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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 570918, 9 pages
http://dx.doi.org/10.1155/2013/570918
Research Article

Strict Efficiency in Vector Optimization with Nearly Convexlike Set-Valued Maps

1Department of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
2Chongqing Police College, Chongqing 401331, China
3Basic Course Department, Nanchang Institute of Science and Technology, Nanchang 330108, China

Received 10 December 2012; Revised 6 March 2013; Accepted 11 March 2013

Academic Editor: Sheng-Jie Li

Copyright © 2013 Xiaohong Hu 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.

Abstract

The concept of the well posedness for a special scalar problem is linked with strictly efficient solutions of vector optimization problem involving nearly convexlike set-valued maps. Two scalarization theorems and two Lagrange multiplier theorems for strict efficiency in vector optimization involving nearly convexlike set-valued maps are established. A dual is proposed and duality results are obtained in terms of strictly efficient solutions. A new type of saddle point, called strict saddle point, of an appropriate set-valued Lagrange map is introduced and is used to characterize strict efficiency.

1. Introduction

One important problem in vector optimization is to find the efficient points of a set. As observed by Kuhn, Tucker, and later by Geoffrion, some efficient points exhibit certain abnormal properties. To eliminate such abnormal efficient points, various concepts of proper efficiency have been introduced. The original concept was introduced by Kuhn and Tucker [1] and Geoffrion [2] and was later modified and formulated in a more general framework by Borwein [3], Hartley [4], Benson [5], Henig [6], and Borwein and Zhuang [7]; also see the references therein. In particular, the concept of strict efficiency was first introduced by Bednarczak and Song [8] in order to obtain upper semicontinuity of the section mapping at an efficient point. Zaffaroni [9] used a special scalar function to characterize the strict efficiency and obtained some properties of strict efficiency, which includes well posedness.

Recently, several authors have turned their interests to vector optimization of set-valued maps. For instance, see [1016]. Li [17] extended the concept of Benson proper efficiency to set-valued maps and presented two scalarization theorems and Lagrange multiplier theorems for set-valued vector optimization problem under cone subconvexlikeness. Mehra [18] and Xia and Qiu [19] discussed the super efficiency in vector optimization problem involving nearly cone-convexlike set-valued maps and nearly cone-subconvexlike set-valued maps, respectively. Miglierina [20] linked the properly efficient solutions of set-valued vector optimization with well-posedness hypothesis of a special scalar problem.

In this paper, inspired by [8, 17, 18], we study strict efficiency for vector optimization problem involving nearly cone-convexlike set-valued maps in the framework of real normed locally convex spaces. The paper is organized as follows. In Section 2, we recall some basic concepts and lemmas. In Section 3, the well posedness of a special scalar problems on strict efficiency involving nearly cone-convexlike set-valued maps is discussed. In Section 4, two scalarization theorems for strict efficiency in vector optimization problems involving nearly cone-convexlike set-valued maps are obtained. In Section 5, we establish two Lagrange multiplier theorems which show that strictly efficient solution of the constrained vector optimization problem is equivalent to strictly efficient solution of an appropriate unconstrained vector optimization problem. In Section 6, some results on strict duality are given. In Section 7, a new concept of strict saddle point for set-valued Lagrangian map is introduced and is then utilized to characterize strict efficiency.

2. Preliminaries

Throughout this paper, let be a linear space and and two real normed locally convex spaces, with topological dual spaces and . For a set , , , , and denote the closure, the interior, the boundary, and the complement of , respectively. Moreover, we will denote by the closed unit ball of . A set is said to be a cone if for any and . A cone  is said to be convex if , and it is said to be pointed if . The generated cone of is defined by The dual cone of is defined as The quasi-interior of is the set Recall that a base of a cone  is a convex subset of such that Of course, is pointed whenever has a base. Furthermore, if is a nonempty closed convex pointed cone in , then if and only if has a base.

