Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 649756, 5 pages http://dx.doi.org/10.1155/2014/649756

Research Article

A Characterization of / -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 Gopfert 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 inefficiency 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

Gopfert 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 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 dA, respectively. The cone generated by A is defined as

cone A = y aA.

A cone AcY is pointed if An (-A) = {0}. Let K be a closed convex pointed cone in Y with nonempty topological interior. For any x,yeY, we define

x<K y ^^ y - x e K.

In this paper, we consider the following vector optimization problem:

min/ (x),

where f:X ^ Y and 0 = DcX.

Definition 1 (see [4, 6]). Let E cY.H0 £ E and E + K = E, then E is said to be an improvement set with respect to K.

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 cY be an improvement set with respect to K. A feasible point x0 e D is said to be an £-Benson proper efficient solution of (VP) if

cl (cone (f(D)+E-f (x0))) n (-K) = {0}. (3)

We denote the set of all £-Benson proper efficient solutions by x0 e PAE(f, E).

Consider the following scalar optimization problem:

min0 (x), (P)

xtZ v '

where $ : X ^ R, 0 = Z c X. Let e > 0 and x0 e Z. If <p(x) > $(x0) - e, for all x e Z, then x0 is called an e-minimal solution of (P). The set of all e-minimal solutions is denoted by AMin(0, e). Moreover, if <p(x) > $>(x0) - e, for all x e Z, then x0 is called a strictly e-minimal solution of (P). The set of all strictly e-minimal solutions is denoted by SAMin(0, e).

3. A Characterization of E-Benson Proper Efficiency

In this section, we give a characterization of £-Benson proper efficiency of (VP) via a kind of nonlinear scalarization function proposed by Gopfert et al.

Let :Y ^ R U {±ra} be defined by

Zq,E (y) = inf jse R | y esq -E}, y eY, (4) with inf 0 =

Lemma 4. Let E cY be a closed improvement set with respect to K and q e int K. Then ^qE is continuous and

[y eY I $qE (y) <c} =cq- int E, Vc e R,

[yeYI (y) = c\=cq- dE, Vc e R, (5)

$q,E (-E) < 0, ^q E (-dE) = 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:

min^ (f(x)-y), (pJ

where y eY, q e int K. Denote Zq,E(f(x) - y) by (Zq,E,y ° /)(x),the setof e-minimal solutions of (P) by AMin(^qEy0 f, e), and the set of strictly e-minimal solutions of (P) by SAMintf^ of,e). ' □

Theorem 5. Let E c Y be a closed improvement set with respect to K, q e int(£ n K) and e = inf {s e R++ | sq e int(£ n K)}. Then

(i) ^ e PAE (f,E) ^ X0 e AMin tfq,E,f(Xo) ° f,e);

(ii) additionally, if cone(f(D) + E - f(x0)) is a closed set, then

X0 e SAMin (tq,E,f{Xo) °f,e)=^X0 e PAE (f, E). (6)

Proof. We first prove (i). Assume that x0 e PAE(f,E). Then we have

cl (cone (f(D) + E-f (X0))) n (-K) = {0}. (7) Therefore,

(f(D) + E-f(X0))n(- int K) = 0. (8)

We can prove that

(f(X0)- int E)nf(D) = 0. (9)

On the contrary, there exists x e D such that

f(x)-f(x0)e- int E. (10)

Hence, from Theorem 3.1 in [8], it follows that

f(x)-f(X0) e -E- int K. (11)

Therefore,

f(x)-f(x0) + Ec- int K, (12)

which contradicts (8) and so (9) holds. From Lemma 4, we obtain

[yeYIZq,E (>0<0} = - int E. (13)

From (9), we have

(f(D)-f(x0))n(-int E) = 0. (14)

By using (13) and (14), we deduce that

(f (D) - f (X0)) n[yeYI Zq,E (y) < 0} = 0. (15)

(W*»)0 f) (x) = ^e (f (*) - f (x0)) >0, Vxe D.

In addition, since {s e R++ | sq e int(£ n K)} c{se R I sq e E},

(Zq,E,f(Xo) 0 f) (X0) = (0) = inf js e R IsqeE}< e.

It follows from (16) that

(W*0) 0 f) (X) > (^q,E,f(x0) 0 f) (X0) - e. (18)

Therefore, X0 e AMin(^EJ{Xo) 0 f, e).

Next, we prove (ii). Suppose that x0 e SAMin(^qEj(x») 0 f,e) and x0 £ PAE(f,E). Since cone(f(D) + E - f(x0)) is

a closed set, there exist 0 = d e -K, X > 0, x e D, and e e E such that

d = X(f(x)-f(xQ) + e). (19)

Since K is a cone,

f(5e)-f(Xo) + ee-K. (20)

Therefore, we can obtain that

f(x)-f(x0)e-e-Kc-E-K = -E. (21) Moreover, by Lemma 4, we have, for every c e R, cq + f (x) - f (xQ) e cq- E

that is,

= cq - cl E

Le (c1 + f(x)-f (X0)) < C.

Let c = 0 in (23); then, we have

Zq,E (f&)-f(x0))<0.

