Applied Mathematics, Article ID 649756, 5 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 Göpfert et al. Some examples are given to illustrate the main results. 1. Introduction Itiswellknownthatapproximatesolutionshavebeenplaying 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, 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 approximateproperefficiency,andobtainedsomecharacterizations 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,clA,and A,respectively.The cone generated by A is defined as cone A= αa. (1) α 0 AconeA Yis pointed if A ( A)={0}.LetKbeaclosed convexpointedconeinywith 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), (VP) x D where f:x Yand 0 =D X. Definition 1 (see [4, 6]). Let E Y.If0 Eand E+K=E, then E is said to be an improvement set with respect to K.
2 Applied Mathematics 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 Ybe an improvement set with respect to K.Afeasiblepointx 0 Dis said to be an E-Benson proper efficient solution of (VP) if cl (cone (f (D) +E f(x 0 ))) ( K) = {0}. (3) We denote the set of all E-Benson proper efficient solutions by x 0 PAE(f, E). Consider the following scalar optimization problem: minφ (x), (P) x Z where φ:x R, 0 =Z X.Letε 0and x 0 Z.If φ(x) φ(x 0 ) ε,forallx Z,thenx 0 is called an ε-minimal solution of (P). Thesetofallε-minimal solutions is denoted by AMin(φ, ε). Moreover,ifφ(x) > φ(x 0 ) ε,forallx Z, then x 0 is called a strictly ε-minimal solution of (P). Theset of all strictly ε-minimal solutions is denoted by SAMin(φ, ε). 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 functionproposedby Göpfert et al. Let ξ q,e :Y R {± }be defined by ξ q,e (y) = inf {s R y sq E}, y Y, (4) with inf 0=+. Lemma 4. Let E Ybe a closed improvement set with respect to K and q int K.Thenξ q,e is continuous and {y Y ξ q,e (y)<c}=cq int E, c R, {y Y ξ q,e (y) = c} = cq E, c R, ξ q,e ( E) 0, ξ q,e ( E) =0. Proof. This can be easily seen from Proposition 2.3.4 and Theorem 2.3.1 in [9]. Consider the following scalar optimization problem: (5) min x D ξ q,e (f (x) y), (P q,y ) where y Y, q int K. Denoteξ q,e (f(x) y) by (ξ q,e,y f)(x),thesetofε-minimal solutions of (P q,y ) by AMin(ξ q,e,y f, ε),andthesetofstrictlyε-minimal solutions of (P q,y ) by SAMin(ξ q,e,y f,ε). Theorem 5. 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) x 0 PAE (f,e) x 0 AMin (ξ q,e,f(x0 ) f,ε); (ii) additionally, if cone(f(d) + E f(x 0 )) is a closed set, then x 0 SAMin (ξ q,e,f(x0 ) f,ε) x 0 PAE (f, E). (6) Proof. We first prove (i). Assume that x 0 PAE(f, E). Then we have cl (cone (f (D) +E f(x 0 ))) ( K) = {0}. (7) We can prove that (f (D) +E f(x 0 )) ( int K) =0. (8) (f (x 0 ) int E) f(d) =0. (9) On the contrary, there exists x Dsuch that f ( x) f(x 0 ) int E. (10) Hence, from Theorem 3.1 in [8], it follows that f ( x) f(x 0 ) E int K. (11) f ( x) f(x 0 ) +E int K, (12) which contradicts (8) andso(9) holds.fromlemma 4, we obtain From (9), we have {y Y ξ q,e (y) < 0} = int E. (13) (f (D) f(x 0 )) ( int E) =0. (14) By using (13)and(14), we deduce that Thus, (f (D) f(x 0 )) {y Y ξ q,e (y)<0}=0. (15) (ξ q,e,f(x0 ) f)(x) =ξ q,e (f (x) f(x 0 )) 0, x D. (16) In addition, since {s R ++ sq int(e K)} {s R sq E}, (ξ q,e,f(x0 ) f)(x 0 )=ξ q,e (0) = inf {s R sq E} ε. (17) It follows from (16)that (ξ q,e,f(x0 ) f)(x) (ξ q,e,f(x0 ) f)(x 0 ) ε. (18) x 0 AMin(ξ q,e,f(x0 ) f,ε). Next, we prove (ii). Suppose that x 0 SAMin(ξ q,e,f(x0 ) f, ε) and x 0 PAE(f, E). Since cone(f(d) + E f(x 0 )) is
