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

- 974 Downloads

## 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

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 .

- (i)
Let Open image in new window . Then Open image in new window if and only if there exists a neighborhood Open image in new window of Open image in new window such that Open image in new window .

- (ii)
If Open image in new window is a bounded base of Open image in new window , then Open image in new window .

Definition 2.3.

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 .

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]).

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.

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

Example 2.8.

Remark 2.9.

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]).

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.

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.

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.

which implies that Open image in new window .

Proposition 3.3.

Proof.

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 .

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

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).

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

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

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.

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.

Case 3.

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]).

Theorem 5.2.

Proof.

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.

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.

Definition 6.2.

Definition 6.3.

We now can establish the following dual theorems.

Theorem 6.4 (weak duality).

Proof.

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).

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.

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.

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

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

that is condition (i) holds.

*Sufficiency*. From Open image in new window , Open image in new window , and condition (ii), we get

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 .

- (i)By the necessity of Lemma 7.3, we have(7.30)

- (ii)From the assumption, and by Theorem 5.2, there exists Open image in new window such that(7.31)

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).

### References

- 1.Kuhn HW, Tucker AW:
**Nonlinear programming.**In*Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability, 1951, Berkeley, Calif, USA*. University of California Press; 481–492.Google Scholar - 2.Geoffrion AM:
**Proper efficiency and the theory of vector maximization.***Journal of Mathematical Analysis and Applications*1968,**22:**618–630. 10.1016/0022-247X(68)90201-1MathSciNetCrossRefMATHGoogle Scholar - 3.Borwein J:
**Proper efficient points for maximizations with respect to cones.***SIAM Journal on Control and Optimization*1977,**15**(1):57–63. 10.1137/0315004MathSciNetCrossRefMATHGoogle Scholar - 4.Hartley R:
**On cone-efficiency, cone-convexity and cone-compactness.***SIAM Journal on Applied Mathematics*1978,**34**(2):211–222. 10.1137/0134018MathSciNetCrossRefMATHGoogle Scholar - 5.Benson HP:
**An improved definition of proper efficiency for vector maximization with respect to cones.***Journal of Mathematical Analysis and Applications*1979,**71**(1):232–241. 10.1016/0022-247X(79)90226-9MathSciNetCrossRefMATHGoogle Scholar - 6.Henig MI:
**Proper efficiency with respect to cones.***Journal of Optimization Theory and Applications*1982,**36**(3):387–407. 10.1007/BF00934353MathSciNetCrossRefMATHGoogle Scholar - 7.Borwein JM, Zhuang D:
**Super efficiency in vector optimization.***Transactions of the American Mathematical Society*1993,**338**(1):105–122. 10.2307/2154446MathSciNetCrossRefMATHGoogle Scholar - 8.Fu WT:
**The strictly efficient points of a set in a normed linear space.***Journal of Systems Science and Mathematical Sciences*1997,**17**(4):324–329.MathSciNetMATHGoogle Scholar - 9.Zaffaroni A:
**Degrees of efficiency and degrees of minimality.***SIAM Journal on Control and Optimization*2003,**42**(3):1071–1086. 10.1137/S0363012902411532MathSciNetCrossRefMATHGoogle Scholar - 10.Zheng XY:
**Proper efficiency in locally convex topological vector spaces.***Journal of Optimization Theory and Applications*1997,**94**(2):469–486. 