First Order Conditions for Ideal Minimization of Matrix-Valued Problems
L. M. Gra, Alfredo Noel Iusem, Estrada Dona Castorina · 2003
The aim of this paper is to study first order optimality conditions for ideal efficient points in the Lowner 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.