Abstract and Applied Analysis

Volume 2014 (2014), Article ID 636907, 5 pages

http://dx.doi.org/10.1155/2014/636907

Research Article

A Kind of Unified Proper Efficiency in Vector Optimization

Ke Quan Zhao and Yuan Mei Xia

College of Mathematics Science, Chongqing Normal University, Chongqing 401331, China

Received 22 February 2014; Accepted 14 April 2014; Published 30 April 2014

Academic Editor: Chong Li

Copyright © 2014 Ke Quan Zhao and Yuan Mei Xia. 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.

Based on the ideas of the classical Benson proper efficiency, a new kind of unified proper efficiency named S-Benson proper efficiency is introduced by using Assumption (B) proposed by Flores-Bazán and Hernández, which unifies some known exact and approximate proper efficiency including -proper efficiency and E-Benson proper efficiency in vector optimization. Furthermore, a characterization of S-Benson proper efficiency is established via a kind of nonlinear scalarization functions introduced by Göpfert et al.

1. Introduction

It is well known that approximate solutions have been playing an important role in vector optimization. Since Kutateladze initially introduced the concept of approximate solutions in [1], a lot of research achievements of approximate solutions have been obtained for vector optimization problems. Loridan proposed -efficient solutions of vector optimization problems and gave some properties in [2]. In a general topological vector space, Rong and Wu proposed -weak efficient solutions of vector optimization problems with set-valued maps and obtained some linear scalarization theorems, Lagrangian multipliers theorems, saddle point theorems, and duality theorems in [3]. Recently, Gutiérrez et al. introduced the concept of coradiant set and proposed -efficient solutions which extend and unify some known different notions of approximate solutions in [4]. Gao et al. proposed the concept of properly approximate efficient solutions by means of coradiant set and established some linear and nonlinear scalarization results in [5]. Furthermore, Gutiérrez et al. obtained some characterizations of this kind of approximate solutions in terms of linear scalarization in [6].

Moreover, Debreu introduced the concept of free disposal sets to deal with mathematical economic problems in [7]. In a finite dimensional space, Chicco et al. introduced the concepts of improvement sets and -efficient solutions and obtained some characterizations in [8]. Improvement sets are close to free disposal sets and can be applied to study vector optimization problems as an important tool. In particular, Zhao and Yang obtained a unified stability result with perturbations by means of improvement sets in [9]. Furthermore, Gutiérrez et al. generalized the concepts of improvement sets and -efficient solutions to a general real locally convex Hausdorff topological vector space and studied some linear scalarization results in [10]. Zhao and Yang proposed -weak efficient solutions of vector optimization problems with set-valued maps and established some linear scalarization theorems, Lagrange multiplier theorems, saddle point criteria, and duality in [11]. Zhao and Yang introduced the concept of -Benson proper efficiency which unifies some proper efficiency and obtained some linear scalarization theorems and Lagrange multiplier theorems of this kind of proper efficiency in [12]. Flores-Bazán and Hernández proposed Assumption (B) and obtained some complete scalarizations of solution sets of a class of unified vector optimization problems via nonlinear scalarization in [13]. In addition, Flores-Bazán and Hernández obtained some optimality conditions of a class of unified vector optimization problems under Assumption (B) in [14].

Motivated by the works of [4, 5, 12, 13], we present a new kind of unified proper efficiency named -Benson proper efficiency by using Assumption (B) proposed by Flores-Bazán and Hernández. This kind of proper efficiency unifies some known exact and approximate proper efficiency including -proper efficiency and -Benson proper efficiency in vector optimization. Furthermore, we also give a characterization of -Benson proper efficiency via nonlinear scalarization.

2. Preliminaries

Let be a linear space and a real Hausdorff locally convex topological linear space. For a subset of , we denote the topological interior, the topological closure, the boundary, and the complement of by , , , and , respectively. A set is solid if and is proper if is nonempty and . The cone generated by is defined as

Let denote the topological dual space of . The positive dual cone of a subset is defined as

Let be a closed convex pointed cone in with nonempty topological interior. For any , we define

In this paper, we consider the following vector optimization problem:

where and .

We say that is a coradiant set if satisfies for every , . Let be a proper solid coradiant set and define

Lemma 1 (see [5]). Let be a proper solid convex coradiant set. Then,

(i);(ii).

