A Characterization of E -Benson Proper Efficiency via Nonlinear Scalarization in Vector Optimization

Abstract

A class of vector optimization problems is considered and a characterization of E -Benson proper efficiency is obtained by using a nonlinear scalarization function proposed by Göpfert et al. Some examples are given to illustrate the main results.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Academic Editor:Xian-Jun Long

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

Received 22 February 2014; Accepted 14 April 2014; 28 April 2014

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.

1. Introduction

It is well known that approximate solutions have been playing an important role in vector optimization theory and applications. During the recent years, there are a lot of works related to vector optimization and some concepts of approximate solutions of vector optimization problems are proposed and some characterizations of these approximate solutions are studied; see, for example, [1-3] and the references therein.

Recently, Chicoo et al. proposed the concept of E -efficiency by means of improvement sets in a finite dimensional Euclidean space in [4]. E -efficiency unifies some known exact and approximate solutions of vector optimization problems. Zhao and Yang proposed a unified stability result with perturbations by virtue of improvement sets under the convergence of a sequence of sets in the sense of Wijsman in [5]. Furthermore, Gutiérrez et al. generalized the concepts of improvement sets and E -efficiency to a general Hausdorff locally convex topological linear space in [6]. Zhao et al. established linear scalarization theorem and Lagrange multiplier theorem of weak E -efficient solutions under the nearly E -subconvexlikeness in [7]. Moreover, Zhao and Yang also introduced a kind of proper efficiency, named E -Benson proper efficiency which unifies some proper efficiency and approximate proper efficiency, and obtained some characterizations of E -Benson proper efficiency in terms of linear scalarization in [8].

Motivated by the works of [8, 9], by making use of a kind of nonlinear scalarization functions proposed by Göpfert et al., we establish nonlinear scalarization results of E -Benson proper efficiency in vector optimization. We also give some examples to illustrate the main results.

2. Preliminaries

Let X be a linear space and let Y be a real Hausdorff locally convex topological linear space. For a nonempty subset A in Y , we denote the topological interior, the topological closure, and the boundary of A by int ... A , cl ... A , and ∂ A , respectively. The cone generated by A is defined as [figure omitted; refer to PDF] A cone A ⊂ Y is pointed if A ∩ ( - A ) = { 0 } . Let K be a closed convex pointed cone in Y with nonempty topological interior. For any x , y ∈ Y , we define [figure omitted; refer to PDF] In this paper, we consider the following vector optimization problem: [figure omitted; refer to PDF] where f : X [arrow right] Y and ∅ ...0; D ⊂ X .

Definition 1 (see [4, 6]).

Let E ⊂ Y . If 0 ∉ E and E + K = E , then E is said to be an improvement set with respect to K .

Remark 2.

If E ...0; ∅ , then, from Theorem 3.1 in [8], it is clear that int ... E ...0; ∅ . Throughout this paper, we assume that E ...0; ∅ .

Definition 3 (see [8]).

Let E ⊂ Y be an improvement set with respect to K . A feasible point _{x 0} ∈ D is said to be an E -Benson proper efficient solution of ( VP ) if [figure omitted; refer to PDF] We denote the set of all E -Benson proper efficient solutions by _{x 0} ∈ PAE ( f , E ) .

Consider the following scalar optimization problem: [figure omitted; refer to PDF] where [varphi] : X [arrow right] ... , ∅ ...0; Z ⊂ X . Let ... ...5; 0 and _{x 0} ∈ Z . If [varphi] ( x ) ...5; [varphi] ( _{x 0} ) - ... , for all x ∈ Z , then _{x 0} is called an ... -minimal solution of ( P ) . The set of all ... -minimal solutions is denoted by AMin ( [varphi] , ... ) . Moreover, if [varphi] ( x ) > [varphi] ( _{x 0} ) - ... , for all x ∈ Z , then _{x 0} is called a strictly ... -minimal solution of ( P ) . The set of all strictly ... -minimal solutions is denoted by SAMin ( [varphi] , ... ) .

3. A Characterization of E -Benson Proper Efficiency

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

Let _{ξ q , E} : Y [arrow right] ... ∪ { ± ∞ } be defined by [figure omitted; refer to PDF] with inf ... ∅ = + ∞ .

Lemma 4.

Let E ⊂ Y be a closed improvement set with respect to K and q ∈ int ... K . Then _{ξ q , E} is continuous and [figure omitted; refer to PDF]

Proof.

This can be easily seen from Proposition 2.3.4 and Theorem 2.3.1 in [9].

Consider the following scalar optimization problem: [figure omitted; refer to PDF] where y ∈ Y , q ∈ int ... K . Denote _{ξ q , E} ( f ( x ) - y ) by ( _{ξ q , E , y} [composite function] f ) ( x ) , the set of ... -minimal solutions of ( _{P q , y} ) by AMin ( _{ξ q , E , y} [composite function] f , ... ) , and the set of strictly ... -minimal solutions of ( _{P q , y} ) by SAMin ( _{ξ q , E , y} [composite function] f , ... ) .

Theorem 5.

Let E ⊂ Y be a closed improvement set with respect to K , q ∈ int ... ( E ∩ K ) and ... = inf ... { s ∈ _{... + +} |" s q ∈ int ... ( E ∩ K ) } . Then

(i) _{x 0} ∈ PAE ( f , E ) [implies] _{x 0} ∈ AMin ( _{ξ q , E , f ( x 0 )} [composite function] f , ... ) ;

(ii) additionally, if cone ... ( f ( D ) + E - f ( _{x 0} ) ) is a closed set, then [figure omitted; refer to PDF]

Proof.

