Calm local optimality for couple-constrained minimax problems
Xiaoxiao Ma, Jane Ye
TL;DR
The paper addresses calm local optimality in minimax problems with inner sets that depend on the outer variable (coupled constraints), extending the calm local minimax concept to nonsmooth, nonconvex, nonconcave settings. It develops first- and second-order necessary and sufficient optimality conditions using a localized value function $V_{\tau}$ and a calm radius $\tau$, together with dual formulations via a Lagrangian $L(x,y,\alpha,\beta)$. The results cover set-constrained systems and inequality/equality constraints under MSCQ/MFCQ/RS, providing a unifying framework that weakens prior assumptions and offers explicit directional information for sensitivity analysis. Together, these contributions enable practical verification and guide algorithmic development in minimax contexts such as adversarial training and resource allocation where the inner feasible set depends on the outer variable.
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.