The Continuity and Convexity of a Nonlinear Scalarization Function with Applications in Set Optimization Problems Involving a Partial Order Relation

In this paper, we deal with the properties and applications of a nonlinear scalarization function for sets by using the Minkowski difference. Under some suitable assumptions, the continuity and convexity concerned with the nonlinear scalarization function for sets are presented. As applications, the path connectedness of the solution sets to set optimization problems and the continuity of the solution mappings of parametric set optimization problems are established. The results achieved do not impose the monotonicity of the set-valued objective mapping, which are obviously different from the related ones in the literature.