PATH CONNECTEDNESS OF SOLUTION SETS FOR PARTIALLY ORDERED SET OPTIMIZATION PROBLEMS

The aim of this paper is to study the path connectedness of the minimal and weak minimal solution sets for set optimization problems equipped with partial set order relations defined on the family of bounded sets by means of Minkowski difference. We propose some new notions of generalized convexity for set-valued maps and a lower level map. Further, by investigating several properties of this level map, we study path connectedness of the solution sets for set optimization problems with respect to partial order relations. Finally, we give an application of the derived results to Nash equilibrium games with vector-valued maps under uncertainty.