Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

L. M. Graña Drummond

Programa de Engenharia de Sistemas de Computação, COPPE-UFRJ, CP 68511, Rio
de Janeiro, RJ 21945-970, 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

Keywords

  • Vector optimization
  • Löwner order
  • ideal efficiency
  • first order optimality conditions
  • convex programming
  • duality

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Suggested citation

L. M. Graña Drummond, A. N. Iusem. “First Order Conditions for Ideal Minimization of Matrix-Valued Problems.” Journal of Convex Analysis 10 (2003), No. 1, 129–147. https://doi.org/10.68381/jca1006

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Published by Heldermann Verlag, 2003. Rights now held by Banach Press LLC.