epsilon -Optimality conditions of vector optimization problems with set-valued maps based on the algebraic interior in real linear spaces

In this paper, firstly, the necessary and sufficient optimality conditions for $epsilon $ -global properly efficient elements of set-valued optimization problems, respectively, are established in linear spaces. Secondly, an equivalent characterization of $epsilon $ -global proper saddle point is presented. Finally, the necessary and sufficient conditions for $epsilon $ -global properly saddle point of a Lagrangian set-valued map are obtained. The results in this paper generalize some known results in the literature.