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Tightly Proper Efficiency in Vector Optimization with Nearly ConeSubconvexlike SetValued Maps
Journal of Inequalities and Applications volumeÂ 2011, ArticleÂ number:Â 839679 (2011)
Abstract
A scalarization theorem and two Lagrange multiplier theorems are established for tightly proper efficiency in vector optimization involving nearly conesubconvexlike setvalued maps. A dual is proposed, and some duality results are obtained in terms of tightly properly efficient solutions. A new type of saddle point, which is called tightly proper saddle point of an appropriate setvalued Lagrange map, is introduced and is used to characterize tightly proper efficiency.
1. Introduction
One important problem in vector optimization is to find 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, there are many papers to introduce various concepts of proper efficiency; see [1â€“8]. Particularly, Zaffaroni [9] introduced the concept of tightly proper efficiency and used a special scalar function to characterize the tightly proper efficiency, and obtained some properties of tightly proper efficiency. Zheng [10] extended the concept of superefficiency from normed spaces to locally convex topological vector spaces. Guerraggio et al. [11] and Liu and Song [12] made a survey on a number of definitions of proper efficiency and discussed the relationships among these efficiencies, respectively.
Recently, several authors have turned their interests to vector optimization of setvalued maps, for instance, see [13â€“18]. Gong [19] discussed setvalued constrained vector optimization problems under the constraint ordering cone with empty interior. Sach [20] discussed the efficiency, weak efficiency and Benson proper efficiency in vector optimization problem involving icconeconvexlike setvalued maps. Li [21] extended the concept of Benson proper efficiency to setvalued maps and presented two scalarization theorems and Lagrange mulitplier theorems for setvalued vector optimization problem under conesubconvexlikeness. Mehra [22], Xia and Qiu [23] discussed the superefficiency in vector optimization problem involving nearly coneconvexlike setvalued maps, nearly conesubconvexlike setvalued maps, respectively. For other results for proper efficiencies in optimization problems with generalized convexity and generalized constraints, we refer to [24â€“26] and the references therein.
In this paper, inspired by [10, 21â€“23], we extend the concept of tight properness from normed linear spaces to locally convex topological vector spaces, and study tightly proper efficiency for vector optimization problem involving nearly conesubconvexlike setvalued maps and with nonempty interior of constraint cone in the framework of locally convex topological vector spaces.
The paper is organized as follows. Some concepts about tightly proper efficiency, superefficiency and strict efficiency are introduced and a lemma is given in Section 2. In Section 3, the relationships among the concepts of tightly proper efficiency, strict efficiency and superefficiency in local convex topological vector spaces are clarified. In Section 4, the concept of tightly proper efficiency for setvalued vector optimization problem is introduced and a scalarization theorem for tightly proper efficiency in vector optimization problems involving nearly conesubconvexlike setvalued maps is obtained. In Section 5, we establish two Lagrange multiplier theorems which show that tightly properly efficient solution of the constrained vector optimization problem is equivalent to tightly properly efficient solution of an appropriate unconstrained vector optimization problem. In Section 6, some results on tightly proper duality are given. Finally, a new concept of tightly proper saddle point for setvalued Lagrangian map is introduced and is then utilized to characterize tightly proper efficiency in Section 7. Section 8 contains some remarks and conclusions.
2. Preliminaries
Throughout this paper, let be a linear space, and be two real locally convex topological spaces (in brief, LCTS), with topological dual spaces and , respectively. For a set , , , , and denote the closure, the interior, the boundary, and the complement of , respectively. Moreover, by we denote 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
and the quasiinterior 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.
Also, in this paper, we assume that, unless indicated otherwise, and are pointed closed convex cones with and , respectively.
Definition 2.1 (see [27]).
Let be a base of . Define
Cheng and Fu in [27] discussed the propositions of , and the following remark also gives some propositions of .
Remark 2.2 (see [27]).

