Home

 

Recent Issues

Volume 52

1

2

3

4

Volume 51

1

2

3

4

Volume 50

1

2

3

4

Volume 49

1

2

3

4

Volume 48

1

2

3

4

Past Issues

 

The Journal

Cover

Aims and Scope

Subscription Information

Editorial Board

Instructions for Author

Contact Us

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Vietnam Journal of Mathematics 40: 2&3(2012) 285-304

 

Generalized Dubovitskii-Milyutin Approach in

Set-Valued Optimization

Akhtar A. Khan1 and Christiane Tammer2

1Center for Applied and Computational Mathematics, School of Mathematical

Sciences, Rochester Institute of Technology, 85 Lomb Memorial Drive,

Rochester, New York, 14623, USA

2Institute of Mathematics, Martin-Luther-University of Halle-Wittenberg,

Theodor-Lieser-Str. 5, D-06120 Halle-Salle, Germany

Dedicated to Professor Phan Quoc Khanh on the occasion of his 65th birthday

Received December 3, 2011

Revised July 3, 2012

Abstract. The primary objective of this paper is to employ the Dubovitskii-Milyutin approach to give first and second-order optimality conditions for set-valued optimization problems with more general set-valued constraints. In the particular case when some constraints are single-valued, we recover the case of set-valued optimization problems with many set-valued inequality constraints and many single-valued equality constraints. It is known that the classical Dubovitskii-Milyutin approach is not suitable for optimization problems with multi-equality constraints. The main reason for this deficiency is the fact that the separation arguments used in the classical Dubovitskii-Milyutin approach are applicable to an empty intersection of cones in which at most one cone can be closed. However, a proper formulation of multi-equality constraints leads to an empty intersection with more than one closed cones. To study optimization problems with multi-equality constraints, a generalized Dubovitskii-Milyutin theory has been developed. In this work we present an extension of the generalized Dubovitskii-Milyutin theory to the set-valued optimization problems. In this process, we also obtain new applications of this theory to nonsmooth optimization and to more general vector optimization problems. New second-order asymptotic derivatives of set-valued maps are introduced and used to give the optimality conditions.

2000 Mathematics Subject Classification. 90C26, 90C29, 90C30.

Keywords. Set-valued optimization, lower Dini derivative, second-order lower Dini asymptotic derivative, asymptotic contingent cones, interiorly tangent cones, contingent cones, optimality conditions.

 

 

Established by Vietnam Academy of Science and Technology & Vietnam Mathematical Society
Published by Springer since January 2013