Tightly Proper Efficiency in Vector Optimization with Nearly Cone-Subconvexlike Set-Valued Maps

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Abstract

A scalarization theorem and two Lagrange multiplier theorems are established for tightly proper efficiency in vector optimization involving nearly cone-subconvexlike set-valued 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 set-valued Lagrange map, is introduced and is used to characterize tightly proper efficiency.

Keywords

Convex Cone Topological Vector Space Normed Linear Space Vector Optimization Problem Efficient Point 

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 [1, 2, 3, 4, 5, 6, 7, 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 set-valued maps, for instance, see [13, 14, 15, 16, 17, 18]. Gong [19] discussed set-valued 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 ic-cone-convexlike set-valued maps. Li [21] extended the concept of Benson proper efficiency to set-valued maps and presented two scalarization theorems and Lagrange mulitplier theorems for set-valued vector optimization problem under cone-subconvexlikeness. Mehra [22], Xia and Qiu [23] discussed the superefficiency in vector optimization problem involving nearly cone-convexlike set-valued maps, nearly cone-subconvexlike set-valued maps, respectively. For other results for proper efficiencies in optimization problems with generalized convexity and generalized constraints, we refer to [24, 25, 26] and the references therein.

In this paper, inspired by [10, 21, 22, 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 cone-subconvexlike set-valued 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 set-valued vector optimization problem is introduced and a scalarization theorem for tightly proper efficiency in vector optimization problems involving nearly cone-subconvexlike set-valued 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 set-valued 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 Open image in new window be a linear space, Open image in new window and Open image in new window be two real locally convex topological spaces (in brief, LCTS), with topological dual spaces Open image in new window and Open image in new window , respectively. For a set Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window denote the closure, the interior, the boundary, and the complement of Open image in new window , respectively. Moreover, by Open image in new window we denote the closed unit ball of Open image in new window . A set Open image in new window is said to be a cone if Open image in new window for any Open image in new window and Open image in new window . A cone Open image in new window is said to be convex if Open image in new window , and it is said to be pointed if Open image in new window . The generated cone of Open image in new window is defined by
The dual cone of Open image in new window is defined as
and the quasi-interior of Open image in new window is the set
Recall that a base of a cone Open image in new window is a convex subset of Open image in new window such that

Of course, Open image in new window is pointed whenever Open image in new window has a base. Furthermore, if Open image in new window is a nonempty closed convex pointed cone in Open image in new window , then Open image in new window if and only if Open image in new window has a base.

Also, in this paper, we assume that, unless indicated otherwise, Open image in new window and Open image in new window are pointed closed convex cones with Open image in new window and Open image in new window , respectively.

Definition 2.1 (see [27]).

Cheng and Fu in [27] discussed the propositions of Open image in new window , and the following remark also gives some propositions of Open image in new window .

Remark 2.2 (see [27]).

Definition 2.3.

A point Open image in new window is said to be efficient with respect to Open image in new window (denoted Open image in new window ) if

Remark 2.4 (see [28]).

If Open image in new window is a closed convex pointed cone and Open image in new window , then Open image in new window .

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 Open image in new window be an ordering cone with a base Open image in new window . Then Open image in new window , by the Hahn Banach separation theorem, there are a Open image in new window and an Open image in new window such that

It is clear that, for each convex neighborhood Open image in new window of Open image in new window with Open image in new window , Open image in new window is convex and Open image in new window . Obviously, Open image in new window is convex pointed cone, indeed, Open image in new window is also a base of Open image in new window .

Definition 2.5 (see [8]).

Suppose that Open image in new window is a subset of Open image in new window and Open image in new window denotes the family of all bases of Open image in new window . Open image in new window is said to be a strictly efficient point with respect to Open image in new window , written as Open image in new window , if there is a convex neighborhood Open image in new window of Open image in new window such that
Open image in new window is said to be a strictly efficient point with respect to Open image in new window , written as, Open image in new window if

Remark 2.6.

