until now, there are only a few papers to deal with higher-order optimality conditions and duality of set-valued optimization problems by virtue of the higher-order derivatives or epiderivatives introduced by the higher-order tangent sets. Jahn et al. [12] introduced second-order contingent epiderivative and generalized contingent epiderivative for a set-valued map and obtained some second-order optimality conditions based on these concepts. In [13], Li et al. studied some properties of higher-order tangent sets and higher-order derivatives introduced in [3] and then obtained higher-order Fritz John type necessary and sufficient optimality conditions for set-valued optimization problems in terms of the higher-order derivatives. By using these concepts, they also discussed higher-order Mond-Weir duality for a set-valued optimization problem in [14]. In general, since the epigraph of a set-valued map has nicer properties than the graph of a set-valued map, it is advantageous to employ the epiderivatives in set-valued optimization. Li and Chen [15] introduced the definitions of higher-order generalized contingent epiderivative and higher-order generalized adjacent epiderivative and obtained higher-order optimality conditions for Henig properly efficient solutions of a set-valued optimization problem with constraints based on the higher-order generalized adjacent epiderivative and contingent epiderivative. Chen et al. [16] introduced the concepts of higher-order weak contingent epiderivative and higher-order weak adjacent epiderivative for set-valued maps and obtained higher order Mond-Weir type duality, higher-order Wolfe type duality, and higher-order Kuhn-Tucker type optimality conditions to a constrained set-valued optimization problem based on the higher-order weak adjacent (contingent) epiderivatives and Henig efficiency. Since the existence of the higher-order contingent (adjacent) derivative and the higher-order generalized contingent (adjacent) epiderivative of a set-valued map in a general setting may not exist, Wang and Li [17] introduced the generalized higher-order contingent (adjacent) epiderivative of a set-valued map and obtained the optimality conditions for Henig efficient solutions to set-valued optimization problems. Wang et al. [18] introduced the generalized higher-order contingent (adjacent) derivatives of set-valued maps, and necessary and sufficient optimality conditions are obtained for weakly efficient solutions of set-valued optimization problems under no convexity assumptions.

To the best of our knowledge, by virtue of the second-order derivatives or epiderivatives introduced by the higher-order tangent sets, the second-order optimality conditions have not been studied for Benson proper efficient solutions of set-valued optimization problems until now. This paper aims to investigate the second-order optimality conditions for Benson proper efficient solutions of set-valued optimization problems by virtue of the generalized second-order epiderivatives.

The rest of the paper is organized as follows. In Section 2, we collect some of the concepts and introduce two kinds of set-valued optimization problem models. In Section 3, we recall the concepts and properties of generalized second-order contingent (adjacent) epiderivatives of set-valued maps and discuss some new properties of them. In Section 4, we establish second-order necessary and sufficient conditions for Benson proper efficient solutions to a set-valued optimization problem, whose constraint set is determined by a fixed set. In Section 5, we establish second-order Kuhn-Tuck type optimality conditions for Benson proper efficient solutions to a set-valued optimization problem, whose constraint set is determined by a set-valued map.

2. Preliminaries

Throughout this paper, let , and be three real normed spaces, where the spaces and are partially ordered by nontrivial closed convex pointed cones and with , respectively. Denote by the topological dual space of and by the zero element in . The dual cone of is defined as The quasi-interior of is the set Recall that a base of a cone is a convex subset of such that Of course, is pointed whenever has a base. Furthermore, if is a nonempty closed convex pointed cone in , then if and only if has a base.

Let and be set-valued maps. The effective domain, the graph, and the epigraph of are defined by , and , respectively. Denote . Let .

Definition 2.1 (see [19]). is called -convex on a nonempty convex subset , if, for any and , such that .

Definition 2.2 (see [20]). Let be a set-valued map, . is said to be generalized -convex at on , if is convex.

Remark 2.3 (see [20]). If is -convex on convex set , then is generalized -convex at on . But the converse may not hold.

Definition 2.4 (see [6]). Let be a subset of . is said to be a Benson proper efficient point of if . The set of all Benson proper efficient points of is denoted by .

Definition 2.5 (see [21]). Let be a real normed space, be a base of , and be a nonempty subset. is said to be a Henig efficient point of , written as , if for some , where and is the closed unit ball of .

Lemma 2.6 (see [22]). Let and be two closed convex cones in a locally convex vector space, and let be pointed and have a compact base. If , then there exists such that .

Definition 2.7 (see [6]). A set-valued map is said to be (i)strictly positive homogeneous if (ii)subadditive if

In this paper, consider the following set-valued optimization problem:

that is, to find a pair with and is called a Benson proper efficient element of problem (OP) if . We also consider the following constraint set-valued optimization problem:

Set . A pair with and is called a Benson proper efficient element of problem (SOP) if .