(i)
Let . Then if and only if there exists a neighborhood of such that .

(ii)
If is a bounded base of , then .
Definition 2.3.
A point is said to be efficient with respect to (denoted ) if
Remark 2.4 (see [28]).
If is a closed convex pointed cone and , then .
In [10], Zheng generalized two kinds of proper efficiency, namely, Henig proper efficiency and superefficiency, from normed linear spaces to LCTS. And Fu [8] generalized a kind of proper efficiency, namely strict efficiency, from normed linear spaces to LCTS. Let be an ordering cone with a base . Then , by the Hahn Banach separation theorem, there are a and an such that
Let . Then is a neighborhood of and
It is clear that, for each convex neighborhood of with , is convex and . Obviously, is convex pointed cone, indeed, is also a base of .
Definition 2.5 (see [8]).
Suppose that is a subset of and denotes the family of all bases of . is said to be a strictly efficient point with respect to , written as , if there is a convex neighborhood of such that
is said to be a strictly efficient point with respect to , written as, if
Remark 2.6.
Since is open in , thus is equivalent to .
Definition 2.7.
The point is called tightly properly efficient with respect to (denoted ) if there exists a convex cone with satisfying and there exists a neighborhood of such that
is said to be a tightly properly efficient point with respect to , written as, if
Now, we give the following example to illustrate Definition 2.7.
Example 2.8.
Let , . Given (see Figure 1). Thus, it follows from the direct computation and Definition 2.7 that
Remark 2.9.
By Definitions 2.7 and 2.3, it is easy to verify that
but, in general, the converse is not valid. The following example illustrates this case.
Example 2.10.
, , and . Then, by Definitions 2.3 and 2.7, we get
thus, .
Definition 2.11 (see [10]).
is called a superefficient point of a subset of with respect to ordering cone , written as , if, for each neighborhood of , there is neighborhood of such that
Definition 2.12 (see [29, 30]).
A setvalued map is said to be nearly subconvexlike on if is convex.
Given the two setvalued maps , , let
The product is called nearly subconvexlike on if is nearly subconvexlike on . Let be the space of continuous linear operators from to , and let
Denote by the setvalued map from to defined by
If , , we also define and by
respectively.
Lemma 2.13 (see [23]).
If is nearly subconvexlike on , then:
(i)for each , is nearly subconvexlike on ;
(ii)for each , is nearly subconvexlike on .
3. Tightly Proper Efficiency, Strict Efficiency, and Superefficiency
In [11, 12], the authors introduced many concepts of proper efficiency (tightly proper efficiency except) for normed spaces and for topological vector spaces, respectively. Furthermore, they discussed the relationships between superefficiency and other proper efficiencies. If we can get the relationship between tightly proper efficiency and superefficiency, then we can get the relationships between tightly proper efficiency and other proper efficiencies. So, in this section, the aim is to get the equivalent relationships between tightly proper efficiency and superefficiency under suitable assumption by virtue of strict efficiency.
Lemma 3.1.
If has a bounded base , then
Proof.
From the definition of and , we only need prove that for any . Indeed, for each , by the separation theorem, there exists such that
Hence, . Since is bounded, there exists such that
It is clear that and . If there exists such that , then for any convex cone with satisfying and for any neighborhood of such that
It implies that there exists such that
Then there is and such that , since , then there exists and such that . By (3.2) and (3.3), we see that . Therefore, and , it is a contradiction. Therefore, for each .
Proposition 3.2.
If has a bounded base , then
Proof.
By Definition 2.11, for any , there exists a convex neighborhood of with such that
It is easy to verify that
Now, let and by Lemma 3.1, we have
which implies that .
Proposition 3.3.
Let . Then
Proof.
For each , there exists a convex cone with satisfying
and there exists a neighborhood of such that
Since expression (3.11) can be equivalently expressed as
, and by (3.12), we have
Since is open in , we get
It implies that . Therefore this proof is completed.
