Open Access

Higher-Order Weakly Generalized Adjacent Epiderivatives and Applications to Duality of Set-Valued Optimization

Journal of Inequalities and Applications20092009:462637

DOI: 10.1155/2009/462637

Received: 6 February 2009

Accepted: 8 July 2009

Published: 4 August 2009

The Erratum to this article has been published in Journal of Inequalities and Applications 2011 2011:817965

Abstract

A new notion of higher-order weakly generalized adjacent epiderivative for a set-valued map is introduced. By virtue of the epiderivative and weak minimality, a higher-order Mond-Weir type dual problem and a higher-order Wolfe type dual problem are introduced for a constrained set-valued optimization problem, respectively. Then, corresponding weak duality, strong duality, and converse duality theorems are established.

1. Introduction

In the last several decades, several notions of derivatives of set-valued maps have been proposed and used for the formulation of optimality conditions and duality in set-valued optimization problems. By using a contingent epiderivative of a set-valued map, Jahn and Rauh [1] obtained a unified necessary and sufficient optimality condition. Chen and Jahn [2] introduced a notion of a generalized contingent epiderivative of a set-valued map and obtained a unified necessary and sufficient conditions for a set-valued optimization problem. Lalitha and Arora [3] introduced a notion of a weak Clarke epiderivative and use it to establish optimality criteria for a constrained set-valued optimization problem. On the other hand, various kinds of differentiable type dual problems for set-valued optimization problems, such as Mond-Weir type and Wolfe type dual problems, have been investigated. By virtue of the tangent derivative of a set-valued map introduced in [4], Sach and Craven [5] discussed Wolfe type duality and Mond-Weir type duality problems for a set-valued optimization problem. By virtue of the codifferential of a set-valued map introduced in [6], Sach et al. [7] obtained Mond-Weir type and Wolfe type weak duality and strong duality theorems of set-valued optimization problems. As to other concepts of derivatives (epiderivatives) of set-valued maps and their applications, one can refer to [815]. Recently, Second-order derivatives have also been proposed, for example, see [16, 17] and so on.

Since higher-order tangent sets introduced in [4], in general, are not cones and convex sets, there are some difficulties in studying higher-order optimality conditions and duality for general set-valued optimization problems. Until now, there are only a few papers to deal with higher-order optimality conditions and duality of set-valued optimization problems by virtue of the higher-order derivatives or epiderivatives introduced by the higher-order tangent sets. Li et al. [18] studied some properties of higher-order tangent sets and higher-order derivatives introduced in [4], and then obtained higher-order necessary and sufficient optimality conditions for set-valued optimization problems under cone-concavity assumptions. By using these higher-order derivatives, they also discussed a higher-order Mond-Weir duality for a set-valued optimization problem in [19]. Li and Chen [20] introduced higher-order generalized contingent(adjacent) epiderivatives of set-valued maps, and obtained higher-order Fritz John type necessary and sufficient conditions for Henig efficient solutions to a constrained set-valued optimization problem.

Motivated by the work reported in [3, 5, 1820], we introduce a notion of higher-order weakly generalized adjacent epiderivative for a set-valued map. Then, by virtue of the epiderivative, we discuss a higher-order Mond-Weir type duality problem and a higher-order Wolfe type duality problem to a constrained set-valued optimization problem, respectively.

The rest of the paper is organized as follows. In Section 2, we collect some of the concepts and some of their properties required for the paper. In Section 3, we introduce a generalized higher-order adjacent set of a set and a higher-order weakly generalized adjacent epiderivative of a set-valued map, and study some of their properties. In Sections 4 and 5, we introduce a higher-order Mond-Weir type dual problem and a higher-order Wolfe type dual problem to a constrained set-valued optimization problem and establish corresponding weak duality, strong duality and converse duality theorems, respectively.

2. Preliminaries and Notations

Throughout this paper, let , and be three real normed spaces, where the spaces and are partially ordered by nontrivial pointed closed convex cones and with and , respectively. We assume that denote the origins of , respectively, denotes the topological dual space of and denotes the dual cone of , defined by . Let be a nonempty set in . The cone hull of is defined by . Let be a nonempty subset of , and be two given nonempty set-valued maps. The effective domain, the graph and the epigraph of are defined respectively by , and The profile map is defined by , for every . Let ,

Definition 2.1.