3. Properties of Generalized Second-Order Contingent (Adjacent) Epiderivatives

In this section, we discuss some new properties of generalized second-order contingent and adjacent epiderivatives. Moreover, we give an example to show these properties. Let be a set-valued map from to .

Definition 3.1 (see [17]). Let be a set-valued map from to , .(i)The generalized second-order contingent epiderivative of at for vectors is the set-valued map from to defined by (ii)The generalized second-order adjacent epiderivative of at for vectors is the set-valued map from to defined by

Remark 3.2. If , then

From [17, Proposition 3.9 and Corollary 3.11], we have the following result.

Proposition 3.3. Let , , . Then, (i); (ii).

Proposition 3.4. Let be a nonempty subset of , , . Let be generalized -convex at on , . Then,

Proof. Take a arbitrary sequence with , any , and . Since is generalized -convex at on , is convex, and then It follows from , and is a convex cone that Then, Thus, and the proof of the proposition is complete.

Corollary 3.5. Let be a nonempty subset of , , . If is generalized -convex at on , , then

Proposition 3.6. Let . Then, (i) is strictly positive homogeneous. Moreover, if is generalized -convex at on , then (ii) is subadditive.

Proof. (i) Since   is strictly positive homogeneous.(ii) Let , , . Then one has . Since is generalized -convex at on , is convex, and then it follows from the proof of [13, Proposition 3.2], that is convex. Thus, we have and then it follows from (i) that which implies The proof of the proposition is complete.

For the sake of comparison, we recall some notions in [13, 15, 16].

Definition 3.7 (see [13]). The second-order adjacent derivative of at for vector is the set-valued map from to defined by

Definition 3.8 (see [15]). The second-order generalized adjacent epiderivative of at for vectors is the set-valued map from to defined by

Definition 3.9 (see [16]). The second-order weak adjacent epiderivative , of at for vector is the set-valued map from to defined by

Remark 3.10. Notice that in Proposition 3.6, we establish a special property of , which is similar to the corresponding property of the generalized contingent epiderivative in [6, Theorem 1]. But we cannot obtain the similar property for ,, , and ,, even though is a -convex map. The following example explains the case.

Example 3.11. Let , . Take and . Then, Therefore, for any , one has Naturally, , , and are not strictly positive homogeneous, but is strictly positive homogeneous here.

It follows from [13, Corollary 3.1], and the proof of Proposition 3.6 that the following result holds.

Corollary 3.12. Let . Then (i) is strictly positive homogeneous. Moreover, if is generalized -convex at on , and , then (ii) is subadditive.

Remark 3.13. Under the assumption of cone-convex maps, Chen and Jahn [6, Theorem 1] obtained the subadditivity of . However, it is under the assumption of generalized cone-convex maps that we obtain the subadditivity of in Corollary 3.12.

By Propositions 3.3–3.6 and Corollary 3.12, we obtain the following results.

Corollary 3.14. Let . Then,

Corollary 3.15. Let and . If is generalized -convex at on , then (i) is a closed convex cone. Moreover, if , then (ii) is a closed convex cone.

4. Second-Order Optimality Conditions of (OP)

In this section, by employing the generalized second-order adjacent epiderivative, we will discuss the second-order optimality conditions for Benson proper efficient solutions of (OP).

Theorem 4.1. Let , and . If is a Benson proper efficient element of (OP), then, for every , one has

Proof. Since is a Benson proper efficient element of (OP), Assume that there exists an such that (4.1) does not hold, that is, there exists and then, by the definition of generalized second-order adjacent epiderivatives, for arbitrary sequence with , there exists a sequence such that Since and is a convex cone, . Then, by (4.4), we get Thus, from (4.5) and (4.3), one obtains which contradicts (4.2). So (4.1) holds and the proof is complete.

Theorem 4.2. Let , and let be locally compact. Suppose that the following conditions are satisfied: (i) is generalized -convex at on ; (ii)the pair is a Benson proper efficient element of (OP). Then, there exists such that where .

Proof. Define By the similar line of proof for convexity of in [15], Theorem 5.1, we obtain that is a convex set. It follows from Theorem 4.1, that Thus, by Lemma 2.6, there exists such that . So, we have It follows from Corollary 3.15 that , so and the proof of the theorem is complete.

Theorem 4.3. Let and . If there exists such that then the pair is a Benson proper efficient solution of (OP).