Since Open image in new window is open in Open image in new window , thus Open image in new window is equivalent to Open image in new window .

Definition 2.7.

The point Open image in new window is called tightly properly efficient with respect to Open image in new window (denoted Open image in new window ) if there exists a convex cone Open image in new window with Open image in new window satisfying Open image in new window and there exists a neighborhood Open image in new window of Open image in new window such that
Open image in new window is said to be a tightly properly efficient point with respect to Open image in new window , written as, Open image in new window if

Now, we give the following example to illustrate Definition 2.7.

Example 2.8.

Let Open image in new window , Open image in new window . Given Open image in new window (see Figure 1). Thus, it follows from the direct computation and Definition 2.7 that
Figure 1

The set C .

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.

thus, Open image in new window .

Definition 2.11 (see [10]).

Open image in new window is called a superefficient point of a subset Open image in new window of Open image in new window with respect to ordering cone Open image in new window , written as Open image in new window , if, for each neighborhood Open image in new window of Open image in new window , there is neighborhood Open image in new window of Open image in new window such that

Definition 2.12 (see [29, 30]).

A set-valued map Open image in new window is said to be nearly Open image in new window -subconvexlike on Open image in new window if Open image in new window is convex.

The product Open image in new window is called nearly Open image in new window -subconvexlike on Open image in new window if Open image in new window is nearly Open image in new window -subconvexlike on Open image in new window . Let Open image in new window be the space of continuous linear operators from Open image in new window to Open image in new window , and let

respectively.

Lemma 2.13 (see [23]).

If Open image in new window is nearly Open image in new window -subconvexlike on Open image in new window , then:

(i)for each Open image in new window , Open image in new window is nearly Open image in new window -subconvexlike on Open image in new window ;

(ii)for each Open image in new window , Open image in new window is nearly Open image in new window -subconvexlike on Open image in new window .

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.

Proof.

From the definition of Open image in new window and Open image in new window , we only need prove that Open image in new window for any Open image in new window . Indeed, for each Open image in new window , by the separation theorem, there exists Open image in new window such that
It implies that there exists Open image in new window such that

Then there is Open image in new window and Open image in new window such that Open image in new window , since Open image in new window , then there exists Open image in new window and Open image in new window such that Open image in new window . By (3.2) and (3.3), we see that Open image in new window . Therefore, Open image in new window and Open image in new window , it is a contradiction. Therefore, Open image in new window for each Open image in new window .

Proposition 3.2.

Proof.

By Definition 2.11, for any Open image in new window , there exists a convex neighborhood Open image in new window of Open image in new window with Open image in new window such that
It is easy to verify that
Now, let Open image in new window and by Lemma 3.1, we have

which implies that Open image in new window .

Proposition 3.3.

Proof.

For each Open image in new window , there exists a convex cone Open image in new window with Open image in new window satisfying
and there exists a neighborhood Open image in new window of Open image in new window such that
Since expression (3.11) can be equivalently expressed as
Open image in new window , and by (3.12), we have

It implies that Open image in new window . Therefore this proof is completed.

Remark 3.4.

If Open image in new window 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 Open image in new window , Open image in new window (see Figure 2) and Open image in new window .

respectively. Thus, the converse of Proposition 3.3 is not valid.
Figure 2

The set S .

Figure 3

The set F(A) .

Proposition 3.6 (see [8]).

From Propositions 3.2, 3.3, and 3.6, we can get immediately the following corollary.

Corollary 3.7.

Example 3.8.

Lemma 3.9 (see [23]).

Let Open image in new window be a closed convex pointed cone with a bounded base Open image in new window and Open image in new window . Then, Open image in new window .

From Corollary 3.7 and Lemma 3.9, we can get the following proposition.

Proposition 3.10.

If Open image in new window has a bounded base Open image in new window and Open image in new window is a nonempty subset of Open image in new window , then Open image in new window .

