Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 649756, pages http://dx.doi.org/10.1155/2014/649756 Research Article A Characterization of 𝐸-Benson Proper Efficiency via Nonlinear Scalarization in Vector Optimization Ke Quan Zhao, Yuan Mei Xia, and Hui Guo College of Mathematics Science, Chongqing Normal University, Chongqing 401331, China Correspondence should be addressed to Ke Quan Zhao; kequanz@163.com Received 22 February 2014; Accepted 14 April 2014; Published 28 April 2014 Academic Editor: Xian-Jun Long Copyright © 2014 Ke Quan Zhao et al 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 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 Găopfert et al Some examples are given to illustrate the main results 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 𝐸efficiency 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, Guti´errez 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 Găopfert 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 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 cone 𝐴 = ⋃ 𝛼𝐴 𝛼≥0 (1) 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 𝑥 ≤𝐾 𝑦 ⇐⇒ 𝑦 − 𝑥 ∈ 𝐾 (2) In this paper, we consider the following vector optimization problem: min𝑓 (𝑥) , 𝑥∈𝐷 (VP) where 𝑓 : 𝑋 → 𝑌 and ≠ 𝐷 ⊂ 𝑋 Definition (see [4, 6]) Let 𝐸 ⊂ 𝑌 If ∉ 𝐸 and 𝐸 + 𝐾 = 𝐸, then 𝐸 is said to be an improvement set with respect to 𝐾 2 Journal of Applied Mathematics Remark If 𝐸 ≠ 0, then, from Theorem 3.1 in [8], it is clear that int 𝐸 ≠ Throughout this paper, we assume that 𝐸 ≠ (ii) additionally, if cone(𝑓(𝐷) + 𝐸 − 𝑓(𝑥0 )) is a closed set, then Definition (see [8]) Let 𝐸 ⊂ 𝑌 be an improvement set with respect to 𝐾 A feasible point 𝑥0 ∈ 𝐷 is said to be an 𝐸-Benson proper efficient solution of (VP) if 𝑥0 ∈ SAMin (𝜉𝑞,𝐸,𝑓(𝑥0 ) ∘ 