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Journal of Convex Analysis 10 (2003), No. 1, 129--147
Copyright Heldermann Verlag 2003
First Order Conditions for Ideal Minimization of Matrix-Valued Problems
L. M. Graña Drummond
Programa de Engenharia des Sistemas de Computação, COPPE-UFRJ, CP 68511, Rio
de Janeiro, RJ 21845-970, Brazil,
lmgd@cos.ufrj.br
A. N. Iusem
Instituto de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina 110,
Rio de Janeiro, RJ, CEP 22460-320, Brazil,
iusp@impa.br
The aim of this paper is to study first order optimality
conditions for ideal efficient points in the Löwner partial
order, when the data functions of the minimization problem are
differentiable and convex with respect to the cone of symmetric
semidefinite matrices. We develop two sets of first order
necessary and sufficient conditions. The first one, formally very
similar to the classical Karush-Kuhn-Tucker conditions for
optimization of real-valued functions, requires two constraint
qualifications, while the second one holds just under a
Slater-type one. We also develop duality schemes for both sets of
optimality conditions.
Keywords: Vector optimization, Löwner order, ideal efficiency, first order
optimality conditions, convex programming, duality.
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