Optimal conditions in uncertain set-valued optimization problems via second-order subdifferential involving Minkowski difference with application to zero-sum matrix games

Abstract

The primary aim of this paper is to explore new ideas regarding second-order subdifferentials for set-valued maps, utilizing the framework of set order relations involving Minkowski difference. We commence by establishing fundamental properties of these subdifferentials, encompassing convexity, closure and the Moreau–Rockafellar theorem. Furthermore, existence theorems of the subdifferentials are derived. In addition, we establish optimality conditions of the $m$-order robust solutions to uncertain set optimization via the subdifferential. Moreover, we formulate duality theorems between the primal and the Wolfe dual problems. Finally, the paper concludes with an application of our current methodology to the context of two-player zero-sum matrix games.