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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. Gra˝a Drummond
Programa de Engenharia des Sistemas de ComputašŃo, COPPE-UFRJ, CP 68511, Rio de Janeiro, RJ 21845-970, Brazil, lmgd@cos.ufrj.br

A. N. Iusem
Instituto de Matemßtica 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 L÷wner 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, L÷wner order, ideal efficiency, first order optimality conditions, convex programming, duality.

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