Existence and connectedness of the l-minimal approximate solutions for set optimization problems: an application in generalized multiobjective robustness

The goal of this paper is to study the existence and connectedness of the l-minimal approximate solutions of the set-valued optimization problem using an extended signed distance function. The existence of the l-minimal approximate solutions is established by virtue of the FAN-KKM and Cantor's intersection theorems. A scalarization result of the set of l-minimal approximate solutions is proposed without adopting the convexity notion of the objective function. By using this scalarization result, we explore the (path) connectedness of the l-minimal approximate solutions. Moreover, we apply our approach to generalized multiobjective robustness problems. Some necessary examples are illustrated to validate our main results.