Definition 1. Let be a nonempty subset of . The set of all strictly efficient points and the set of all efficient points with respect to the convex cone  with nonempty interior are defined as

It is easy to verify that Also, in this paper, we assume that and are pointed closed convex cone with nonempty interior. and are set-valued maps with nonempty value. Let be the space of continuous linear operations from to , and let Let be the set-valued map from to , denoted by If , , we also define and by and , respectively.

Definition 2 (see [12]). A set-valued map is said to be nearly -convexlike on if is convex in .

Lemma 3 (see [12]). Let be nearly -convexlike set-valued map on . Then exactly one of the following statement holds:(i) such that ,(ii) such that .

Lemma 4 (see [12]). If is nearly -convexlike on , then(i)for each , is nearly -convexlike on ;(ii)for each , is nearly -convexlike on .

Lemma 5 (see [21]). Let be a closed convex subset of and a closed convex pointed cone. Then , if and only if there exists such that , for all .

3. Strict Efficiency and Well Posedness

Consider the following vector optimization problem with set-valued maps: Denote the feasible solution set of by And denote the image of under by

Definition 6. A point is said to be a strictly efficient solution of , if there exists such that , and the point is said to be a strictly efficient minimizer of .

Definition 7. For a set , let the function be defined as where with .

The function was first introduced in [22], and its main properties are gathered together in the following proposition.

Proposition 8 (see [9]). If the set is nonempty and , then(1)   is real valued; (2)   is 1-Lipschitzian; (3)   for every ,   for every   , and     for every   ; (4)if   is closed, then it holds that   ;(5)if   is a convex, then   is convex;(6)if   is a cone, then   is positively homogeneous;(7)if   is a closed convex cone, then   is nonincreasing with respect to the ordering relation induced on   .

We consider the following parameterized scalar problem:

The following theorem characterize the relation between strictly efficient points of and the parameterized scalar problem .

Theorem 9. Let , . Then is a strictly efficient minimizer of if and only if there exists a nondecreasing function with and for all , such that for all .

Proof. Since is a strictly efficient minimizer of vector optimization problem can be rephrased as follows: for every there exists such that for every with . So suppose that the point is a strictly efficient minimizer of and consider the following functions: It is evident that is nondecreasing, null at the origin, and positive elsewhere; moreover, for every it holds that If, on the other hand, there exists a nondecreasing function with the above properties and such that for all , then it holds that for all with . To show that is a strictly efficient minimizer of , for every , we can let , and it implies that the proof is completed.

The scalar problem is Tikhonov well posed if for all with and

Theorem 10. Let , . The is a strictly efficient minimizer of if and only if is a solution of and the scalar problem is Tikhonov well posed.

Proof. If is a strictly efficient minimizer of , then, by Theorem 9, is the unique solution of and there exists a forcing function such that , for all . Since implies , hence for any sequence such that , then it must converge to .
Conversely, if the scalar problem is Tikhonov well posed, then holds for every . Thus, we consider the function ; it holds by the construction that , and it is to see that is nondecreasing on [0, ) with and for all . Hence, again, by the Theorem 9, we get that is a strictly efficient minimizer of .

4. Strict Efficiency and Linear Scalarization

In association with the vector optimization problem involving set-valued maps, we consider the following linearly scalar optimization problem with a set-valued map: where .

Definition 11. If , and then and are called a minimal solution and a minimizer of , respectively.

Lemma 12. Let , and is a closed convex cone with having a compact base . Then is a strictly efficient point of if and only if .

Proof. The definition of strict efficiency of can be rephrased as follows: for every , there exists such that for every with . Hence, if there exists sequence , , and some such that , then ; hence, . Since , , and is pointed, is outside some small ball around the origin. This shows that is not the strictly efficient point of , this contraction shows that .
Conversely, if is not a strictly efficient point of , then there exists and a sequence , and such that We write with and , then, by (16) and as is compact, there exists and such that . Indeed, by (16), we have ; furthermore, since is compact, thus does not converse to 0, and it implies that there exists a real number and such that for all . Now, we define , .
Thus, we obtain Hence, This contradiction shows that is a strictly efficient point of .

