A Kind of Unified Proper Efficiency in Vector Optimization

and Applied Analysis 3 Theorem 10. Let K ⊂ Y be a pointed closed convex cone, S = intK, and q ∈ intK. Then, S-Benson proper efficiency reduces to the Benson proper efficiency. Proof. Since K is a convex cone, then we have intK + K = intK and hence, by 0 ∉ S, we can obtain that S is an improvement set with respect to K. Then, it follows from Remark 3.2 in [12] that cl (Y \ (−S)) +R ++ q ⊂ int (Y \ (−S)) . (13) For 0 ∈ ∂S, q and S satisfy Assumption B. Assume that x is a S-Benson proper efficient solution of (VP) and then, from Proposition 4.1 in [16], we have cl (cone (f (D) + K − f (x))) ∩ (−K) = cl (cone (f (D) + intK − f (x))) ∩ (−K) = cl (cone (f (D) + S − f (x))) ∩ (− cl (cone (conv (S)))) = {0} , (14) which implies that x is a Benson proper efficient solution of (VP). Theorem 11. Let K ⊂ Y be a pointed closed convex set and q ∈ intK. If S = E ⊂ K is an improvement set with respect to K and 0 ∈ ∂S, then S-Benson proper efficiency reduces to the E-Benson proper efficiency. Proof. From Remark 3.2 in [12], we know that q and S satisfy Assumption B. Assume that x is S-Benson proper efficient solution of (VP). We first point out that cl (cone (conv S)) = K. (15) In fact, since S ⊂ K, then we only need to prove K ⊂ cl (cone (conv S)) . (16) Suppose that there exists k 0 ∈ K such that k 0 ∉ cl (cone (conv S)). By applying separation theorem for convex sets, it follows that there exists λ ∈ Y∗ \ {0 Y ∗} such that ⟨λ, k 0 ⟩ > ⟨λ, e⟩ , ∀e ∈ cl (cone (convS)) . (17) Let e = 0; we have ⟨λ, k 0 ⟩ > 0. (18) Furthermore, we can show that −λ ∈ (cl(cone(convS)))+ = S. Since S is an improvement set with respect to K and by Lemma 4, we can obtain −λ ∈ E + = K + , (19) which implies ⟨λ, k 0 ⟩ ≤ 0. This contradicts (18) and then (15) holds. Hence, cl (cone (f (D) + E − f (x))) ∩ (−K) = cl (cone (f (D) + S − f (x))) ∩ (− cl (cone (convS))) = {0} . (20) This means that x is an E-Benson proper efficient solution of (VP). Theorem 12. Let C be a proper solid convex coradiant set, q ∈ intC(0), ε ≥ 0, S = C(ε), and 0 ∈ ∂S. Then, S-Benson proper efficiency reduces to (C, ε)-proper efficiency. Proof. From the convexity of S and Lemma 1(i), we have cl S + clC (0) = clC (ε)+clC (0) ⊂ cl (C (ε)+C (0)) ⊂ clC (ε) = cl S, (21) and so, from 0 ∈ clC(0), it follows that cl S + clC (0) = cl S. (22) We first point out that q and S satisfy Assumption B. In fact, we only need to prove Y \ (− int S) +R ++ q ⊂ Y \ (− cl S) . (23) For any x ∈ Y \ (− int S) + R ++ q, we only need to prove x ∉ − cl S. On the contrary, suppose that −x ∈ cl S. Since x ∈ Y \ (− int S) +R ++ q, then there exist x 1 ∈ Y \ (− int S) , x 2 ∈ R ++ q (24) such that x = x 1 + x 2 ; that is, −x 1 = −x + x 2 . Hence, from Lemma 1(ii) and (22), we have −x 1 ∈ cl S +R ++ q ⊂ cl S + intC (0) ⊂ int (cl S + C (0)) ⊂ int (cl S + clC (0)) = int (cl S) = int S, (25) which contradicts x 1 ∈ Y \ (− int S) and so q and S satisfy Assumption B. Furthermore, from S ⊂ C(0) and by means of (22), similar with the proof of (15), we have cl (cone (conv (cl S))) = clC (0) . (26) From Lemma 7, it follows that cl (cone (convS)) = cl (cone S) = cl (cone (cl S)) = cl (cone (conv (cl S))) = clC (0) . (27) If x is S-Benson proper efficient solution of (VP), then cl (cone (f (D) + C (ε) − f (x))) ∩ (− clC (0)) = cl (cone (f (D) + S − f (x))) ∩ (− cl (cone (convS))) = {0} . (28) It follows that cl (cone (f (D) + C (ε) − f (x))) ∩ (−C (0)) ⊂ {0} , (29) which implies that x is a (C, ε)-proper efficient solution of (VP). 4 Abstract and Applied Analysis 4. A Characterization via Nonlinear Scalarization In this section, we give a characterization of S-Benson proper efficiency of (VP) via a kind of nonlinear scalarization function proposed by Göpfert et al. Definition 13. Let ξ q,S : Y → R ∪ {±∞} be defined by ξ q,S (y) = inf {t ∈ R | y ∈ tq − S} , y ∈ Y, (30) with inf 0 = +∞. Flores-Bazán and Hernández obtained the following nonconvex separation theorem. Lemma 14 (see [13]). Let q and S satisfy Assumption B. Then, {y ∈ Y | ξ q,S (y) < c} = cq − int S, ∀c ∈ R, {y ∈ Y | ξ q,S (y) = c} = cq − ∂S, ∀c ∈ R, ξ q,S (−S) ≤ 0, ξ q,S (−∂S) = 0. (31) We consider the following scalar optimization problem