10.1023/A:1022648115446MathSciNetCrossRefMATHGoogle Scholar - 11.Guerraggio A, Molho E, Zaffaroni A:
**On the notion of proper efficiency in vector optimization.***Journal of Optimization Theory and Applications*1994,**82**(1):1–21. 10.1007/BF02191776MathSciNetCrossRefMATHGoogle Scholar - 12.Liu J, Song W:
**On proper efficiencies in locally convex spaces—a survey.***Acta Mathematica Vietnamica*2001,**26**(3):301–312.MathSciNetMATHGoogle Scholar - 13.Corley HW:
**Existence and Lagrangian duality for maximizations of set-valued functions.***Journal of Optimization Theory and Applications*1987,**54**(3):489–501. 10.1007/BF00940198MathSciNetCrossRefMATHGoogle Scholar - 14.Li Z-F, Chen G-Y:
**Lagrangian multipliers, saddle points, and duality in vector optimization of set-valued maps.***Journal of Mathematical Analysis and Applications*1997,**215**(2):297–316. 10.1006/jmaa.1997.5568MathSciNetCrossRefMATHGoogle Scholar - 15.Song W:
**Lagrangian duality for minimization of nonconvex multifunctions.***Journal of Optimization Theory and Applications*1997,**93**(1):167–182. 10.1023/A:1022658019642MathSciNetCrossRefMATHGoogle Scholar - 16.Chen GY, Jahn J:
**Optimality conditions for set-valued optimization problems.***Mathematical Methods of Operations Research*1998,**48**(2):187–200. 10.1007/s001860050021MathSciNetCrossRefMATHGoogle Scholar - 17.Rong WD, Wu YN:
**Characterizations of super efficiency in cone-convexlike vector optimization with set-valued maps.***Mathematical Methods of Operations Research*1998,**48**(2):247–258. 10.1007/s001860050026MathSciNetCrossRefMATHGoogle Scholar - 18.Li SJ, Yang XQ, Chen GY:
**Nonconvex vector optimization of set-valued mappings.***Journal of Mathematical Analysis and Applications*2003,**283**(2):337–350. 10.1016/S0022-247X(02)00410-9MathSciNetCrossRefMATHGoogle Scholar - 19.Gong X-H:
**Optimality conditions for Henig and globally proper efficient solutions with ordering cone has empty interior.***Journal of Mathematical Analysis and Applications*2005,**307**(1):12–31. 10.1016/j.jmaa.2004.10.001MathSciNetCrossRefMATHGoogle Scholar - 20.Sach PH:
**New generalized convexity notion for set-valued maps and application to vector optimization.***Journal of Optimization Theory and Applications*2005,**125**(1):157–179. 10.1007/s10957-004-1716-4MathSciNetCrossRefMATHGoogle Scholar - 21.Li ZF:
**Benson proper efficiency in the vector optimization of set-valued maps.***Journal of Optimization Theory and Applications*1998,**98**(3):623–649. 10.1023/A:1022676013609MathSciNetCrossRefMATHGoogle Scholar - 22.Mehra A:
**Super efficiency in vector optimization with nearly convexlike set-valued maps.***Journal of Mathematical Analysis and Applications*2002,**276**(2):815–832. 10.1016/S0022-247X(02)00452-3MathSciNetCrossRefMATHGoogle Scholar - 23.Xia LY, Qiu JH:
**Superefficiency in vector optimization with nearly subconvexlike set-valued maps.***Journal of Optimization Theory and Applications*2008,**136**(1):125–137. 10.1007/s10957-007-9291-0MathSciNetCrossRefMATHGoogle Scholar - 24.Kim DS, Lee GM, Sach PH:
**Hartley proper efficiency in multifunction optimization.***Journal of Optimization Theory and Applications*2004,**120**(1):129–145.MathSciNetCrossRefMATHGoogle Scholar - 25.Sach PH:
**Hartley proper efficiency in multiobjective optimization problems with locally Lipschitz set-valued objectives and constraints.***Journal of Global Optimization*2006,**35**(1):1–25. 10.1007/s10898-005-1652-3MathSciNetCrossRefMATHGoogle Scholar - 26.Huang XX, Yang XQ:
**On characterizations of proper efficiency for nonconvex multiobjective optimization.***Journal of Global Optimization*2002,**23**(3–4):213–231.CrossRefMathSciNetMATHGoogle Scholar - 27.Cheng YH, Fu WT:
**Strong efficiency in a locally convex space.***Mathematical Methods of Operations Research*1999,**50**(3):373–384. 10.1007/s001860050076MathSciNetCrossRefMATHGoogle Scholar - 28.Zhuang D:
**Density results for proper efficiencies.***SIAM Journal on Control and Optimization*1994,**32**(1):51–58. 10.1137/S0363012989171518MathSciNetCrossRefMATHGoogle Scholar - 29.Yang XM, Li D, Wang SY:
**Near-subconvexlikeness in vector optimization with set-valued functions.***Journal of Optimization Theory and Applications*2001,**110**(2):413–427. 10.1023/A:1017535631418MathSciNetCrossRefMATHGoogle Scholar - 30.Sach PH:
**Nearly subconvexlike set-valued maps and vector optimization problems.***Journal of Optimization Theory and Applications*2003,**119**(2):335–356.MathSciNetCrossRefMATHGoogle Scholar

## Copyright information

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.