Applied Mathematics 3 a closed set, there exist 0 =d K, λ>0, x D,and e E such that Since K is a cone, we can obtain that d=λ(f( x) f(x 0 )+ e). (19) f ( x) f(x 0 )+ e K. (20) f ( x) f(x 0 ) e K E K= E. (21) Moreover, by Lemma4, wehave, foreveryc R, that is, cq + f ( x) f(x 0 ) cq E Let c=0in (23); then, we have On the other hand, from x 0 follows that =cq cl E ={y Y ξ q,e (y) c} ; (22) ξ q,e (cq + f ( x) f(x 0 )) c. (23) ξ q,e (f ( x) f(x 0 )) 0. (24) SAMin(ξ q,e,f(x0 ) f,ε),it ξ q,e (f ( x) f(x 0 )) > ξ q,e (f (x 0 ) f(x 0 )) ε In the following, we prove (25) ξ q,e (0) =ε. (26) We first point out that, for any s 0, sq E.Itisobviousthat 0 Ewhen s=0. Assume that there exists s <0such that sq E.Sinceq int(e K) K and sq K,wehave 0= sq sq E + K = E, (27) which contradicts the fact that E is an improvement set with respect to K.Hence, ξ q,e (0) = inf {s R 0 sq E} = inf {s R ++ sq E}. (28) Moreover, since q int(e K) K,wehave,foranys R ++, sq K.Itfollowsfrom(28)that ξ q,e (0) = inf {s R ++ sq E K}. (29) Hence (26) holdsandthus,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(x0 ) f,ε). Example 7. Let X=Y=R 2, K=R 2 +, f(x) = x,and E = {(x 1,x 2 ) x 1 +x 2 1,x 1 0,x 2 0}, D={(x 1,x 2 ) x 1 x 2 =0, 1 2 x 1 0}. (30) Clearly, K is a closed convex cone and E is a closed improvement set with respect to K. Letx 0 = (0, 0) D and q=(1,1) int(e K).Thenε=1/2since cl (cone (f (D) +E f(x 0 ))) ( K) (31) ={(x 1,x 2 ) x 1 +x 2 0} ( R 2 + )={(0, 0)}. Hence For any x D, ξ q,e (f (x) f(x 0 )) = ξ q,e (f (x)) x 0 PAE (f, E). (32) = inf {s R f(x) sq E} 0= 1 2 1 2 (33) x 0 AMin (ξ q,e,f(x0 ) f,ε). (34) However, there exists x = ( 1/2, 1/2) D such that ξ q,e (f ( x) f(x 0 )) = ξ q,e (f ( x)) Hence = inf {s R f( x) sq E} =0= 1 2 1 2 (35) x 0 SAMin (ξ q,e,f(x0 ) f,ε). (36) 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=R 2, K=R 2 +, f(x) = x,and E={(x 1,x 2 ) x 1 +x 2 1,x 1 0,x 2 1 2 }, D = {(x 1,x 2 ) x 1 0,x 2 =0}. (37) Clearly, K is a closed convex cone and E is a closed improvement set with respect to K. Letx 0 = (0, 0) D and q=(1,1) int(e K).Thenε=1/2and cone (f (D) +E f(x 0 )) ={(x 1,x 2 ) x 1 R,x 2 >0} {(0, 0)} (38)
4 Applied Mathematics is not a closed set, since for any x D However, there exists x = ( 1/2, 1/2) D such that ξ q,e (f (x) f(x 0 )) = ξ q,e (f (x)) ξ q,e (f ( x) f(x 0 )) = ξ q,e (f ( x)) = inf {s R f(x) sq E} (39) = inf {s R f( x) sq E} (47) = 1 2 > 1 2 1 2 =0= 1 2 1 2 However, x 0 SAMin (ξ q,e,f(x0 ) f,ε). (40) cl (cone (f (D) +E f(x 0 ))) ( K) ={(x 1,x 2 ) x 1 R,x 2 0} ( R 2 + ) ={(x 1,x 2 ) x 1 0,x 2 =0} = {(0, 0)}. (41) x 0 PAE (f, E). (42) Remark 10. Theorem 5(ii) may not be true if x 0 SAMin(ξ q,e,f(x0 ) f,ε)is replaced by x 0 AMin(ξ q,e,f(x0 ) f, ε) and the following example can illustrate it. Example 11. Let X=Y=R 2, K=R 2 +, f(x) = x,and E={(x 1,x 2 ) x 1 +x 2 1,x 1 1 2,x 2 0} {(x 1,x 2 ) x 1 1 2,x 2 1 2 }, D={(x 1,x 2 ) x 1 x 2 =0, 1 2 x 1 0}. (43) Clearly, K is a closed convex cone and E is a closed improvement set with respect to K. Letx 0 = (0, 0) D and q=(1,1) int(e K).Thenε=1/2and cone (f (D) +E f(x 0 )) ={(x 1,x 2 ) x 1 R,x 2 0} {(x 1,x 2 ) x 1 +x 2 0,x 1 0,x 2 0} is a closed set, since for any x D ξ q,e (f (x) f(x 0 )) = ξ q,e (f (x)) = inf {s R f(x) sq E} 0= 1 2 1 2 (44) (45) x 0 AMin (ξ q,e,f(x0 ) f,ε). (46) Hence, Moreover, x 0 SAMin (ξ q,e,f(x0 ) f,ε). (48) cl (cone (f (D) +E f(x 0 ))) ( K) ={(x 1,x 2 ) x 1 R,x 2 0} {(x 1,x 2 ) x 1 +x 2 0,x 1 0,x 2 0} ( R 2 + ) ={(x 1,x 2 ) x 1 0,x 2 =0} Conflict of Interests = {(0, 0)}. (49) x 0 PAE (f, E). (50) 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). References [1] C. Gutiérrez, B. Jiménez, 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,andK.L.Teo, Optimalityconditionsfor approximate solutions of vector optimization problems, Journal of Industrial and Management Optimization,vol.7,no.2,pp. 483 496, 2011. [3] F. Flores-Bazán and E. Hernández, 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, Optimization Theory and Applications, vol.150,no.3,pp.516 529, 2011.
Applied Mathematics 5 [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. Gutiérrez, B. Jiménez, and V. Novo, Improvement sets and vector optimization, European 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 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. Göpfert, C. Tammer, H. Riahi, and C. Zălinescu, Variational Methods in Partially Ordered Spaces, Springer, New York, NY, USA, 2003.
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