An element is said to be a minimal point resp., weakly minimal point of if resp., . The set of all minimal point (resp., weakly minimal point) of is denoted by (resp., .

Definition 2.2.

Let be a set-valued map.

(i) is said to be -convex on a convex set , if for any and ,

(2.1)

(ii) is said to be -convex like on a nonempty subset , if for any and , there exists such that

Remark 2.3.

(i)??If is -convex on a convex set , then is -convex like on . But the converse does not hold.

(ii)?? If is -convex like on a nonempty subset , then is convex.

Suppose that is a positive integer, is a normed space supplied with a distance and is a subset of . We denote by the distance from to , where we set .

Definition 2.4 (see [4]).

Let belong to a subset K of a normed space and let be elements of . We say that the subset
(2.2)

is the th-order adjacent set of at

From [18, Propositions ?3.2], we have the following result.

Proposition 2.5.

If is convex, , and , then is convex.

3. Higher-Order Weakly Generalized Adjacent Epiderivatives

Definition 3.1.

Let belong to a subset K of and let be elements of . The subset
(3.1)

is said to be the th-order generalized adjacent set of at

Definition 3.2.

The th-order weakly generalized adjacent epiderivative of at with respect to vectors is the set-valued map from to defined by
(3.2)

Definition 3.3 (see [3, 21]).

The weak domination property resp., domination property is said to hold for a subset of if resp., .

To compare our derivative with well-known derivatives, we recall some notions.

Definition 3.4 (see [4]).

The th-order adjacent derivative of at with respect to vectors is the set-valued map from to defined by
(3.3)

Definition 3.5 (see [19]).

The -directed th-order adjacent derivative of at with respect to vectors is the th-order adjacent derivative of set-valued mapping at with respect to .

Definition 3.6 (See [20]).

The th-order generalized adjacent epiderivative of at with respect to vectors is the set-valued map from to defined by
(3.4)

Using properties of higher-order adjacent sets [4], we have the following result.

Proposition 3.7.

Let . If and the set - fulfills the weak domination property for all , then for any ,
  1. (i)
    (3.5)
     
  1. (ii)
    (3.6)
     
  1. (iii)
    (3.7)
     

Remark 3.8.

The reverse inclusions in Proposition 3.7 may not hold. The following examples explain the case, where we only take .

Example 3.9.

Let , , and . Then for any , and - Therefore, for any , , and do not exist, but
(3.8)

Example 3.10.

Let , , and . Then, , - . Hence, for any , , and do not exist. But
(3.9)

Example 3.11.

Suppose that . Let be a set-valued map with and . Then , - Therefore for any ,
(3.10)

Now we discuss some crucial propositions of the th-order weakly generalized adjacent epiderivative.

Proposition 3.12.

Let , , . If the set - fulfills the weak domination property for all , then for all ,
(3.11)

Proof.

Take any and an arbitrary sequence with . Since ,
(3.12)
It follows from , and is a convex cone that
(3.13)
We get
(3.14)
which implies that
(3.15)
that is, . By the definition of th-order weakly generalized adjacent epiderivative and the weak domination property, we have
(3.16)

Thus .

Remark 3.13.

Since the cone-convexity and cone-concavity assumptions are omitted, Proposition 3.12 improves [18, Theorem ] and [20, Proposition ].

Proposition 3.14.

Let be a nonempty convex subset of , , . Let be -convex like on , . If the set - fulfills the weak domination property for all , then
(3.17)

Proof.

Take any , and an arbitrary sequence with . Since is convex and be -convex like on , we get that is a convex subset and is a convex cone. Therefore
(3.18)
It follows from , is convex and is a convex cone that
(3.19)
We obtain that
(3.20)
which implies that
(3.21)
that is, . By the definition of th-order weakly generalized adjacent epiderivative and the weak domination property, we have
(3.22)

Thus and the proof is complete.

Remark 3.15.

Since the cone-convexity assumptions are replaced by cone-convex likeness assumptions, Proposition 3.14 improves [20, Proposition ].

4. Higher-Order Mond-Weir Type Duality

In this section, we introduce a higher-order Mond-Weir type dual problem for a constrained set-valued optimization problem by virtue of the higher-order weakly generalized adjacent epiderivative and discuss its weak duality, strong duality and converse duality properties. The notation is used to denote . Firstly, we recall the definition of interior tangent cone of a set and state a result regarding it from [16].

The interior tangent cone of at is defined as

(4.1)

where stands for the closed ball centered at and of radius .