Remark 3.4.
If does not have a bounded base, then the converse of Proposition 3.3 may not hold. The following example illustrates this case.
Example 3.5.
Let , (see Figure 2) and .
Then, let , we have . It follows from the definitions of and that
respectively. Thus, the converse of Proposition 3.3 is not valid.
Proposition 3.6 (see [8]).
If has a bounded base , then
From Propositions 3.2, 3.3, and 3.6, we can get immediately the following corollary.
Corollary 3.7.
If has a bounded base , then
Example 3.8.
Let , be given in Example 3.5 and . Then
Lemma 3.9 (see [23]).
Let be a closed convex pointed cone with a bounded base and . Then, .
From Corollary 3.7 and Lemma 3.9, we can get the following proposition.
Proposition 3.10.
If has a bounded base and is a nonempty subset of , then .
4. Tightly Proper Efficiency and Scalarization
Let be a closed convex pointed cone. We consider the following vector optimization problem with setvalued maps
where , are setvalued maps with nonempty values. Let be the set of all feasible solutions of (VP).
Definition 4.1.
is said to be a tightly properly efficient solution of (VP), if there exists such that .
We call is a tightly properly efficient minimizer of (VP). The set of all tightly properly efficient solutions of (VP) is denoted by TPE(VP).
In association with the vector optimization problem (VP) of setvalued maps, we consider the following scalar optimization problem with setvalued map :
where . The set of all optimal solutions of ( ) is denoted by , that is,
The fundamental results characterize tightly properly efficient solution of (VP) in terms of the solutions of ( ) are given below.
Theorem 4.2.
Let the cone have a bounded base . Let , , and be nearly subconvexlike on . Then if and only if there exists such that .
Proof.
Necessity. Let . Then, by Lemma 3.1 and Proposition 3.10, we have . Hence, there exists a convex cone with satisfying and there exists a convex neighborhood of such that
From the above expression and , we have
Since is open in , we have
By the assumption that is nearly subconvexlike on , thus is convex set. By the HahnBanach separation theorem, there exists such that
It is easy to see that
Hence, we obtain
Furthermore, according to Remark 2.2, we have .
Sufficiency. Suppose that there exists such that . Since has a bounded base , thus by Remark 2.2(ii), we know that . And by Remark 2.2(i), we can take a convex neighborhood of such that
By , we have
From the above expression and (4.8), we get
Therefore, . Noting that has a bounded base and by Lemma 3.1, we have .
Now, we give the following example to illustrate Theorem 4.2.
Example 4.3.
Let , and . Given , . Let
Thus, feasible set of (VP)
By Definition 4.1, we get
For any point , there exists such that
Indeed, for any , we consider the following three cases.
Case 1.
If is in the first quadrant, then for any such that .
Case 2.
If is in the second quadrant, then there exists such that . Let such that
Then, we have
Case 3.
If in the fourth quadrant, then there exists such that . Let such that
Then, we have
Therefore, if follows from Cases 1, 2, and 3 that there exists such that .
From Theorem 4.2, we can get immediately the following corollary.
Corollary 4.4.
Let the cone have a bounded base . For any if is nearly subconvexlike on . Then
5. Tightly Proper Efficiency and the Lagrange Multipliers
In this section, we establish two Lagrange multiplier theorems which show that tightly properly efficient solution of the constrained vector optimization problem (VP), is equivalent to tightly properly efficient solution of an appropriate unconstrained vector optimization problem.
Definition 5.1 (see [17]).
Let be a closed convex pointed cone with . We say that (VP) satisfies the generalized Slater constraint qualification, if there exists such that
Theorem 5.2.
Let have a bounded base and . Let , and is nearly subconvexlike on . Furthermore, let (VP) satisfies the generalized Slater constraint qualification. If and , then there exists such that
Proof.
