Journal of Applied Mathematics

Volume 2012 (2012), Article ID 370654, 9 pages

doi:10.1155/2012/370654

Research Article

A Generalized Alternative Theorem of Partial and Generalized Cone Subconvexlike Set-Valued Maps and Its Applications in Linear Spaces

Zhi-Ang Zhou¹ and Jian-Wen Peng²

¹College of Mathematics and Statistics, Chongqing University of Technology, Chongqing 400054, China

²School of Mathematics, Chongqing Normal University, Chongqing 400047, China

Received 24 June 2012; Accepted 26 July 2012

Academic Editor: Nan-Jing Huang

Copyright © 2012 Zhi-Ang Zhou and Jian-Wen Peng. 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 first introduce a new notion of the partial and generalized cone subconvexlike set-valued map and give an equivalent characterization of the partial and generalized cone subconvexlike set-valued map in linear spaces. Secondly, a generalized alternative theorem of the partial and generalized cone subconvexlike set-valued map was presented. Finally, Kuhn-Tucker conditions of set-valued optimization problems were established in the sense of globally proper efficiency.

1. Introduction

Generalized convexity plays an important role in set-valued optimization. The generalization of convexity from vector-valued maps to set-valued maps happened in the 1970s. Borwein [1] and Giannessi [2] introduced and studied the cone convexity of set-valued maps. Based on Borwein and Giannessi’s work, some authors [3–7] established a series of optimality conditions of set-valued optimization problems under different types of generalized convexity of set-valued maps in topological spaces. Since linear spaces are wider than topological spaces, generalizing some results of the above mentioned references from topological spaces to linear spaces is an interesting topic. Li [8] introduced a cone subconvexlike set-valued map involving the algebraic interior and established Kuhn-Tucker conditions. Huang and Li [9] studied Lagrangian multiplier rules of set-valued optimization problems with generalized cone subconvexlike set-valued maps in linear spaces. When the algebraic interior of the convex cone is empty, Hernández et al. [10] used the relative algebraic interior of the convex cone to introduce cone subconvexlikeness of set-valued maps and investigated Benson proper efficiency of set-valued optimization problems in linear spaces.

The aim of this paper is to study globally proper efficiency of set-valued optimization problems in linear spaces. This paper is organized as follows. In Section 2, we recalled some basic notions and gave some lemmas. In Section 3, we presented a generalized alternative theorem of the partial and generalized cone subconvexlike set-valued map and established Kuhn-Tucker conditions of set-valued optimization problems in the sense of globally proper efficiency.

2. Preliminaries

In this paper, let and be two real-ordered linear spaces, and let denote the zero element of every space. Let be a nonempty subset in . The cone hull of is defined as is called a convex cone if and only if

A cone is said to be pointed if and only if . A cone is said to be nontrivial if and only if and .

Let and stand for the algebraic dual spaces of and , respectively. Let and be nontrivial, pointed, and convex cones in and , respectively. The algebraic dual cone of is defined as , and the strictly algebraic dual cone of is defined as , where denotes the value of the linear functional at the point . The meaning of is similar to that of .

Let be a nonempty subset of . The linear hull of is defined as , and the affine hull aff of is defined as aff. The generated linear subspace of is defined as .

Definition 2.1 (see [11]). Let be a nonempty subset of . The algebraic interior of is the set

Definition 2.2 (see [12]). Let be a nonempty subset of . The relative algebraic interior of is the set

Clearly, . Therefore, Definition 2.2 is consistent with the definition of the relative algebraic interior of in [13, 14]. However, Definition 2.2 seems to be more convenient than the ones in [13, 14].

It is worth noting that if is a nontrivial and pointed cone in , then , and if is a convex cone, then is a convex set, and is a convex cone.

Lemma 2.3 (see [13]). If is a convex cone in , then .

Lemma 2.4 (see [10, 12, 14]). If is a nonempty subset in , then

(a) ;

if is convex in and , then

(b) ; (c) .

Lemma 2.5 (see [12]). Let be a convex set with in . If , then there exists such that

3. Main Results

Let be a nonempty set, and let and be two set-valued maps on . Write and . The meanings of and are similar to those of and .

Now, we introduce a new notion of the partial and generalized cone subconvexlike set-valued map.

Definition 3.1. A set-valued map is called partial and generalized -subconvexlike on if and only if is a convex set in .

The following theorem will give some equivalent characterizations of the partial and generalized -subconvexlike set-valued map in linear spaces.

