Abstract
A class of vector optimization problems is considered and a characterization of -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.
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 -efficiency by means of improvement sets in a finite dimensional Euclidean space in [4]. -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 -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 -efficient solutions under the nearly -subconvexlikeness in [7]. Moreover, Zhao and Yang also introduced a kind of proper efficiency, named -Benson proper efficiency which unifies some proper efficiency and approximate proper efficiency, and obtained some characterizations of -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 -Benson proper efficiency in vector optimization. We also give some examples to illustrate the main results.
2. Preliminaries
Let be a linear space and let be a real Hausdorff locally convex topological linear space. For a nonempty subset in , we denote the topological interior, the topological closure, and the boundary of by , , and , respectively. The cone generated by is defined as A cone is pointed if . 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 .
Definition 1 (see [4, 6]). Let . If and , then is said to be an improvement set with respect to .
Remark 2. If , then, from Theorem 3.1 in [8], it is clear that . Throughout this paper, we assume that .
Definition 3 (see [8]). Let be an improvement set with respect to . A feasible point is said to be an -Benson proper efficient solution of if
We denote the set of all -Benson proper efficient solutions by .
Consider the following scalar optimization problem:
where , . Let and . If , for all , then is called an -minimal solution of . The set of all -minimal solutions is denoted by . Moreover, if , for all , then is called a strictly -minimal solution of . The set of all strictly -minimal solutions is denoted by .
3. A Characterization of -Benson Proper Efficiency
In this section, we give a characterization of -Benson proper efficiency of via a kind of nonlinear scalarization function proposed by Göpfert et al.
Let be defined by with .
Lemma 4. Let be a closed improvement set with respect to and . Then is continuous and
Proof. This can be easily seen from Proposition 2.3.4 and Theorem 2.3.1 in [9].
Consider the following scalar optimization problem:
where , . Denote by , the set of -minimal solutions of by , and the set of strictly -minimal solutions of by .
Theorem 5. Let be a closed improvement set with respect to , and . Then(i);(ii)additionally, if is a closed set, then
Proof. We first prove (i). Assume that . Then we have
Therefore,
We can prove that
On the contrary, there exists such that
Hence, from Theorem 3.1 in [8], it follows that
Therefore,
which contradicts (8) and so (9) holds. From Lemma 4, we obtain
From (9), we have
By using (13) and (14), we deduce that
Thus,
In addition, since ,
It follows from (16) that
Therefore, .
Next, we prove (ii). Suppose that and . Since is a closed set, there exist , , , and such that
Since is a cone,
Therefore, we can obtain that
Moreover, by Lemma 4, we have, for every ,
that is,
Let in (23); then, we have
On the other hand, from , it follows that
In the following, we prove
We first point out that, for any , . It is obvious that when . Assume that there exists such that . Since and , we have
which contradicts the fact that is an improvement set with respect to . Hence,
Moreover, since , we have, for any , . It follows from (28) that
Hence (26) holds and thus, by (25), we obtain , which contradicts (24) and so .
Remark 6. does not imply .
Example 7. Let , , , and Clearly, is a closed convex cone and is a closed improvement set with respect to . Let and . Then since Hence For any , Therefore, However, there exists such that Hence
Remark 8. Theorem 5(ii) may not be true if the closedness of is removed and the following example can illustrate it.
Example 9. Let , , , and Clearly, is a closed convex cone and is a closed improvement set with respect to . Let and . Then and is not a closed set, since for any Therefore, However, Therefore,
Remark 10. Theorem 5(ii) may not be true if is replaced by and the following example can illustrate it.
Example 11. Let , , , and Clearly, is a closed convex cone and is a closed improvement set with respect to . Let and . Then and is a closed set, since for any Therefore, However, there exists such that Hence, Moreover, Therefore,
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).