Definition 2 (see [5]). Let . A feasible point is said to be a -proper efficient solution of if

Definition 3 (see [10]). A nonempty set is said to be an improvement set with respect to if and .

Lemma 4 (see [10]). Let be a nonempty set. If is an improvement set with respect to , then . Additionally, if , then .

Definition 5 (see [12]). Let be an improvement set with respect to . A feasible point is said to be an -Benson proper efficient solution of if

Flores-Bazán and Hernández introduced Assumption B as follows.

Assumption B (see [13]). Consider that and is a proper (not necessary closed) set such that and .

Remark 6. From Assumption B, we have the equivalence

Lemma 7 (see [15]). Let be any nonempty subset. Then, .

3. A Kind of Unified Proper Efficiency

In this section, we propose a kind of unified proper efficiency of by means of Assumption B by using the idea of the classical Benson proper efficiency and discuss some relations with other proper efficiency.

Definition 8. Let and satisfy Assumption B. One says that is a -Benson proper efficient solution of if

Denote by the set of -Benson proper efficient solutions of .

Example 9. Let , , , , and

Since

then, from Remark 6, it follows that and satisfy Assumption B. Let . Since

then

Therefore, .

In the following, we discuss some relations between -Benson proper efficiency and some other proper efficiency.

Theorem 10. Let be a pointed closed convex cone, , and . Then, -Benson proper efficiency reduces to the Benson proper efficiency.

Proof. Since is a convex cone, then we have and hence, by , we can obtain that is an improvement set with respect to . Then, it follows from Remark 3.2 in [12] that

For , and satisfy Assumption B. Assume that is a -Benson proper efficient solution of and then, from Proposition 4.1 in [16], we have

which implies that is a Benson proper efficient solution of .

Theorem 11. Let be a pointed closed convex set and . If is an improvement set with respect to and , then -Benson proper efficiency reduces to the -Benson proper efficiency.

Proof. From Remark 3.2 in [12], we know that and satisfy Assumption B. Assume that is -Benson proper efficient solution of . We first point out that

In fact, since , then we only need to prove

Suppose that there exists such that . By applying separation theorem for convex sets, it follows that there exists such that

Let ; we have

Furthermore, we can show that . Since is an improvement set with respect to and by Lemma 4, we can obtain

which implies . This contradicts (18) and then (15) holds. Hence,

This means that is an -Benson proper efficient solution of .

Theorem 12. Let be a proper solid convex coradiant set, , , , and . Then, -Benson proper efficiency reduces to -proper efficiency.

Proof. From the convexity of and Lemma 1(i), we have

and so, from , it follows that

We first point out that and satisfy Assumption B. In fact, we only need to prove

For any , we only need to prove . On the contrary, suppose that . Since , then there exist

such that ; that is, . Hence, from Lemma 1(ii) and (22), we have

which contradicts and so and satisfy Assumption B. Furthermore, from and by means of (22), similar with the proof of (15), we have

From Lemma 7, it follows that

If is -Benson proper efficient solution of , then

It follows that

which implies that is a -proper efficient solution of .

4. A Characterization via Nonlinear Scalarization

In this section, we give a characterization of -Benson proper efficiency of via a kind of nonlinear scalarization function proposed by Göpfert et al.

Definition 13. Let be defined by

with .

Flores-Bazán and Hernández obtained the following nonconvex separation theorem.

Lemma 14 (see [13]). Let and satisfy Assumption B. Then,

We consider the following scalar optimization problem

where and . Denote by . Let and . If

then is called an -minimal solution of . Denote the set of -minimal solutions of by .

Theorem 15. Let and satisfy Assumption B and . Then,

Proof. Since , then

and it follows that

Therefore,

Furthermore, we can verify that

In fact, from Lemma 2.5 in [17], we have

Hence, from (37), we deduce that

From Lemma 14, we can obtain that, for all ,

Let in (41); then we have

It follows from (40) that

Therefore,

Now, we calculate . In fact,

Hence, from (44), we have

which means that

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is partially supported by the National Natural Science Foundation of China (Grant nos. 11301574, 11271391, and 11171363), the Natural Science Foundation Project of Chongqing (Grant no. CSTC2012jjA00002), and the Research Fund for the Doctoral Program of Chongqing Normal University (13XLB029).

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