Since has bounded base , by Lemma 2.13, we have . Thus, there is a convex cone with satisfying
and there exists an absolutely convex open neighborhood of such that
Since (5.3) is equivalent to , and from (5.4) we see that
Moreover, for any , we have . Therefore,
Since is open in , thus, we get
By the assumption that is nearly subconvexlike on , we have
is convex. Hence, it follows from the HahnBanach separation theorem that there exists such that
Thus, we obtain
Since is a cone, we get
Since , . Choose . By (5.13), we know that , thus
Letting and noting that , in (5.10), we get
Thus, , which implies
Now, we claim that . If this is not the case, then
By the generalized Slater constraint qualification, then there exists such that
and so there exists such that . Hence, . But substituting into (5.10), and by taking , and in (5.10), we have
This contradiction shows that . Therefore . From (5.12) and Remark 2.2, we have . And since is a bounded base of , so . Hence, we can choose such that and define the operator by
Clearly, and by (5.16), we see that
Therefore,
From (5.10) and (5.20), we obtain
Since is nearly subconvexlike on , by Lemma 2.13, we have is nearly subconvexlike on . From (5.22), Theorem 4.2 and the above expression, we have
Therefore, the proof is completed.
Theorem 5.3.
Let be a closed convex pointed cone with a bounded base , and . If there exists such that and , then and .
Proof.
Since has a bounded base, and , we have . Thus, there exists a convex cone with satisfying
and there exits a convex neighborhood of such that
By , we have
Thus,
Therefore, by the definition of and , we get and , respectively.
6. Tightly Proper Efficiency and Duality
Definition 6.1.
The setvalued Lagrangian map for problem (VP) is defined by
Definition 6.2.
The setvalued map , defined by
is called a tightly properly dual map for (VP). We now associate the following Lagrange dual problem with (VP):
Definition 6.3.
A point is said to be an efficient point of (VD) if
We now can establish the following dual theorems.
Theorem 6.4 (weak duality).
If and . Then
Proof.
One has
Then, there exists such that
Hence,
Particularly,
Noting that
and taking in (6.8), we have
Hence, from and , we get
This completes the proof.
Theorem 6.5 (strong duality).
Let be a closed convex pointed cone with a bounded base in and be a closed convex pointed cone with in . Let , , be nearly subconvexlike on . Furthermore, let (VP) satisfy the generalized Slater constraint qualification. Then, and if and only if is an efficient point of (VD).
Proof.
Let and , then according to Theorem 5.2, there exists such that and . Hence
By Theorem 6.4, we know that is an efficient point of (VD).
Conversely, Since is an efficient point of (VD), then . Hence, there exists such that
Since has a bounded base , by Lemma 3.1 and Proposition 3.10, we have
Hence, there exists a convex cone with satisfying and there exists an absolutely open convex neighborhood of such that
Hence, we have
Since, is open subset of , we have
Since is nearly subconvexlike on , by Lemma 2.13, we have is nearly subconvexlike on , which implies that
is convex. From (6.17) and by the HahnBanach separation theorem, there exists such that
From this, we have
From (6.21), we know that . And by is bounded base of , it implies that . For any , there exists . Since , we have and hence . From this and (6.20), we have
that is . By Theorem 4.2, we have and .
7. Tightly Proper Efficiency and Tightly Proper Saddle Point
We now introduce a new concept of tightly proper saddle point for a setvalued Lagrange map and use it to characterize tightly proper efficiency.
Definition 7.1.
Let , is a closed convex pointed cone of and . if there exists a convex cone with satisfying and there is a convex neighborhood of such that
is said to be a tightly properly efficient point with respect to , written as, if
It is easy to find that if and only if , and if is bounded, then we also have .
Definition 7.2.
A pair is said to be a tightly proper saddle point of Lagrangian map if
We first present an important equivalent characterization for a tightly proper saddle point of the Lagrange map .
Lemma 7.3.
is said to be a tight proper saddle point of Lagrange map if only if there exist and such that
(i),
(ii).