On the other hand, from x0 e SAMin^^^) o f,e), it follows that

Zq,E (f (e - f (xq)) > Zq,E (f (x0) - f (x0)) - e = Zq,E (0)-e. In the following, we prove

Zq,E (0) = e.

We first point out that, for any s < 0, sqi E. It is obvious that 0 i E when s = 0. Assume that there exists e < 0 such that eeq e E. Since q e int(£ n K) c K and -eeq e K, we have

0 = sq - sq e E + K = E,

which contradicts the fact that E is an improvement set with respect to K. Hence,

(0) = inf jse R \ 0 esq-E} = inf js e R++ \ sqe E}.

Moreover, since q e int(EnK) c K,wehave, forany s e R++, sq e K. It follows from (28) that

(0) = inf jse R++ \sqeEnK]. (29)

Hence (26) holds and thus, by (25), we obtain £,qyE(f(x) -f(x0)) > 0, which contradicts (24) and so x0 e PAE(f, E). □

Remark 6. x0 e PAE(f, E) does not imply x0 e

SAMin(^EJ{Xo) of,e).

Example 7. Let X = Y = R2, K= R+, f(x) = x, and E = x2) \ x1 + x2 > 1,x1 > 0, x2 > 0},

D = {(x1, x2) \ x1 - x2 = 0, -1 < x1 < 0} .

Clearly, K is a closed convex cone and E is a closed improvement set with respect to K. Let x0 = (0,0) e D and q = (1,1) e int(EnK). Thene = 1/2 since

cl (cone (f(D)+E-f(x0)))n(-K)

= j(X1,X2) \ X1 +X2 > 0} n (-R+) = {(0, 0)}.

x0 e PAE (f, E).

For any x e D,

Zq,E (f(x)-f(xQ))=Zq,E (f(x))

= inf js e R \ f (x) esq-E}

>0=1--1-22

= Zq,E (0)-e.

Therefore,

x0e AMin ($q,E,f(Xo) o f,e). (34)

However, there exists x = (-1/2,-1/2) e D such that Zq,E (f(x)-f(Xo)) = ^q,E (f(x))

= inf js e R \ f (x) esq-E}

=0=1-1 2 2

= Zq,E (0)-e.

x0l SAMin (tq,EJ(Xo) °f,e).

Remark 8. 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.

Example 9. Let X = Y = R2, K = R+, f(x) = x, and E = {(x1, x2) \ x1 + x2 > 1,x1 > 0,x2 > , D = jj(x1,x2) \ x1 < 0,x2 = 0}.

Clearly, K is a closed convex cone and E is a closed improvement set with respect to K. Let x0 = (0,0) e D and q = (1,1) e int(EnK). Thene = 1/2 and

cone (f(D)+E-f(x0))

= j(xx,x2) \x1 e R,x2 > 0} u {(0,0)}

is not a closed set, since for any x e D

tq,E (f(x)-f(Xo))=HqJ! (f(*))

= inf jse R | f (x) esq-E}

_1 11 = 2 2 2

_Zq,E (0)-e.

Therefore,

e SAMin (Zq,E,f{Xo) °f,e). (40)

However,

cl (cone (f(D)+E-f(x0)))n(-K)

_j(x1,x2)lx1 e R,x2 >0}n(-R+) (41) _j(xi,x2)lxi <0,X2 _ 0} _ {(0,0)}.

Therefore,

x0$. PAE (f,E). (42)

Remark 10. Theorem 5(ii) may not be true if x0 e

SAMin(^,E,/(*0) ° f' e) is rePlaced by X0 e AMin(^EJ(Xo) ° f, e) and the following example can illustrate it.

Example 11. Let X_Y _ R2, K_ R+, f(x) _ x, and E _ {(%!' x2) l x1 + x2 > 1,x1 > 2, x2 > 0}

U{(X1'X2)lX1 <1,X2 >1}' (43)

D _ {(%!, x2) l x1 - x2 _ 0, < x1 < 0} .

Clearly, K is a closed convex cone and E is a closed improvement set with respect to K. Let x0 _ (0,0) e D and q _ (1,1) e int(EnK). Thene _ 1/2 and

cone (f(D) + E-f(x0))

_j(x1,x2) l x1 e R,x2 > 0} (44)

U jj(x1,x2) l x1 + x2 > 0,x1 > 0, x2 < 0}

is a closed set, since for any x e D

tq,E (f(x)-f(Xo))_Zq,E (fM)

_ inf js e R l f (x) esq-E}

>o=1--1-2 2

Therefore,

However, there exists x _ (-1/2,-1/2) e D such that Zq,E (f(x)-f(Xo))_^q,E (f(x))

_ inf js e R l f (x) esq-E}

_0_1--1-22

_Zq,E (0)-e.

Hence,

x0t SAMin° f,e) . (48)

Moreover, cl (cone (f(D)+E-f(x0)))n(-K) _ j(x1,x2) l x1 e R,x2 > 0}

U j(x1,x2) l x1 +x2 >0,x1 > 0, x2 < 0} n (-R+) _ j(x1,x2) lX1 <0,X2 _ 0} _ {(0,0)}.

Therefore,

X0l PAE (f,E).

^ e AMin (^q,E,f(Xo) °f,e).

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).

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