Theorem 13. Let , , let have a compact base, and let be fixed. If is a minimizer of , then is a strictly efficient minimizer of .

Proof. By Lemma 12, we need only to prove that Indeed, let . Then there exists , such that hence, Since is a minimizer of and , we have , while and imply that . Hence, . On the other hand, since , we have . Thus . Again, by and , we must have .
Hence, we have shown that . Therefore, this proof is completed.

Theorem 14. Let be nearly -convexlike on , , and , and let have a compact base. If is a strictly efficient minimizer of , then there exists such that is a minimizer of .

Proof. Since is a strictly efficient minimizer of , thus by Lemma 12, we have
By the definition of nearly -convexlike set-valued map , we have that is closed convex set in , since is a closed, convex, pointed, compact cone. Thus, by Lemma 5, there exists such that Since we obtain Therefore, is a minimizer of .

If we denote by the set of strictly efficient minimizer of and by the set of minimizer of , then from Theorems 13 and 14, we get immediately the following corollary.

Corollary 15. Let be nearly -convexlike on . Then,

5. Strict Efficiency and Lagrange Multipliers

In this section, we establish two Lagrange multiplier theorems which show that the set of strictly efficient minimizer of the constrained set-valued vector optimization problem , it is equivalent to the set of an appropriate unconstrained vector optimization problem.

The following concept is a generalization of Slater constraint qualification in mathematical programming and in vector optimization.

Definition 16. We say that satisfies the generalized Slater constraint qualification if there exists such that .

Theorem 17. Let be nearly -convexlike on . Let be nearly -convexlike on and let have a compact base. Furthermore, let satisfy the generalized Slater constraint qualification. If is a strictly efficient minimizer of , then there exists such that and is a strictly efficient minimizer of the following unconstrained vector optimization problem:

Proof. Since is a strictly efficient minimizer of , by Theorem 14, there exists such that Define by Since is nearly -convexlike on , by Lemma 4, we have that is nearly -convexlike on , while (27) implies that has no solution, and hence, by Lemma 3, there exists such that Since , that is, , this implies that there exists such that . Then, since , we get Also, let in (30) and noting that , and , we get .
Hence, We now claim that . If this is not the case, then By the generalized Slater constraint qualification, there exists such that and so there exists such that Hence, . But substituting into (30), and by taking and in (30), we have This contradiction shows that . From this and , we can choose such that and define the operator by Obviously, Thus, From (30) and (37), we obtain Dividing the above inequality by , we obtain Since, is nearly -convexlike on , by Lemma 4, is nearly -convexlike on . Therefore, by Theorem 13 and , we have that is a strictly efficient minimizer of .

Theorem 18. Let , and let have a compact base. If there exists such that and is a strictly efficient minimizer of , then is a strictly efficient minimizer of .

Proof. From the assumption, we have Since , we have . Thus there exists . Then, which implies that ; that is, , . So, we have This together with (43) implies Noting that , and by Lemma 12, we have that is a strictly efficient minimizer of .

6. Strict Efficiency and Duality

Definition 19. The Lagrange map for is the set-valued map defined by

We denote

Definition 20. The set-valued map is defined as which is called a strict dual map of .

Using this definition, we define the Lagrange dual problem associated with the primal problem as follows:

Definition 21. is called an efficient point of if

We can now establish the following dual theorems.

Theorem 22 (weak duality). If and , then

Proof. From , there exists such that Hence, In particular, Noting that , we choose . Then, , and taking in (54), we have Hence, from and , we obtain This completes the proof.

Theorem 23 (strong duality). Let be nearly C-convexlike on . Let be nearly ( )-convexlike on and let have a compact base. Furthermore, let satisfy the generalized Slater constraint qualification. If is a strictly efficient solution of , then there exists that is an efficient point of .

