Optimality conditions for global minimizers to a class of convex set optimization problem subjected to geometric constraints

In this paper, we study optimality conditions for both global and approximate minimizers to convex set optimization problems with geometric constraints. We first consider a form of Gerstewitz’s nonlinear scalarization function concerning the set-less relation introduced by Kuroiwa. Then, it is employed to construct a type of directional derivative and sub-gradient for cone-convex set-valued maps. We also give some properties and usual calculus rules for these concepts. Later, some necessary and sufficient conditions for global and approximate solutions are established. Examples are provided for analyzing and illustrating the obtained results.