Lemma 4.1 (see [16]).

If is convex, and , then
(4.2)

Consider the following set-valued optimization problem:

(4.3)

Set . A point is said to be a feasible solution of if and .

Definition 4.2.

A point is said to be a weakly minimal solution of if satisfying and .

Suppose that , , , and . We introduce a higher-order Mond-Weir type dual problem of as follows:

(4.4)
(4.5)
(4.6)
(4.7)

Let satisfy conditions (4.4)–(4.7) . A point satisfying (4.4)–(4.7) is called a feasible solution of . A feasible solution is called a weakly maximal solution of if .

Theorem 4.3 (weak duality).

Let and . Let the set - fulfill the weak domination property for all . If is a feasible solution of and is a feasible solution of , then
(4.8)

Proof.

It follows from Proposition 3.12 that
(4.9)

Since is a feasible solution of , . Take . Then, it follows from (4.5) and (4.7) that

(4.10)
By (4.4), (4.6), (4.7), (4.9) and (4.10), we get
(4.11)

Thus, the proof is complete.

Remark 4.4.

In Theorem 4.3, cone-convexity assumptions of [19, Theorem ?4.1] are omitted.

By the similiar proof method of Theorem 4.3, it follows from Proposition 3.14 that the following theorem holds.

Theorem 4.5 (weak duality).

Let , and . Suppose that is -convex like on a nonempty convext subset . Let the set - fulfill the weak domination property for all . If is a feasible solution of and is a feasible solution of , then
(4.12)

Lemma 4.6.

Let , , . Let the set - fulfill the weak domination property for all . If is a weakly minimal solution of , then
(4.13)

for all

Proof.

Since is a weakly minimal solution of , . Then,
(4.14)
Assume that the result (4.13) does not hold. Then there exist , and with such that
(4.15)
(4.16)
It follows from (4.15) and the definition of th-order weakly generalized adjacent epiderivative that
(4.17)
Thus, for an arbitrary sequence with , there exists a sequence such that
(4.18)
From (4.16) and (4.18), there exists a sufficiently large such that
(4.19)
(4.20)
Since , and is a convex cone,
(4.21)
It follows from (4.19) and (4.21) that
(4.22)
By (4.20) and Lemma 4.1, we have . Then, it follows from the definition of that Since , there exists a sufficiently large such that
(4.23)
Then, from (4.20), we have
(4.24)
It follows from , and is a convex cone that
(4.25)
Since , there exist such that It follows from (4.25) that , for , and then
(4.26)
It follows from (4.22) that
(4.27)

which contradicts (4.14). Thus (4.13) holds and the proof is complete.

Theorem 4.7 (strong duality).

Suppose that , and the following conditions are satisfied:

(i) , ;

(ii) is -convex like on a nonempty convex subset ;

(iii) is a weakly minimal solution of ;

(iv) - fulfills the weak domination property for all and ;

(v)There exists an such that .

Then there exist and such that is a weakly maximal solution of .

Proof.

Define
(4.28)
By the similar proof method for the convexity of in [20, Theorem ?5.1], just replacing -order generalized adjacent epiderivative by -order weakly generalized adjacent epiderivative, we have that is a convex set. It follows from Lemma 4.6 that
(4.29)
By the separation theorem of convex sets, there exist and , not both zero functionals, such that
(4.30)
It follows from (4.30) that
(4.31)
(4.32)

From (4.31), we obtain that is bounded below on the . Then, , for all . Naturally, . By the similar proof method for , we get .

Now we show that . Suppose that . Then . By Proposition 3.14 and condition (v), there exists a point such that and

(4.33)

Thus it follows from (4.32) that . Since and , we have , which leads to a contradiction. So .

From (4.32) and assumption (iv), we have . Since and , . Therefore

(4.34)

It follows from (4.32), (4.34), and that , for all So is a feasible solution of .

Finally, we prove that is a weakly maximal solution of .

Suppose that is not a weakly maximal solution of . Then there exists a feasible solution of such that

(4.35)
According to , we get
(4.36)
Since is a weakly minimal solution of , it follows from Theorem 4.5 that
(4.37)

which contradicts (4.36). Thus the conclusion holds and the proof is complete.

Now we give an example to illustrate the Strong Duality. we only take .

Example 4.8.

Let . Let be a set-valued map with
(4.38)
and be a set-valued map with
(4.39)

Naturally, is a -convex like map on the convex set .

