Nonsmooth Nonconvex-Concave Minimax Optimization: Convergence Criteria and Algorithms

This paper considers constrained stochastic nonsmooth minimax optimization problem of the form minx∈Xmaxy∈Yfleft(x,yright)=mathbbE[F(x,y;xi)], where the objective f(x,y) is concave in y but possibly nonconvex in x, the stochastic component F(x,y;xi) indexed by random variable xi is mean-squared Lipschitz continuous, and the feasible sets mathcal X and mathcal Y are convex and compact. We introduce the notion of (ηx,ηy,δ,ε)-Goldstein saddle stationary point (GSSP) to characterize the convergence for solving constrained nonsmooth minimax problems. We then develop projected gradient-free descent ascent methods for finding (ηx,ηy,δ,ε)-GSSPs of the objective function f(x,y) with non-asymptotic convergence rates. We further propose nested-loop projected gradient-free descent ascent methods to establish the non-asymptotic convergence for finding (η,δ,ε)-generalized Goldstein stationary points (GGSP) [Liu et al., 2024] of the primal function Phi(x)triangleqmaxy∈Yfleft(x,yright). It is worth noting that our algorithm designs and theoretical analyses do not require additional assumptions such as the weak convexity used in prior works on nonsmooth minimax optimization [Lin et al., 2025, Boct and B"ohm, 2023].