Proof. It follows from Proposition 3.3 that for all . Then, by (4.13), (4.14), and , we get Thus, by the sufficient condition of [23, Theorem 4.1], we get that is a Benson proper efficient solution of (OP), and the proof of the theorem is complete.

Remark 4.4. In Theorem 4.3, no assumption of generalized convexity is imposed.

Remark 4.5. If we use instead of in Theorem 4.3, then the corresponding result for may not hold. The following example explains the case.

Example 4.6. Suppose that , . Let be a set-valued map with Consider the following constrained set-valued optimization problem (4.1): Take . We have Then, for any , we have Naturally, the pair is not a Benson proper efficient solution of (4.1).

From Proposition 3.4, we know that the following theorem holds.

Theorem 4.7. Let . Suppose that the following conditions are satisfied: (i), (ii) is generalized -convex at on , (iii)there exists such that Then the pair is a Benson proper efficient solution of (OP).

5. Second-Order Kuhn-Tuck Type Optimality Conditions of (SOP)

In this section, by employing the generalized second-order adjacent epiderivative, we will discuss the second-order Kuhn-Tuck type optimality conditions for Benson proper efficient solutions of (SOP). The notation is used to denote . Let . Firstly, we recall a result in [24].

The interior tangent cone of at is defined as where stands for the closed ball centered at and of radius .

Lemma 5.1 (see [24]). If is convex, and , then

Theorem 5.2. Let , . If is a Benson proper efficient element of , then for all .

Proof. Since is a Benson proper efficient element of , Suppose that there exists some such that (5.3) does not hold, that is, there exists some It follows from (5.5) and the definition of generalized second-order contingent epiderivatives that there exist sequences with and such that From (5.6) and (5.8), there exists a sufficiently large natural number such that According to (5.9) and Lemma 5.1, we obtain . Then, it follows from the definitions of that , ,,. Since and (5.9), there exists a sufficiently large natural number such that ,for and , for , that is, It follows from and is a convex cone that Then from (5.7), , there exist , , , , and such that and . It follows from (5.11) that , for , and then Since and is a convex cone, . Then by (5.7), (5.12), and (5.8), we get Then it follows from (5.6) that which contradicts (5.4). Thus, (5.3) holds and the proof of the theorem is complete.

Theorem 5.3. Let , . Suppose that the following conditions are satisfied: (i) has a compact base; (ii) is generalized -convex at on ; (iii)the pair is a Benson proper efficient element of (SOP). Then, there exist and such that where and .

Proof. Define By the similar line of proof for convexity of in [15, Theorem 5.1], we obtain that is a convex set. It follows from Theorem 5.2 that Thus, by employing Corollary 3.15, it follows from a standard separation theorem of convex sets and the similar proof method of [8, Theorem 1] that there exist and such that The proof is complete.

Remark 5.4. It follows from Remarks 2.3 and 3.2 that the necessary optimality condition in Theorem 5.3 is obtained under weaker assumptions than those assumed of [8, Theorem 1].

Now we give an example to illustrate the necessary optimality conditions for generalized second-order contingent epiderivatives.

Example 5.5. Suppose that , . Let be a set-valued map with and be a set-valued map with Consider the following constrained set-valued optimization problem (4.2): Take and . Naturally, is generalized -convex at on , and is a Benson proper efficient solution of (4.2). By directly calculation, we have Take and . Naturally, necessary optimality condition of Theorem 5.3 holds here.

Theorem 5.6. Let . Suppose that the following conditions are satisfied: (i), (ii) there exist and such that where . Then, the pair is a Benson proper efficient solution of (SOP).

Proof. It follows from Proposition 3.3 that for all . Then, by assumption (ii), we have Thus, there exists a with such that . It follows from and (5.25) that It follows from the sufficient condition of [23, Theorem 4.1], that is a Benson proper efficient solution of (SOP) and the proof of the theorem is complete.

Remark 5.7. Since Theorem 5.6 does not involve the assumption of convexity, it improves and generalizes [8, Theorem 2].

From Proposition 3.4 and the proof of Theorem 5.6, we easily obtain that the following Theorem holds.

Theorem 5.8. Let . Suppose that the following conditions are satisfied: (i), (ii) is generalized -convex at on ;(iii) there exist and such that where . Then, the pair is a Benson proper efficient solution of (SOP).

Acknowledgments

The authors would like to thank anonymous referees for their valuable comments and suggestions, which helped to improve the paper. This research was partially supported by the National Natural Science Foundation of China (10901004, 11171362, and 11071267), Natural Science Foundation Project of CQ CSTC (cstcjjA00019), and Science and Technology Research Project of Chong Qing Municipal Education Commission (KJ100419).

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