About Hahn–Banach extension theorems and applications to set-valued optimization

In this paper we revise several generalizations of the algebraic Hanh–Banach extension theorem for set-valued maps and give new versions in terms of set-relations and selections. We apply our results by establishing optimality conditions for a set-valued constrained optimization problem considering several types of solutions. Finally, we give necessary conditions for the existence of affine selections via affinelike maps in the framework of linear spaces.