Theorem 3.2. Let . Then the following statements are equivalent:

(a)the set-valued map is partial and generalized -subconvexlike on ,(b),

(c),

(d),

Proof. . Let , and . Clearly,

Since is partial and generalized -subconvexlike on , it follows from (3.4) that

which implies that (3.1) holds.

The implications are clear.

. Let . Then there exist , and such that .

Case one: if or , we have .

Case two: if and , we have

where .

By Lemma 2.4, we obtain

Since , there exists such that

By (3.3), (3.6), (3.8), and Lemma 2.3, we have

Cases one and two imply that is a convex set in . Therefore, (a) holds.

Remark 3.3. Theorem 3.2 generalizes the sixth item of Proposition  2.4 in [14], Lemma  2.1 in [15], and Lemma  2 in [16].

Now, we will give a generalized alternative theorem of the partial and generalized -subconvexlike map. We consider the following two systems.

System 1. There exists such that .

System 2. There exists such that

Theorem 3.4 (generalized alternative theorem). Let , and let the set-valued map be partial and generalized -subconvexlike on . Then,

(i)if System 1 has no solutions, then System 2 has a solution; (ii)if is a solution of System 2, then System 1 has no solutions.

Proof. (i) Firstly, we assert that . Otherwise, there exist and such that .

Case one: if , then . Since is a nontrivial, pointed, and convex cone, . Thus, we obtain a contradiction.

Case two: if , then there exists such that

which contradicts that System 1 has no solutions.

Cases one and two show that our assertion is true. Since the set-valued map is partial and generalized -subconvexlike on is a convex set in . Note that . Thus, all conditions of Lemma 2.5 are satisfied. Therefore, there exists such that

Letting in (3.12), we have

We again assert that . Otherwise, there exists such that . Let , and be fixed. Then there exists sufficiently large positive number such that , that is,

By Lemma 2.3, . Thus, (3.14) contradicts (3.13). Therefore, . Similarly, we can prove that .

Let be fixed in (3.13). Then, . Letting in (3.13), we have

Letting in (3.15), we obtain

which implies that System 2 has a solution.

(ii) If is a solution of System 2, then

We assert that System 1 has no solutions. Otherwise, there exist and such that and . Therefore, we have , which contradicts (3.17). Therefore, our assertion is true.

Remark 3.5. If is a finite-dimensional space, then the partial and generalized -subconvexlikeness of implies that is a nonempty convex in , which in turn implies that the condition holds trivially.

Remark 3.6. Theorem 3.4 generalizes Theorem  3.7 in [14], Theorem  2.1 in [15], and Theorem  1 in [16].

From now on, we suppose that .

Definition 3.7 (see [17]). Let be called a global properly efficient point with respect to (denoted by if and only if there exists a nontrivial, pointed, and convex cone with such that .

Now, we consider the following set-valued optimization problem:

The feasible set of (3.18) is defined by .

Definition 3.8. Let be called a global properly efficient solution of (3.18) if and only if there exists such that . The pair is called a global properly efficient element of (3.18).

Now, we will establish Kuhn-Tucker conditions of set-valued optimization problem (3.18) in the sense of globally proper efficiency.

Theorem 3.9. Suppose that the following conditions hold:

(i) is a global properly efficient element of (3.18); (ii)the set-valued map is partial and generalized -subconvexlike on , where .

Then, there exists such that

Proof. Since is a global properly efficient element of (3.18), there exists a nontrivial, pointed, and convex cone with such that

It follows from (3.20) that

By (3.21), we obtain

Since is partial and generalized -subconvexlike on , it follows from (3.22) and Theorem 3.4 that there exists such that

that is

Because , there exists such that . Since , we have

Letting in (3.24), we obtain

It follows from (3.25) and (3.26) that

Therefore, we have

By (3.24) and (3.28), we have . Letting in (3.24), we have

It follows from (3.27) and (3.29) that .

The following theorem, which can be found in [17], is a sufficient condition of global properly efficient elements of (3.18).

Theorem 3.10. Suppose that the following conditions hold:

(i),(ii)there exist and such that

Then, is a global properly efficient element of (3.18).

Acknowledgments

This paper was supported by the National Natural Science Foundation of China (Grants 11171363 and 11271391), the Natural Science Foundation of Chongqing (CSTC 2011jjA00022 and CSTC 2011BA0030), the Special Fund of Chongqing Key Laboratory (CSTC 2011KLORSE01), and the project of the Third Batch Support Program for Excellent Talents of Chongqing City High Colleges.

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