Calm local optimality for couple-constrained minimax problems

Abstract: Recently, a new local optimality concept for minimax problems, termed calm local minimax points, has been introduced. In this paper, we extend this concept to a general class of nonsmooth, nonconvex nonconcave minimax problems with coupled constraints, where the inner feasible set depends on the outer variable. We derive comprehensive first-order and second-order necessary and sufficient optimality conditions for calm local minimax points in the setting of nonsmooth, nonconvex nonconcave minimax problems with coupled constraints. Furthermore, we show how these conditions apply to problems with set constraints, as well as those involving systems of inequalities and equalities. By unifying existing formulations that often rely on stronger assumptions within the framework of calm local minimax points, we show that our results hold under weaker assumptions than those previously required.