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HigherOrder Weakly Generalized Adjacent Epiderivatives and Applications to Duality of SetValued Optimization
Journal of Inequalities and Applications volumeÂ 2009, ArticleÂ number:Â 462637 (2009)
Abstract
A new notion of higherorder weakly generalized adjacent epiderivative for a setvalued map is introduced. By virtue of the epiderivative and weak minimality, a higherorder MondWeir type dual problem and a higherorder Wolfe type dual problem are introduced for a constrained setvalued 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 setvalued maps have been proposed and used for the formulation of optimality conditions and duality in setvalued optimization problems. By using a contingent epiderivative of a setvalued 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 setvalued map and obtained a unified necessary and sufficient conditions for a setvalued optimization problem. Lalitha and Arora [3] introduced a notion of a weak Clarke epiderivative and use it to establish optimality criteria for a constrained setvalued optimization problem. On the other hand, various kinds of differentiable type dual problems for setvalued optimization problems, such as MondWeir type and Wolfe type dual problems, have been investigated. By virtue of the tangent derivative of a setvalued map introduced in [4], Sach and Craven [5] discussed Wolfe type duality and MondWeir type duality problems for a setvalued optimization problem. By virtue of the codifferential of a setvalued map introduced in [6], Sach et al. [7] obtained MondWeir type and Wolfe type weak duality and strong duality theorems of setvalued optimization problems. As to other concepts of derivatives (epiderivatives) of setvalued maps and their applications, one can refer to [8â€“15]. Recently, Secondorder derivatives have also been proposed, for example, see [16, 17] and so on.
Since higherorder tangent sets introduced in [4], in general, are not cones and convex sets, there are some difficulties in studying higherorder optimality conditions and duality for general setvalued optimization problems. Until now, there are only a few papers to deal with higherorder optimality conditions and duality of setvalued optimization problems by virtue of the higherorder derivatives or epiderivatives introduced by the higherorder tangent sets. Li et al. [18] studied some properties of higherorder tangent sets and higherorder derivatives introduced in [4], and then obtained higherorder necessary and sufficient optimality conditions for setvalued optimization problems under coneconcavity assumptions. By using these higherorder derivatives, they also discussed a higherorder MondWeir duality for a setvalued optimization problem in [19]. Li and Chen [20] introduced higherorder generalized contingent(adjacent) epiderivatives of setvalued maps, and obtained higherorder Fritz John type necessary and sufficient conditions for Henig efficient solutions to a constrained setvalued optimization problem.
Motivated by the work reported in [3, 5, 18â€“20], we introduce a notion of higherorder weakly generalized adjacent epiderivative for a setvalued map. Then, by virtue of the epiderivative, we discuss a higherorder MondWeir type duality problem and a higherorder Wolfe type duality problem to a constrained setvalued 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 higherorder adjacent set of a set and a higherorder weakly generalized adjacent epiderivative of a setvalued map, and study some of their properties. In Sections 4 and 5, we introduce a higherorder MondWeir type dual problem and a higherorder Wolfe type dual problem to a constrained setvalued 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 setvalued 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 setvalued map.
(i) is said to be convex on a convex set , if for any and ,
(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
is the thorder 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. HigherOrder Weakly Generalized Adjacent Epiderivatives
Definition 3.1.
Let belong to a subset K of and let be elements of . The subset
is said to be the thorder generalized adjacent set of at
Definition 3.2.
The thorder weakly generalized adjacent epiderivative of at with respect to vectors is the setvalued map from to defined by
The weak domination property resp., domination property is said to hold for a subset of if resp., .
To compare our derivative with wellknown derivatives, we recall some notions.
Definition 3.4 (see [4]).
The thorder adjacent derivative of at with respect to vectors is the setvalued map from to defined by
Definition 3.5 (see [19]).
The directed thorder adjacent derivative of at with respect to vectors is the thorder adjacent derivative of setvalued mapping at with respect to .
Definition 3.6 (See [20]).
The thorder generalized adjacent epiderivative of at with respect to vectors is the setvalued map from to defined by
Using properties of higherorder 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 ,

