A Minimization Approach for Minimax Optimization with Coupled Constraints

TL;DR

This paper introduces a novel minimization approach for nonconvex-strongly-concave minimax problems with coupled constraints, transforming them into more tractable forms and enabling the use of existing minimization methods.

Contribution

It proposes a dual variable-based reformulation and a partial forward-backward envelope technique to convert complex minimax problems into explicit minimization problems with equivalent stationary points.

Findings

The reformulation preserves first-order minimax points.

The proposed method enables efficient solution of constrained minimax problems.

Preliminary experiments show promising results.

Abstract

In this paper, we focus on the nonconvex-strongly-concave minimax optimization problem (MCC), where the inner maximization subproblem contains constraints that couple the primal variable of the outer minimization problem. We prove that by introducing the dual variable of the inner maximization subproblem, (MCC) has the same first-order minimax points as a nonconvex-strongly-concave minimax optimization problem without coupled constraints (MOL). We then extend our focus to a class of nonconvex-strongly-concave minimax optimization problems (MM) that generalize (MOL). By performing the partial forward-backward envelope to the primal variable of the inner maximization subproblem, we propose a minimization problem (MMPen), where its objective function is explicitly formulated. We prove that the first-order stationary points of (MMPen) coincide with the first-order minimax points of (MM). Therefore, various efficient minimization methods and their convergence guarantees can be directly employed to solve (MM), hence solving (MCC) through (MOL). Preliminary numerical experiments demonstrate the great potential of our proposed approach.