Proof.
Necessity. Since is a tightly proper saddle point of , by Definition 7.2 there exist and such that
From (7.5) and the definition of , then there exists a convex cone with satisfying
and there is a convex neighborhood of such that
Since, for every ,
We have
Thus, from (7.6), we have
Let be defined by
Then, (7.10) can be written as
By (7.7) and the above expression show that is a tightly properly efficient point of the vector optimization problem
Since is a linear map, of course, is nearly subconvexlike on . Hence, by Theorem 4.2, 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, 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, . And taking in (7.20), we obtain
Hence,
which contradicts (7.14). Therefore,
Thus, , and since . If , then
hence , by . But, taking in (7.14) leads to
This contradiction shows that , that is, condition (ii) holds.
Therefore, by (7.4) and (7.5), we know
that is condition (i) holds.
Sufficiency. From , , and condition (ii), we get
And by condition (i), we obtain
Therefore, is a tightly proper saddle point of , and the proof is completed.
The following saddlepoint theorem allows us to express a tightly properly efficient solution of (VP) as a tightly proper saddle of the setvalued Lagrange map .
Theorem 7.4.
Let be nearly convexlike on . If for any point such that is nearly convexlike on , and (VP) satisfy generalized Slater constraint qualification.
(i)If is a tightly proper saddle point of , then is a tightly properly efficient solution of (VP).
(ii)If be a tightly properly efficient minimizer of (VP), . Then there exists such that is a tightly proper saddle point of Lagrange map .
Proof.

(i)
By the necessity of Lemma 7.3, we have
(7.30)
and there exists such that is a tightly properly efficient minimizer of the problem
According to Theorem 5.3, is a tightly properly efficient minimizer of (VP). Therefore, is a tightly properly efficient solution of (VP).

(ii)
From the assumption, and by Theorem 5.2, there exists such that
(7.31)
Therefore there exists such that . Hence, from Lemma 7.3, it follows that is a tightly proper saddle point of Lagrange map .
8. Conclusions
In this paper, we have extended the concept of tightly proper efficiency from normed linear spaces to locally convex topological vector spaces and got the equivalent relations among tightly proper efficiency, strict efficiency and superefficiency. We have also obtained a scalarization theorem and two Lagrange multiplier theorems for tightly proper efficiency in vector optimization involving nearly conesubconvexlike setvalued maps. Then, we have introduced a Lagrange dual problem and got some duality results in terms of tightly properly efficient solutions. To characterize tightly proper efficiency, we have also introduced a new type of saddle point, which is called the tightly proper saddle point of an appropriate setvalued Lagrange map, and obtained its necessary and sufficient optimality conditions. Simultaneously, we have also given some examples to illustrate these concepts and results. On the other hand, by using the results of the Section 3 in this paper, we know that the above results hold for superefficiency and strict efficiency in vector optimization involving nearly coneconvexlike setvalued maps and, by virtue of [12, Theorem 3.11], all the above results also hold for positive proper efficiency, Hurwicz proper efficiency, global Henig proper efficiency and global Borwein proper efficiency in vector optimization with setvalued maps under the conditions that the setvalued and is closed convex and the ordering cone has a weakly compact base.
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Acknowledgments
The authors would like to thank the anonymous referees for their valuable comments and suggestions, which helped to improve the paper. This research was partially supported by the National Natural Science Foundation of China (Grant no. 10871216) and the Fundamental Research Funds for the Central Universities (project no. CDJXS11102212).
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Xu, Y., Li, S. Tightly Proper Efficiency in Vector Optimization with Nearly ConeSubconvexlike SetValued Maps. J Inequal Appl 2011, 839679 (2011). https://doi.org/10.1155/2011/839679
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DOI: https://doi.org/10.1155/2011/839679
Keywords
 Convex Cone
 Topological Vector Space
 Normed Linear Space
 Vector Optimization Problem
 Efficient Point