𝑓, 𝜖) ⇒ 𝑥0 ∈ PAE (𝑓, 𝐸) (6) cl (cone (𝑓 (𝐷) + 𝐸 − 𝑓 (𝑥0 ))) ∩ (−𝐾) = {0} (3) We denote the set of all 𝐸-Benson proper efficient solutions by 𝑥0 ∈ PAE(𝑓, 𝐸) Consider the following scalar optimization problem: min𝜙 (𝑥) , cl (cone (𝑓 (𝐷) + 𝐸 − 𝑓 (𝑥0 ))) ∩ (−𝐾) = {0} where 𝜙 : 𝑋 → R, ≠ 𝑍 ⊂ 𝑋 Let 𝜖 ≥ and 𝑥0 ∈ 𝑍 If 𝜙(𝑥) ≥ 𝜙(𝑥0 ) − 𝜖, for all 𝑥 ∈ 𝑍, then 𝑥0 is called an 𝜖-minimal solution of (P) The set of all 𝜖-minimal solutions is denoted by AMin(𝜙, 𝜖) Moreover, if 𝜙(𝑥) > 𝜙(𝑥0 ) − 𝜖, for all 𝑥 ∈ 𝑍, then 𝑥0 is called a strictly 𝜖-minimal solution of (P) The set of all strictly 𝜖-minimal solutions is denoted by SAMin(𝜙, 𝜖) (𝑓 (𝐷) + 𝐸 − 𝑓 (𝑥0 )) ∩ (− int 𝐾) = (8) We can prove that (𝑓 (𝑥0 ) − int 𝐸) ∩ 𝑓 (𝐷) = (9) On the contrary, there exists 𝑥̂ ∈ 𝐷 such that ̂ − 𝑓 (𝑥0 ) ∈ − int 𝐸 𝑓 (𝑥) (10) Hence, from Theorem 3.1 in [8], it follows that 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 Găopfert et al Let , : R {±∞} be defined by 𝑦 ∈ 𝑌, (4) ̂ − 𝑓 (𝑥0 ) ∈ −𝐸 − int 𝐾 𝑓 (𝑥) (11) ̂ − 𝑓 (𝑥0 ) + 𝐸 ⊂ − int 𝐾, 𝑓 (𝑥) (12) Therefore, which contradicts (8) and so (9) holds From Lemma 4, we obtain {𝑦 ∈ 𝑌 | 𝜉𝑞,𝐸 (𝑦) < 0} = − int 𝐸 with inf = +∞ Lemma Let 𝐸 ⊂ 𝑌 be a closed improvement set with respect to 𝐾 and 𝑞 ∈ int 𝐾 Then 𝜉𝑞,𝐸 is continuous and {𝑦 ∈ 𝑌 | 𝜉𝑞,𝐸 (𝑦) < 𝑐} = 𝑐𝑞 − int 𝐸, {𝑦 ∈ 𝑌 | 𝜉𝑞,𝐸 (𝑦) = 𝑐} = 𝑐𝑞 − 𝜕𝐸, 𝜉𝑞,𝐸 (−𝐸) ≤ 0, ∀𝑐 ∈ R, ∀𝑐 ∈ R, From (9), we have (𝑓 (𝐷) − 𝑓 (𝑥0 )) ∩ (− int 𝐸) = (5) (14) (𝑓 (𝐷) − 𝑓 (𝑥0 )) ∩ {𝑦 ∈ 𝑌 | 𝜉𝑞,𝐸 (𝑦) < 0} = (15) Thus, Proof This can be easily seen from Proposition 2.3.4 and Theorem 2.3.1 in [9] Consider the following scalar optimization problem: 𝑥∈𝐷 (13) By using (13) and (14), we deduce that 𝜉𝑞,𝐸 (−𝜕𝐸) = min𝜉𝑞,𝐸 (𝑓 (𝑥) − 𝑦) , (7) Therefore, (P) 𝑥∈𝑍 𝜉𝑞,𝐸 (𝑦) = inf {𝑠 ∈ R | 𝑦 ∈ 𝑠𝑞 − 𝐸} , Proof We first prove (i) Assume