Optimality theorems for convex semidefinite vector optimization problems

Abstract

In this paper, we consider a convex semidefinite vector optimization problem (SDVP) involving a convex objective vector function, a matrix linear inequality constraint and a geometric constraint, and define (properly, weakly) efficient solutions for SDVP as we do for ordinary vector optimization problems. We present necessary and sufficient optimality conditions for efficient solutions of SDVP, which hold without any constraint qualification. Also, we present necessary and sufficient optimality conditions for properly efficient solutions and weakly efficient solutions for SDVP, which hold without any constraint qualification. Furthermore, we give a sufficient condition for an efficient solution of SDVP to be properly efficient.