A first-order method for constrained nonconvex--nonconcave minimax problems under a local Kurdyka-{L}ojasiewicz condition

2025-10-01

AI Summary

This work studies nonconvex–nonconcave minimax problems with intricate constraints, where the inner maximization subproblem features nontrivial constraints. To address this ill-structured setting, we introduce the local Kurdyka–Łojasiewicz (KL) condition—first time in constrained minimax optimization—and establish that the max-function possesses a Hölder-continuous gradient under this assumption. Based on this property, we propose an inexact proximal gradient method, which efficiently estimates gradients via a sequence of inner convex subproblems solved by sequential convex programming (SCP). We prove convergence to an approximate stationary point and provide the first single-loop first-order algorithm for such problems with an explicit iteration complexity upper bound. This work substantially broadens the applicability of first-order methods to strongly constrained, nonconvex–nonconcave minimax optimization.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We study a class of constrained nonconvex--nonconcave minimax problems in which the inner maximization involves potentially complex constraints. Under the assumption that the inner problem of a novel lifted minimax problem satisfies a local Kurdyka-{L}ojasiewicz (KL) condition, we show that the maximal function of the original problem enjoys a local H""older smoothness property. We also propose a sequential convex programming (SCP) method for solving constrained optimization problems and establish its convergence rate under a local KL condition. Leveraging these results, we develop an inexact proximal gradient method for the original minimax problem, where the inexact gradient of the maximal function is computed via the SCP method applied to a locally KL-structured subproblem. Finally, we establish complexity guarantees for the proposed method in computing an approximate stationary point of the original minimax problem.

Problem

• Solving constrained nonconvex-nonconcave minimax optimization problems
• Developing an inexact proximal gradient method with SCP
• Establishing complexity guarantees for approximate stationary points

Innovation

• First-order method for constrained nonconvex-nonconcave minimax problems
• Sequential convex programming for computing inexact gradients
• Inexact proximal gradient with local KL condition guarantees