4. Tightly Proper Efficiency and Scalarization

Let Open image in new window be a closed convex pointed cone. We consider the following vector optimization problem with set-valued maps

where Open image in new window , Open image in new window are set-valued maps with nonempty values. Let Open image in new window be the set of all feasible solutions of (VP).

Definition 4.1.

Open image in new window is said to be a tightly properly efficient solution of (VP), if there exists Open image in new window such that Open image in new window .

We call Open image in new window 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 set-valued maps, we consider the following scalar optimization problem with set-valued map Open image in new window :
where Open image in new window . The set of all optimal solutions of ( ) is denoted by Open image in new window , 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 Open image in new window have a bounded base Open image in new window . Let Open image in new window , Open image in new window , and Open image in new window be nearly Open image in new window -subconvexlike on Open image in new window . Then Open image in new window if and only if there exists Open image in new window such that Open image in new window .

Proof.

Necessity. Let Open image in new window . Then, by Lemma 3.1 and Proposition 3.10, we have Open image in new window . Hence, there exists a convex cone Open image in new window with Open image in new window satisfying Open image in new window and there exists a convex neighborhood Open image in new window of Open image in new window such that
From the above expression and Open image in new window , we have
By the assumption that Open image in new window is nearly Open image in new window -subconvexlike on Open image in new window , thus Open image in new window is convex set. By the Hahn-Banach separation theorem, there exists Open image in new window such that
It is easy to see that
Hence, we obtain

Furthermore, according to Remark 2.2, we have Open image in new window .

Sufficiency. Suppose that there exists Open image in new window such that Open image in new window . Since Open image in new window has a bounded base Open image in new window , thus by Remark 2.2(ii), we know that Open image in new window . And by Remark 2.2(i), we can take a convex neighborhood Open image in new window of Open image in new window such that
From the above expression and (4.8), we get

Therefore, Open image in new window . Noting that Open image in new window has a bounded base Open image in new window and by Lemma 3.1, we have Open image in new window .

Now, we give the following example to illustrate Theorem 4.2.

Example 4.3.

Thus, feasible set of (VP)
By Definition 4.1, we get
For any point Open image in new window , there exists Open image in new window such that

Indeed, for any Open image in new window , we consider the following three cases.

Case 1.

If Open image in new window is in the first quadrant, then for any Open image in new window such that Open image in new window .

Case 2.

If Open image in new window is in the second quadrant, then there exists Open image in new window such that Open image in new window . Let Open image in new window such that
Then, we have

Case 3.

If Open image in new window in the fourth quadrant, then there exists Open image in new window such that Open image in new window . Let Open image in new window such that
Then, we have

Therefore, if follows from Cases 1, 2, and 3 that there exists Open image in new window such that Open image in new window .

From Theorem 4.2, we can get immediately the following corollary.

Corollary 4.4.

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 Open image in new window be a closed convex pointed cone with Open image in new window . We say that (VP) satisfies the generalized Slater constraint qualification, if there exists Open image in new window such that

Theorem 5.2.

Let Open image in new window have a bounded base Open image in new window and Open image in new window . Let Open image in new window , Open image in new window and Open image in new window is nearly Open image in new window -subconvexlike on Open image in new window . Furthermore, let (VP) satisfies the generalized Slater constraint qualification. If Open image in new window and Open image in new window , then there exists Open image in new window such that

Proof.