Let . Then is a weakly minimal solution of . Take , . Then , , for . The dual problem becomes

(4.40)

Therefore the conditions of Theorem 4.7 are satisfied. Simultaneous, take and . Obviously, is a feasible solution of . It follows from Theorem 4.5 that is a weakly maximal solution of .

Since neither of and is -convex map on the , the assumptions of [19, Theorem ?4.3] are not satisfied. Therefore, [19, Theorem ?4.3] is unusable here.

Theorem 4.9 (converse duality).

Suppose that , , and the following conditions are satisfied:

(i) ;

(ii)the set - fulfills the weak domination property for all ;

(iii)there exist and such that is a weakly maximal solution of .

Then is a weakly minimal solution of .

Proof.

It follows from Proposition 3.12 that
(4.41)
for all Then,
(4.42)
It follows from that there exists such that . So . Then, from (4.5) and (4.42), we get
(4.43)

We now show that is a weakly minimal solution of . Assume that is not a weakly minimal solution of . Then there exists such that . It follows from that , which contradicts (4.43). Thus is a weakly minimal solution of and the proof is complete.

Theorem 4.10 (converse duality).

Suppose that , , and the following conditions are satisfied:

(i) ;

(ii)the set - fulfills the weak domination property for all ;

(iii)there exist and such that is a weakly maximal solution of .

Then is a weakly minimal solution of .

Proof.

By the similar proof method for Theorem 4.9, it follows from Proposition 3.14 that the conclusion holds.

5. Higher-Order Wolfe Type Duality

In this section, we introduce a kind of higher-order Wolf type dual problem for a constrained set-valued optimization problem by virtue of the higher-order weakly generalized adjacent epiderivative and discuss its weak duality, strong duality and converse duality properties.

Suppose that , , , and ? . We introduce a higher-order Wolfe type dual problem of as follows:

(5.1)
(5.2)
(5.3)

A point satisfying (5.1)–(5.3) is called a feasible solution of . A feasible solution is called an optimal solution of if, for any feasible solution , .

Theorem 5.1 (weak duality).

Let , , . Let the set - fulfill the weak domination property for all . If is a feasible solution of and is a feasible solution of , then
(5.4)

Proof.

It follows from Proposition 3.12 that
(5.5)

Since is a feasible solution of , . Take . Then it follows from (5.3) that

(5.6)
From (5.1)–(5.6), we get
(5.7)

and the proof is complete.

Theorem 5.2 (weak duality).

Let , , and and the set - fulfill the weak domination property for all . Suppose that is -convex like on a nonempty convext subset . If is a feasible solution of and is a feasible solution of , then
(5.8)

Proof.

By using similar proof method of Theorem 5.1 and Proposition 3.14, we have that the conclusion holds.

Theorem 5.3 (strong duality).

If the assumptions in Theorem 4.7 are satisfied and , then there exist and such that is an optimal solution of .

Proof.

It follows from the proof of Theorem 4.7 that there exist and such that is a feasible solution of and .

We now prove that is an optimal solution of .

Suppose that is not an optimal solution of . Then there exists a feasible solution such that
(5.9)
Therefore, it follows from that
(5.10)

Since is a weakly minimal solution of , it follows from Theorem 5.2 that . From (5.10), we get , this is impossible since . So is an optimal solution of .

By using similar proof methods for Theorems 4.9 and 4.10, we get the following results.

Theorem 5.4 (converse duality).

Suppose that there exists a such that is an optimal solution of and . Moreover, the assumptions and in Theorem 4.9 are satisfied. Then is a weakly minimal solution of .

Theorem 5.5 (converse duality).

Suppose that there exists a such that is an optimal solution of and . Moreover, the assumptions and in Theorem 4.10 are satisfied. Then is a weakly minimal solution of .

Notes

Declarations

Acknowledgments

The authors thank anonymous referees for their valuable comments and suggestions, which helped them to improve the paper. This research was partially supported by the National Natural Science Foundation of China (10871216), Natural Science Foundation Project of CQ CSTC(2008BB0346) and the Excellent Young Teachers Program (2008EYT-016) of Chongqing Jiaotong University, China.

Authors’ Affiliations

(1)
College of Mathematics and Science, Chongqing University
(2)
College of Sciences, Chongqing Jiaotong University

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Copyright

© Q. L. Wang and S. J. Li. 2009

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.