that 𝑥0 ∈ PAE(𝑓, 𝐸) Then we have (P𝑞,𝑦 ) where 𝑦 ∈ 𝑌, 𝑞 ∈ int 𝐾 Denote 𝜉𝑞,𝐸 (𝑓(𝑥) − 𝑦) by (𝜉𝑞,𝐸,𝑦 ∘ 𝑓)(𝑥), the set of 𝜖-minimal solutions of (P𝑞,𝑦 ) by AMin(𝜉𝑞,𝐸,𝑦 ∘ 𝑓, 𝜖), and the set of strictly 𝜖-minimal solutions of (P𝑞,𝑦 ) by SAMin(𝜉𝑞,𝐸,𝑦 ∘ 𝑓, 𝜖) Theorem Let 𝐸 ⊂ 𝑌 be a closed improvement set with respect to 𝐾, 𝑞 ∈ int(𝐸 ∩ 𝐾) and 𝜖 = inf{𝑠 ∈ R++ | 𝑠𝑞 ∈ int(𝐸 ∩ 𝐾)} Then (i) 𝑥0 ∈ PAE (𝑓, 𝐸) ⇒ 𝑥0 ∈ AMin (𝜉𝑞,𝐸,𝑓(𝑥0 ) ∘ 𝑓, 𝜖); (𝜉𝑞,𝐸,𝑓(𝑥0 ) ∘ 𝑓) (𝑥) = 𝜉𝑞,𝐸 (𝑓 (𝑥) − 𝑓 (𝑥0 )) ≥ 0, ∀𝑥 ∈ 𝐷 (16) In addition, since {𝑠 ∈ R++ | 𝑠𝑞 ∈ int(𝐸 ∩ 𝐾)} ⊂ {𝑠 ∈ R | 𝑠𝑞 ∈ 𝐸}, (𝜉𝑞,𝐸,𝑓(𝑥0 ) ∘ 𝑓) (𝑥0 ) = 𝜉𝑞,𝐸 (0) = inf {𝑠 ∈ R | 𝑠𝑞 ∈ 𝐸} ≤ 𝜖 (17) It follows from (16) that (𝜉𝑞,𝐸,𝑓(𝑥0 ) ∘ 𝑓) (𝑥) ≥ (𝜉𝑞,𝐸,𝑓(𝑥0 ) ∘ 𝑓) (𝑥0 ) − 𝜖 (18) Therefore, 𝑥0 ∈ AMin(𝜉𝑞,𝐸,𝑓(𝑥0 ) ∘ 𝑓, 𝜖) Next, we prove (ii) Suppose that 𝑥0 ∈ SAMin(𝜉𝑞,𝐸,𝑓(𝑥0 ) ∘ 𝑓, 𝜖) and 𝑥0 ∉ PAE(𝑓, 𝐸) Since cone(𝑓(𝐷) + 𝐸 − 𝑓(𝑥0 )) is Journal of Applied Mathematics a closed set, there exist ≠ 𝑑 ∈ −𝐾, 𝜆 > 0, 𝑥̂ ∈ 𝐷, and 𝑒̂ ∈ 𝐸 such that ̂ − 𝑓 (𝑥0 ) + 𝑒̂) 𝑑 = 𝜆 (𝑓 (𝑥) Example Let 𝑋 = 𝑌 = R2 , 𝐾 = R2+ , 𝑓(𝑥) = 𝑥, and 𝐸 = {(𝑥1 , 𝑥2 ) | 𝑥1 + 𝑥2 ≥ 1, 𝑥1 ≥ 0, 𝑥2 ≥ 0} , (19) 𝐷 = {(𝑥1 , 𝑥2 ) | 𝑥1 − 𝑥2 = 0, − ≤ 𝑥1 ≤ 0} Since 𝐾 is a cone, ̂ − 𝑓 (𝑥0 ) + 𝑒̂ ∈ −𝐾 𝑓 (𝑥) (20) Therefore, we can obtain that ̂ − 𝑓 (𝑥0 ) ∈ −̂ 𝑓 (𝑥) 𝑒 − 𝐾 ⊂ −𝐸 − 𝐾 = −𝐸 (21) Clearly, 𝐾 is a closed convex cone and 𝐸 is a closed improvement set with respect to 𝐾 Let 𝑥0 = (0, 0) ∈ 𝐷 and 𝑞 = (1, 1) ∈ int(𝐸 ∩ 𝐾) Then 𝜖 = 1/2 since cl (cone (𝑓 (𝐷) + 𝐸 − 𝑓 (𝑥0 ))) ∩ (−𝐾) = {(𝑥1 , 𝑥2 ) | 𝑥1 + 𝑥2 ≥ 0} ∩ (−R2+ ) = {(0, 0)} Moreover, by Lemma 4, we have, for every 𝑐 ∈ R, 𝑥0 ∈ PAE (𝑓, 𝐸) (22) 𝜉𝑞,𝐸 (𝑓 (𝑥) − 𝑓 (𝑥0 )) = 𝜉𝑞,𝐸 (𝑓 (𝑥)) that is, = inf {𝑠 ∈ R | 𝑓 (𝑥) ∈ 𝑠𝑞 − 𝐸} (23) ≥0= Let 𝑐 = in (23); then, we have ̂ − 𝑓 (𝑥0 )) ≤ 𝜉𝑞,𝐸 (𝑓 (𝑥) = 𝜉𝑞,𝐸 (0) − 𝜖 (25) 𝑥0 ∈ AMin (𝜉𝑞,𝐸,𝑓(𝑥0 ) ∘ 𝑓, 𝜖) However, there exists 𝑥̂ = (−1/2, −1/2) ∈ 𝐷 such that ̂ ∈ 𝑠𝑞 − 𝐸} = inf {𝑠 ∈ R | 𝑓 (𝑥) (26) (27) which contradicts the fact that 𝐸 is an improvement set with respect to 𝐾 Hence, 𝜉𝑞,𝐸 (0) = inf {𝑠 ∈ R | ∈ 𝑠𝑞 − 𝐸} = inf {𝑠 ∈ R++ | 𝑠𝑞 ∈ 𝐸} (28) =0= 1 − 2 Hence 𝑥0 ∉ SAMin (𝜉𝑞,𝐸,𝑓(𝑥0 ) ∘ 𝑓, 𝜖) Example Let 𝑋 = 𝑌 = R2 , 𝐾 = R2+ , 𝑓(𝑥) = 𝑥, and 𝜉𝑞,𝐸 (0) = inf {𝑠 ∈ R++ | 𝑠𝑞 ∈ 𝐸 ∩ 𝐾} 𝐷 = {(𝑥1 , 𝑥2 ) | 𝑥1 ≤ 0, 𝑥2 = 0} ̂ − Hence (26) holds and thus, by (25), we obtain 𝜉𝑞,𝐸 (𝑓(𝑥) 𝑓(𝑥0 )) > 0, which contradicts (24) and so 𝑥0 ∈ PAE(𝑓, 𝐸) ∈ (36) Remark Theorem 5(ii) may not be true if the closedness of cone(𝑓(𝐷)+𝐸−𝑓(𝑥0 )) is removed and the following example can illustrate it 𝐸 = {(𝑥1 , 𝑥2 ) | 𝑥1 + 𝑥2 ≥ 1, 𝑥1 ≥ 0, 𝑥2 ≥ } , (29) (35) = 𝜉𝑞,𝐸 (0) − 𝜖 Moreover, since 𝑞 ∈ int(𝐸∩𝐾) ⊂ 𝐾, we have, for any 𝑠 ∈ R++ , 𝑠𝑞 ∈ 𝐾 It follows from (28) that Remark 𝑥0 ∈ PAE(𝑓, 𝐸) does not imply 𝑥0 SAMin(𝜉𝑞,𝐸,𝑓(𝑥0 ) ∘ 𝑓, 𝜖) (34) ̂ − 𝑓 (𝑥0 )) = 𝜉𝑞,𝐸 (𝑓 (𝑥)) ̂ 𝜉𝑞,𝐸 (𝑓 (𝑥) We first point out that, for any 𝑠 ≤ 0, 𝑠𝑞 ∉ 𝐸 It is obvious that ∉ 𝐸 when 𝑠 = Assume that there exists 𝑠̂ < such that 𝑠̂𝑞 ∈ 𝐸 Since 𝑞 ∈ int(𝐸 ∩ 𝐾) ⊂ 𝐾 and −̂𝑠𝑞 ∈ 𝐾, we have = 𝑠̂𝑞 − 𝑠̂𝑞 ∈ 𝐸 + 𝐾 = 𝐸, (33) Therefore, In the following, we prove 𝜉𝑞,𝐸 (0) = 𝜖 1 − 2 = 𝜉𝑞,𝐸 (0) − 𝜖 (24) On the other hand, from 𝑥0 ∈ SAMin(𝜉𝑞,𝐸,𝑓(𝑥0 ) ∘ 𝑓, 𝜖), it follows that ̂ − 𝑓 (𝑥0 )) > 𝜉𝑞,𝐸 (𝑓 (𝑥0 ) − 𝑓 (𝑥0 )) − 𝜖 𝜉𝑞,𝐸 (𝑓 (𝑥) (32) For any 𝑥 ∈ 𝐷, = {𝑦 ∈ 𝑌 | 𝜉𝑞,𝐸 (𝑦) ≤ 𝑐} ; ̂ − 𝑓 (𝑥0 )) ≤ 𝑐 𝜉𝑞,𝐸 (𝑐𝑞 + 𝑓 (𝑥) (31) Hence ̂ − 𝑓 (𝑥0 ) ∈ 𝑐𝑞 − 𝐸 𝑐𝑞 + 𝑓 (𝑥) = 𝑐𝑞 − cl 𝐸 (30) (37) Clearly, 𝐾 is a closed convex cone and 𝐸 is a closed improvement set with respect to 𝐾 Let 𝑥0 = (0, 0) ∈ 𝐷 and 𝑞 = (1, 1) ∈ int(𝐸 ∩ 𝐾) Then 𝜖 = 1/2 and cone (𝑓 (𝐷) + 𝐸 − 𝑓 (𝑥0 )) = {(𝑥1 , 𝑥2 ) | 𝑥1 ∈ R, 𝑥2 > 0} ∪ {(0, 0)} (38) Journal of Applied Mathematics