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

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 int , cl , , and  \ , respectively.A set  is solid if int  ̸ = 0 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 0 ̸ =  ⊂ .We say that  is a coradiant set if  satisfies  ∈  for every  ∈ ,  > 1.Let  ⊂  be a proper solid coradiant set and define Lemma 1 (see [5]).Let  be a proper solid convex coradiant set.Then, Definition 2 (see [5]).Let  ≥ 0. A feasible point  ∈  is said to be a (, )-proper efficient solution of (VP) Definition 3 (see [10]).A nonempty set  ⊂  is said to be an improvement set with respect to  if 0 ∉  and  +  = .
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 (VP) if Flores-Bazán and Hernández introduced Assumption B as follows.

A Kind of Unified Proper Efficiency
In this section, we propose a kind of unified proper efficiency of (VP) by means of Assumption B by using the idea of the classical Benson proper efficiency and discuss some relations with other proper efficiency.
Denote by PAE(, ) the set of -Benson proper efficient solutions of (VP).
In the following, we discuss some relations between -Benson proper efficiency and some other proper efficiency.Proof.Since  is a convex cone, then we have int  +  = int  and hence, by 0 ∉ , we can obtain that  is an improvement set with respect to .Then, it follows from Remark 3.2 in [12] For 0 ∈ ,  and  satisfy Assumption B. Assume that  is a -Benson proper efficient solution of (VP) and then, from Proposition 4.1 in [16], we have which implies that  is a Benson proper efficient solution of (VP).
In fact, since  ⊂ , then we only need to prove Suppose that there exists  0 ∈  such that  0 ∉ cl (cone (conv )).By applying separation theorem for convex sets, it follows that there exists  ∈  * \ {0  * } such that Let  = 0; we have Furthermore, we can show that − ∈ (cl(cone(conv))) + =  + .Since  is an improvement set with respect to  and by Lemma 4, we can obtain This means that  is an -Benson proper efficient solution of (VP).
Proof.From the convexity of  and Lemma 1(i), we have and so, from 0 ∈ cl (0), it follows that We first point out that  and  satisfy Assumption B. In fact, we only need to prove For any  ∈  \ (− int ) + R ++ , we only need to prove  ∉ − cl .On the contrary, suppose that − ∈ cl .Since  ∈  \ (− int ) + R ++ , then there exist such that  =  1 +  2 ; that is, − 1 = − +  2 .Hence, from Lemma 1(ii) and ( 22 which implies that  is a (, )-proper efficient solution of (VP).

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

Theorem 11 .
Let  ⊂  be a pointed closed convex set and  ∈ int .If  =  ⊂  is an improvement set with respect to  and 0 ∈ , 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 (VP).We first point out that cl (cone (conv )) = .