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).