However, there exists 𝑥̂ = (−1/2, −1/2) ∈ 𝐷 such that is not a closed set, since for any 𝑥 ∈ 𝐷 𝜉𝑞,𝐸 (𝑓 (𝑥) − 𝑓 (𝑥0 )) = 𝜉𝑞,𝐸 (𝑓 (𝑥)) ̂ − 𝑓 (𝑥0 )) = 𝜉𝑞,𝐸 (𝑓 (𝑥)) ̂ 𝜉𝑞,𝐸 (𝑓 (𝑥) = inf {𝑠 ∈ R | 𝑓 (𝑥) ∈ 𝑠𝑞 − 𝐸} = ̂ ∈ 𝑠𝑞 − 𝐸} = inf {𝑠 ∈ R | 𝑓 (𝑥) (39) 1 > − 2 =0= = 𝜉𝑞,𝐸 (0) − 𝜖 1 − 2 (47) = 𝜉𝑞,𝐸 (0) − 𝜖 Therefore, Hence, 𝑥0 ∈ SAMin (𝜉𝑞,𝐸,𝑓(𝑥0 ) ∘ 𝑓, 𝜖) (40) However, 𝑥0 ∉ SAMin (𝜉𝑞,𝐸,𝑓(𝑥0 ) ∘ 𝑓, 𝜖) (48) Moreover, cl (cone (𝑓 (𝐷) + 𝐸 − 𝑓 (𝑥0 ))) ∩ (−𝐾) = {(𝑥1 , 𝑥2 ) | 𝑥1 ∈ R, 𝑥2 ≥ 0} ∩ (−R2+ ) cl (cone (𝑓 (𝐷) + 𝐸 − 𝑓 (𝑥0 ))) ∩ (−𝐾) (41) = {(𝑥1 , 𝑥2 ) | 𝑥1 ≤ 0, 𝑥2 = 0} ≠ {(0, 0)} ∪ {(𝑥1 , 𝑥2 ) | 𝑥1 + 𝑥2 ≥ 0, 𝑥1 ≥ 0, 𝑥2 ≤ 0} ∩ (−R2+ ) Therefore, 𝑥0 ∉ PAE (𝑓, 𝐸) (42) Remark 10 Theorem 5(ii) may not be true if 𝑥0 ∈ SAMin(𝜉𝑞,𝐸,𝑓(𝑥0 ) ∘ 𝑓, 𝜖) is replaced by 𝑥0 ∈ AMin(𝜉𝑞,𝐸,𝑓(𝑥0 ) ∘ 𝑓, 𝜖) and the following example can illustrate it Example 11 Let 𝑋 = 𝑌 = R2 , 𝐾 = R2+ , 𝑓(𝑥) = 𝑥, and (50) 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) (44) References (45) [1] C Guti´errez, B Jim´enez, and V Novo, “A unified approach and optimality conditions for approximate solutions of vector optimization problems,” SIAM Journal on Optimization, vol 17, no 3, pp 688–710, 2006 [2] Y Gao, X Yang, and K L Teo, “Optimality conditions for approximate solutions of vector optimization problems,” Journal of Industrial and Management Optimization, vol 7, no 2, pp 483–496, 2011 [3] F Flores-Baz´an and E Hern´andez, “A unified vector optimization problem: complete scalarizations and applications,” Optimization, vol 60, no 12, pp 1399–1419, 2011 [4] M Chicco, F Mignanego, L Pusillo, and S Tijs, “Vector optimization problem via improvement sets,” Journal of Optimization Theory and Applications, vol 150, no 3, pp 516–529, 2011 ∪ {(𝑥1 , 𝑥2 ) | 𝑥1 + 𝑥2 ≥ 0, 𝑥1 ≥ 0, 𝑥2 ≤ 0} is a