We first prove (i). Assume that _{x 0} ∈ PAE ( f , E ) . Then we have [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF] We can prove that [figure omitted; refer to PDF] On the contrary, there exists x ^ ∈ D such that [figure omitted; refer to PDF] Hence, from Theorem 3.1 in [8], it follows that [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF] which contradicts (8) and so (9) holds. From Lemma 4, we obtain [figure omitted; refer to PDF] From (9), we have [figure omitted; refer to PDF] By using (13) and (14), we deduce that [figure omitted; refer to PDF] Thus, [figure omitted; refer to PDF] In addition, since { s ∈ _{... + +} |" s q ∈ int ... ( E ∩ K ) } ⊂ { s ∈ ... |" s q ∈ E } , [figure omitted; refer to PDF] It follows from (16) that [figure omitted; refer to PDF] Therefore, _{x 0} ∈ AMin ( _{ξ q , E , f ( x 0 )} [composite function] f , ... ) .

Next, we prove (ii). Suppose that _{x 0} ∈ SAMin ( _{ξ q , E , f ( x 0 )} [composite function] f , ... ) and _{x 0} ∉ PAE ( f , E ) . Since cone ... ( f ( D ) + E - f ( _{x 0} ) ) is a closed set, there exist 0 ...0; d ∈ - K , λ > 0 , x ^ ∈ D , and e ^ ∈ E such that [figure omitted; refer to PDF] Since K is a cone, [figure omitted; refer to PDF] Therefore, we can obtain that [figure omitted; refer to PDF] Moreover, by Lemma 4, we have, for every c ∈ ... , [figure omitted; refer to PDF] that is, [figure omitted; refer to PDF] Let c = 0 in (23); then, we have [figure omitted; refer to PDF] On the other hand, from _{x 0} ∈ SAMin ( _{ξ q , E , f ( x 0 )} [composite function] f , ... ) , it follows that [figure omitted; refer to PDF] In the following, we prove [figure omitted; refer to PDF] We first point out that, for any s ...4; 0 , s q ∉ E . It is obvious that 0 ∉ E when s = 0 . Assume that there exists s ^ < 0 such that s ^ q ∈ E . Since q ∈ int ... ( E ∩ K ) ⊂ K and - s ^ q ∈ K , we have [figure omitted; refer to PDF] which contradicts the fact that E is an improvement set with respect to K . Hence, [figure omitted; refer to PDF] Moreover, since q ∈ int ... ( E ∩ K ) ⊂ K , we have, for any s ∈ _{... + +} , s q ∈ K . It follows from (28) that [figure omitted; refer to PDF] Hence (26) holds and thus, by (25), we obtain _{ξ q , E} ( f ( x ^ ) - f ( _{x 0} ) ) > 0 , which contradicts (24) and so _{x 0} ∈ PAE ( f , E ) .

Remark 6.

_{x 0} ∈ PAE ( f , E ) does not imply _{x 0} ∈ SAMin ( _{ξ q , E , f ( x 0 )} [composite function] f , ... ) .

Example 7.

Let X = Y = ^{... 2} , K = ^{... + 2} , f ( x ) = x , and [figure omitted; refer to PDF] Clearly, K is a closed convex cone and E is a closed improvement set with respect to K . Let _{x 0} = ( 0,0 ) ∈ D and q = ( 1,1 ) ∈ int ... ( E ∩ K ) . Then ... = 1 / 2 since [figure omitted; refer to PDF] Hence [figure omitted; refer to PDF] For any x ∈ D , [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF] However, there exists x ^ = ( - 1 / 2 , - 1 / 2 ) ∈ D such that [figure omitted; refer to PDF] Hence [figure omitted; refer to PDF]

Remark 8.

Theorem 5(ii) may not be true if the closedness of cone ... ( f ( D ) + E - f ( _{x 0} ) ) is removed and the following example can illustrate it.

Example 9.

Let X = Y = ^{... 2} , K = ^{... + 2} , f ( x ) = x , and [figure omitted; refer to PDF] Clearly, K is a closed convex cone and E is a closed improvement set with respect to K . Let _{x 0} = ( 0,0 ) ∈ D and q = ( 1,1 ) ∈ int ... ( E ∩ K ) . Then ... = 1 / 2 and [figure omitted; refer to PDF] is not a closed set, since for any x ∈ D [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF] However, [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF]

Remark 10.

Theorem 5(ii) may not be true if _{x 0} ∈ SAMin ( _{ξ q , E , f ( x 0 )} [composite function] f , ... ) is replaced by _{x 0} ∈ AMin ( _{ξ q , E , f ( x 0 )} [composite function] f , ... ) and the following example can illustrate it.

Example 11.

Let X = Y = ^{... 2} , K = ^{... + 2} , f ( x ) = x , and [figure omitted; refer to PDF] Clearly, K is a closed convex cone and E is a closed improvement set with respect to K . Let _{x 0} = ( 0,0 ) ∈ D and q = ( 1,1 ) ∈ int ... ( E ∩ K ) . Then ... = 1 / 2 and [figure omitted; refer to PDF] is a closed set, since for any x ∈ D [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF] However, there exists x ^ = ( - 1 / 2 , - 1 / 2 ) ∈ D such that [figure omitted; refer to PDF] Hence, [figure omitted; refer to PDF] Moreover, [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF]

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).

Conflict of Interests

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

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