Proof. Since is a strictly efficient solution of , then there exists such that is a strictly efficient minimizer of . According to Theorem 17, there exists such that , and is a strictly efficient minimizer of . Hence, . By Theorem 22, we have Therefore, by Definition 20, we know that is an efficient point of .

7. Strict Efficient and Strict Saddle Point

We will now introduce a new concept of strict saddle point for a set-valued Lagrange map and use it to characterize strict efficiency.

For a nonempty subset of , we define a set It is easy to find that if and only if , and if is normed space and has a compact base. Then by Lemma 12, we have if and only if .

Definition 24. A pair is said to be a strict saddle point of Lagrange map if

We first present an important equivalent characterization for a strict saddle point of the Lagrange map .

Lemma 25. Let have a compact base. is said to be a strict saddle point of Lagrange map if only if there exist and such that(i) , (ii) .

Proof (necessity). Since is a strict saddle point of the map , by Definition 24 there exists , such that From (61), we have Since every , we have We have Hence, Thus, from (62), we have Let be defined by Then, (66) can be written as This together with Lemma 12 shows that is a strictly efficient point of the vector optimization problem
Since is a linear map, then of course is nearly -convexlike on . Hence, by Theorem 14, there exists such that Now, we claim that If this is not true, then since is a closed convex cone set, by the strong separation theorem in topological vector space [21], there exists such that In the above expression, taking gets while letting leads to Hence, Let be fixed, and define as It is evident that and that Hence, . Taking in (66), we obtain Hence, which contradicts (70). Therefore, Thus, , since . If , then hence, , by , while taking leads to This contradiction shows that , that is, condition (ii) holds.
Therefore, by (60) and (61), we know that is, condition (i) holds.
Sufficiency. From , , and condition (ii), we get And by condition (i), we obtain Therefore, is a strict saddle point of the set-valued Lagrange map , and the proof is complete.

The following saddle-point theorems allow us to express a strictly efficient solution of as a strict saddle of the set-valued Lagrange map .

Theorem 26. Let be nearly -convexlike on , let be nearly -convexlike on , and let have a compact base. Moreover, satisfies generalized Slater constrained qualification.(i)If is a strict saddle point of the map , then is strictly efficient solution of .(ii)If is a strictly efficient minimizer of , and , then there exists such that is a strict saddle point of the map .

Proof. (i) By the necessity of Lemma 25, we have and there exists such that is a strictly efficient minimizer of the problem
According to Theorem 18, is a strictly efficient minimizer of . Therefore, is strictly efficient solution of .
(ii) From the assumption, and by the Theorem 17, there exists such that Therefore, there exists such that . Hence, from Lemma 25, it follows that is a strict saddle point of the map .

Lemma 27. Let , , and . Then the following conditions:(a) ,(b) ,are equivalent to the following conditions:(i) , (ii) ,(iii) .

Proof. By conditions (a) and (b), it is easy to verify that conditions (i) and (iii) hold. Now, we show that . If this is not true, then there would exist such that Then, since is a closed convex set, by the strong separation theorem in topological vector space (see [21]), there exists such that In the above expression, taking gives and taking leads to Hence, Take and define as Then, and And, by condition (a), we have From and by (95), we have This conflicts with (99). Therefore, .
Conversely, by (i), we have that is, By condition (ii), for every , we have Thus, Therefore, by (99), we have that is, . Therefore, this proof is completed.

By Lemmas 25 and 27 and Theorem 26, we can obtain immediately the following corollary.

Corollary 28. Let be nearly -convexlike on , let be nearly -convexlike on , and let satisfy generalized Slater constrained qualification.(i)If is a strict saddle point of the map , then is strictly efficient solution of . (ii)If is a strictly efficient minimizer of and , , then there exists such that is a strict saddle point of the map .

Acknowledgments

Zhimiao Fang’s research was supported by the Natural Science Foundation Project of CQ CSTC (Grant no. cstc2012jjA00033). The authors would like to thank two anonymous referees for their valuable comments and suggestions, which helped to improve the paper.

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