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Vietnam Journal of Mathematics 40:
2&3(2012) 285-304
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Generalized Dubovitskii-Milyutin Approach in
Set-Valued Optimization
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Akhtar A. Khan1 and Christiane
Tammer2
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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
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Dedicated to Professor
Phan Quoc Khanh on the occasion of his 65th birthday
Received December 3,
2011
Revised July 3, 2012
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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.
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2000 Mathematics
Subject Classification. 90C26, 90C29, 90C30.
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Keywords.
Set-valued optimization, lower Dini derivative, second-order lower Dini
asymptotic derivative, asymptotic contingent cones, interiorly tangent
cones, contingent cones, optimality conditions.
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Established
by Vietnam Academy of Science and Technology &
Vietnam Mathematical Society
Published by
Springer since January 2013
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