closed set, since for any 𝑥 ∈ 𝐷 𝜉𝑞,𝐸 (𝑓 (𝑥) − 𝑓 (𝑥0 )) = 𝜉𝑞,𝐸 (𝑓 (𝑥)) = inf {𝑠 ∈ R | 𝑓 (𝑥) ∈ 𝑠𝑞 − 𝐸} = 𝜉𝑞,𝐸 (0) − 𝜖 Therefore, 𝑥0 ∈ AMin (𝜉𝑞,𝐸,𝑓(𝑥0 ) ∘ 𝑓, 𝜖) 𝑥0 ∉ PAE (𝑓, 𝐸) Acknowledgments cone (𝑓 (𝐷) + 𝐸 − 𝑓 (𝑥0 )) 1 − 2 Therefore, (43) Clearly, 𝐾 is a closed convex cone and 𝐸 is a closed improvement set with respect to 𝐾 Let 𝑥0 = (0, 0) ∈ 𝐷 and 𝑞 = (1, 1) ∈ int(𝐸 ∩ 𝐾) Then 𝜖 = 1/2 and ≥0= (49) The authors declare that there is no conflict of interests regarding the publication of this paper 𝐷 = {(𝑥1 , 𝑥2 ) | 𝑥1 − 𝑥2 = 0, − ≤ 𝑥1 ≤ 0} = {(𝑥1 , 𝑥2 ) | 𝑥1 ∈ R, 𝑥2 ≥ 0} = {(𝑥1 , 𝑥2 ) | 𝑥1 ≤ 0, 𝑥2 = 0} ≠ {(0, 0)} Conflict of Interests 𝐸 = {(𝑥1 , 𝑥2 ) | 𝑥1 + 𝑥2 ≥ 1, 𝑥1 ≥ , 𝑥2 ≥ 0} 1 ∪ {(𝑥1 , 𝑥2 ) | 𝑥1 ≤ , 𝑥2 ≥ } , 2 = {(𝑥1 , 𝑥2 ) | 𝑥1 ∈ R, 𝑥2 ≥ 0} (46) Journal of Applied Mathematics [5] K Q Zhao and X M Yang, “A unified stability result with perturbations in vector optimization,” Optimization Letters, vol 7, no 8, pp 1913–1919, 2013 [6] C Guti´errez, B Jim´enez, and V Novo, “Improvement sets and vector optimization,” European Journal of Operational Research, vol 223, no 2, pp 304–311, 2012 [7] K Q Zhao, X M Yang, and J W Peng, “Weak E-optimal solution in vector optimization,” Taiwanese Journal of Mathematics, vol 17, no 4, pp 1287–1302, 2013 [8] K Q Zhao and X M Yang, “E-Benson proper efficiency in vector optimization, Optimization, 2013 [9] A Găopfert, C Tammer, H Riahi, and C Z˘alinescu, Variational Methods in Partially Ordered Spaces, Springer, New York, NY, USA, 2003 Copyright of Journal of Applied Mathematics is the property of Hindawi Publishing Corporation and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission However, users may print, download, or email articles for individual use ... 3.1 in [8], it follows that A Characterization of