(i)
(3.5)

(ii)
(3.6)

(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
Example 3.10.
Let , , and . Then, , . Hence, for any , , and do not exist. But
Example 3.11.
Suppose that . Let be a setvalued map with and . Then ,  Therefore for any ,
Now we discuss some crucial propositions of the thorder weakly generalized adjacent epiderivative.
Proposition 3.12.
Let , , . If the set  fulfills the weak domination property for all , then for all ,
Proof.
Take any and an arbitrary sequence with . Since ,
It follows from , and is a convex cone that
We get
which implies that
that is, . By the definition of thorder weakly generalized adjacent epiderivative and the weak domination property, we have
Thus .
Remark 3.13.
Since the coneconvexity and coneconcavity 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
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
It follows from , is convex and is a convex cone that
We obtain that
which implies that
that is, . By the definition of thorder weakly generalized adjacent epiderivative and the weak domination property, we have
Thus and the proof is complete.
Remark 3.15.
Since the coneconvexity assumptions are replaced by coneconvex likeness assumptions, Proposition 3.14 improves [20, Proposition ].
4. HigherOrder MondWeir Type Duality
In this section, we introduce a higherorder MondWeir type dual problem for a constrained setvalued optimization problem by virtue of the higherorder 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
where stands for the closed ball centered at and of radius .
Lemma 4.1 (see [16]).
If is convex, and , then
Consider the following setvalued optimization problem:
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 higherorder MondWeir type dual problem of as follows:
Let satisfy conditions(4.4)â€“(4.7). A point satisfying (4.4)â€“(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
Proof.
It follows from Proposition 3.12 that
Since is a feasible solution of , . Take . Then, it follows from (4.5) and (4.7) that
By (4.4), (4.6), (4.7), (4.9) and (4.10), we get
Thus, the proof is complete.
Remark 4.4.
In Theorem 4.3, coneconvexity 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
Lemma 4.6.
Let , , . Let the set  fulfill the weak domination property for all . If is a weakly minimal solution of , then
for all
Proof.
Since is a weakly minimal solution of , . Then,
Assume that the result (4.13) does not hold. Then there exist , and with such that
It follows from (4.15) and the definition of thorder weakly generalized adjacent epiderivative that
Thus, for an arbitrary sequence with , there exists a sequence such that
From (4.16) and (4.18), there exists a sufficiently large such that
Since , and is a convex cone,
It follows from (4.19) and (4.21) that
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
Then, from (4.20), we have
It follows from , and is a convex cone that
Since , there exist such that It follows from (4.25) that , for , and then
It follows from (4.22) that
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
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
By the separation theorem of convex sets, there exist and , not both zero functionals, such that
It follows from (4.30) that
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
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
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
According to , we get
Since is a weakly minimal solution of , it follows from Theorem 4.5 that
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 setvalued map with
and be a setvalued map with
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
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
for all Then,
It follows from that there exists such that . So . Then, from (4.5) and (4.42), we get
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. HigherOrder Wolfe Type Duality
In this section, we introduce a kind of higherorder Wolf type dual problem for a constrained setvalued optimization problem by virtue of the higherorder weakly generalized adjacent epiderivative and discuss its weak duality, strong duality and converse duality properties.
Suppose that , , , and ? . We introduce a higherorder Wolfe type dual problem of as follows:
A point satisfying (5.1)â€“(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
Proof.
It follows from Proposition 3.12 that
Since is a feasible solution of , . Take . Then it follows from (5.3) that
From (5.1)â€“(5.6), we get
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
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
Therefore, it follows from that
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 .
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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 (2008EYT016) of Chongqing Jiaotong University, China.
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An erratum to this article is available at http://dx.doi.org/10.1155/2011/817965.
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Wang, Q.L., Li, S.J. HigherOrder Weakly Generalized Adjacent Epiderivatives and Applications to Duality of SetValued Optimization. J Inequal Appl 2009, 462637 (2009). https://doi.org/10.1155/2009/462637
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DOI: https://doi.org/10.1155/2009/462637