Since Open image in new window has bounded base Open image in new window , by Lemma 2.13, we have Open image in new window . Thus, there is a convex cone Open image in new window with Open image in new window satisfying
and there exists an absolutely convex open neighborhood Open image in new window of Open image in new window such that
Since (5.3) is equivalent to Open image in new window , and from (5.4) we see that
Moreover, for any Open image in new window , we have Open image in new window . Therefore,
By the assumption that Open image in new window is nearly Open image in new window -subconvexlike on Open image in new window , we have
is convex. Hence, it follows from the Hahn-Banach separation theorem that there exists Open image in new window such that
Thus, we obtain
Since Open image in new window is a cone, we get
Now, we claim that Open image in new window . If this is not the case, then
By the generalized Slater constraint qualification, then there exists Open image in new window such that
and so there exists Open image in new window such that Open image in new window . Hence, Open image in new window . But substituting Open image in new window into (5.10), and by taking Open image in new window , and Open image in new window in (5.10), we have
This contradiction shows that Open image in new window . Therefore Open image in new window . From (5.12) and Remark 2.2, we have Open image in new window . And since Open image in new window is a bounded base of Open image in new window , so Open image in new window . Hence, we can choose Open image in new window such that Open image in new window and define the operator Open image in new window by
Clearly, Open image in new window and by (5.16), we see that
Therefore,
From (5.10) and (5.20), we obtain
Since Open image in new window is nearly Open image in new window -subconvexlike on Open image in new window , by Lemma 2.13, we have Open image in new window is nearly Open image in new window -subconvexlike on Open image in new window . From (5.22), Theorem 4.2 and the above expression, we have

Therefore, the proof is completed.

Theorem 5.3.

Let Open image in new window be a closed convex pointed cone with a bounded base Open image in new window , Open image in new window and Open image in new window . If there exists Open image in new window such that Open image in new window and Open image in new window , then Open image in new window and Open image in new window .

Proof.

Since Open image in new window has a bounded base, and Open image in new window , we have Open image in new window . Thus, there exists a convex cone Open image in new window with Open image in new window satisfying
and there exits a convex neighborhood Open image in new window of Open image in new window such that

Therefore, by the definition of Open image in new window and Open image in new window , we get Open image in new window and Open image in new window , respectively.

6. Tightly Proper Efficiency and Duality

Definition 6.1.

The set-valued Lagrangian map Open image in new window for problem (VP) is defined by

Definition 6.2.

The set-valued map Open image in new window , 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 Open image in new window is said to be an efficient point of (VD) if

We now can establish the following dual theorems.

Theorem 6.4 (weak duality).

Proof.

Then, there exists Open image in new window such that
Particularly,
Noting that
and taking Open image in new window in (6.8), we have

This completes the proof.

Theorem 6.5 (strong duality).

Let Open image in new window be a closed convex pointed cone with a bounded base Open image in new window in Open image in new window and Open image in new window be a closed convex pointed cone with Open image in new window in Open image in new window . Let Open image in new window , Open image in new window , Open image in new window be nearly Open image in new window -subconvexlike on Open image in new window . Furthermore, let (VP) satisfy the generalized Slater constraint qualification. Then, Open image in new window and Open image in new window if and only if Open image in new window is an efficient point of (VD).

Proof.

By Theorem 6.4, we know that Open image in new window is an efficient point of (VD).

Conversely, Since Open image in new window is an efficient point of (VD), then Open image in new window . Hence, there exists Open image in new window such that
Since Open image in new window has a bounded base Open image in new window , by Lemma 3.1 and Proposition 3.10, we have
Hence, there exists a convex cone Open image in new window with Open image in new window satisfying Open image in new window and there exists an absolutely open convex neighborhood Open image in new window of Open image in new window such that
Hence, we have
Since Open image in new window is nearly Open image in new window -subconvexlike on Open image in new window , by Lemma 2.13, we have Open image in new window is nearly Open image in new window -subconvexlike on Open image in new window , which implies that
is convex. From (6.17) and by the Hahn-Banach separation theorem, there exists Open image in new window such that
From this, we have
From (6.21), we know that Open image in new window . And by Open image in new window is bounded base of Open image in new window , it implies that Open image in new window . For any Open image in new window , there exists Open image in new window . Since Open image in new window , we have Open image in new window and hence Open image in new window . From this and (6.20), we have

that is Open image in new window . By Theorem 4.2, we have Open image in new window and Open image in new window .

7. Tightly Proper Efficiency and Tightly Proper Saddle Point

We now introduce a new concept of tightly proper saddle point for a set-valued Lagrange map Open image in new window and use it to characterize tightly proper efficiency.

