Contractibility of the solution sets for set optimization problems

Abstract

This paper aims at investigating the contractibility of the solution sets for set optimization problems by utilizing strictly quasi cone-convexlikeness, which is an assumption weaker than strictly cone-convexity, strictly cone-quasiconvexity and strictly naturally quasi cone-convexity. We establish the contractibility of l-minimal, l-weak minimal, u-minimal and u-weak minimal solution sets for set optimization problems by using the star-shape sets and the nonlinear scalarizing functions for sets. Moreover, we also discuss the arcwise connectedness and the contractibility of p-minimal and p-weak minimal solution sets for set optimization problems by using the scalarization technique. Finally, our main results are applied to the contractibility of the solution sets for vector optimization problems.