Contractibility of the solution sets for set optimization problems

TL;DR

This paper investigates the contractibility of solution sets in set optimization problems using weaker convexity assumptions, establishing conditions for various minimal solution sets and applying results to vector optimization.

Contribution

It introduces the use of strictly quasi cone-convexlikeness to analyze solution set contractibility, extending previous convexity assumptions in set optimization.

Findings

Proves contractibility of l-minimal and u-minimal solution sets.

Establishes arcwise connectedness of p-minimal solution sets.

Applies results to vector 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.