Definition 7.1.

Open image in new window is said to be a tightly properly efficient point with respect to Open image in new window , written as, Open image in new window if

It is easy to find that Open image in new window if and only if Open image in new window , and if Open image in new window is bounded, then we also have Open image in new window .

Definition 7.2.

A pair Open image in new window is said to be a tightly proper saddle point of Lagrangian map Open image in new window if

We first present an important equivalent characterization for a tightly proper saddle point of the Lagrange map Open image in new window .

Lemma 7.3.

Open image in new window is said to be a tight proper saddle point of Lagrange map Open image in new window if only if there exist Open image in new window and Open image in new window such that

(i) Open image in new window ,

(ii) Open image in new window .

Proof.

Necessity. Since Open image in new window is a tightly proper saddle point of Open image in new window , by Definition 7.2 there exist Open image in new window and Open image in new window such that
From (7.5) and the definition of Open image in new window , then there exists a convex cone Open image in new window with Open image in new window satisfying
and there is a convex neighborhood Open image in new window of Open image in new window such that
Thus, from (7.6), we have
Then, (7.10) can be written as
By (7.7) and the above expression show that Open image in new window is a tightly properly efficient point of the vector optimization problem
Since Open image in new window is a linear map, of course, Open image in new window is nearly Open image in new window -subconvexlike on Open image in new window . Hence, by Theorem 4.2, there exists Open image in new window such that
Now, we claim that
If this is not true, then since Open image in new window is a closed convex cone set, by the strong separation theorem in topological vector space, there exists Open image in new window such that
In the above expression, taking Open image in new window gets
It is evident that Open image in new window and that
Hence, Open image in new window . And taking Open image in new window in (7.20), we obtain
which contradicts (7.14). Therefore,

This contradiction shows that Open image in new window , that is, condition (ii) holds.

Therefore, by (7.4) and (7.5), we know

that is condition (i) holds.

Sufficiency. From Open image in new window , Open image in new window , and condition (ii), we get
And by condition (i), we obtain

Therefore, Open image in new window is a tightly proper saddle point of Open image in new window , and the proof is completed.

The following saddle-point theorem allows us to express a tightly properly efficient solution of (VP) as a tightly proper saddle of the set-valued Lagrange map Open image in new window .

Theorem 7.4.

Let Open image in new window be nearly Open image in new window -convexlike on Open image in new window . If for any point Open image in new window such that Open image in new window is nearly Open image in new window -convexlike on Open image in new window , and (VP) satisfy generalized Slater constraint qualification.

(i)If Open image in new window is a tightly proper saddle point of Open image in new window , then Open image in new window is a tightly properly efficient solution of (VP).

(ii)If Open image in new window be a tightly properly efficient minimizer of (VP), Open image in new window . Then there exists Open image in new window such that Open image in new window is a tightly proper saddle point of Lagrange map Open image in new window .

Proof.
  1. (i)
    By the necessity of Lemma 7.3, we have
     
and there exists Open image in new window such that Open image in new window is a tightly properly efficient minimizer of the problem
According to Theorem 5.3, Open image in new window is a tightly properly efficient minimizer of (VP). Therefore, Open image in new window is a tightly properly efficient solution of (VP).
  1. (ii)
    From the assumption, and by Theorem 5.2, there exists Open image in new window such that
     

Therefore there exists Open image in new window such that Open image in new window . Hence, from Lemma 7.3, it follows that Open image in new window is a tightly proper saddle point of Lagrange map Open image in new window .

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 cone-subconvexlike set-valued 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 set-valued 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 cone-convexlike set-valued 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 set-valued maps under the conditions that the set-valued Open image in new window and Open image in new window is closed convex and the ordering cone Open image in new window has a weakly compact base.

Notes

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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Copyright information

© Y. D. Xu and S. J. Li. 2011

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  1. 1.College of Mathematics and StatisticsChongqing UniversityChongqingChina

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