Strong Fermat Rules for Constrained Set-Valued Optimization Problems on Banach Spaces

Abstract

In this paper, we propose two kinds of optimality concepts, called the sharp minima and the weak sharp minima, for a constrained set-valued optimization problem. Subsequently, we extend the Fermat rules for the local minima of the constrained set-valued optimization problem to the sharp minima and the weak sharp minima in Banach spaces or Asplund spaces, by means of the Mordukhovich generalized differentiation and the normal cone. As applications, we investigate the generalized inequality systems with constraints, and get some characterizations of error bounds for the constrained generalized inequality systems in convex and nonconvex cases.