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A first-order method for constrained nonconvex-nonconcave minimax optimization

arXiv 2510.01168 ↗First posted 1 Oct 2025math.OC cs.LG cs.NAcited by 0

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Abstract (arXiv)

We study a class of constrained nonconvex-nonconcave minimax optimization problems in which the inner maximization involves potentially complex constraints. Under the assumption that the inner problem of a novel lifted minimax reformulation satisfies a local Kurdyka-Lojasiewicz (KL) condition, we show that the maximal function of the original problem enjoys a local generalized 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.

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References · 19

  1. 2017Towards Deep Learning Models Resistant to Adversarial Attackscited by 6,415
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  3. 2017Deep Decentralized Multi-task Multi-Agent Reinforcement Learning under Partial Observabilitycited by 148
  4. 2019Solving a Class of Non-Convex Min-Max Games Using Iterative First Order Methodscited by 148
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  6. 2021Faster Single-loop Algorithms for Minimax Optimization without Strong Concavitycited by 49
  7. 2018First-order Convergence Theory for Weakly-Convex-Weakly-Concave Min-max Problemscited by 34
  8. 2022Accelerated Single-Call Methods for Constrained Min-Max Optimizationcited by 32
  9. 2022Solving Nonconvex-Nonconcave Min-Max Problems exhibiting Weak Minty Solutionscited by 27
  10. 2024Distributionally Robust Optimization and Robust Statisticscited by 24
  11. 2023Escaping limit cycles: Global convergence for constrained nonconvex-nonconcave minimax problemscited by 21
  12. 2022Nonsmooth Nonconvex-Nonconcave Minimax Optimization: Primal-Dual Balancing and Iteration Complexity Analysiscited by 12
7 more
  1. 2022Universal Gradient Descent Ascent Method for Nonconvex-Nonconcave Minimax Optimizationcited by 12
  2. 2022Zeroth-Order Alternating Gradient Descent Ascent Algorithms for a Class of Nonconvex-Nonconcave Minimax Problemscited by 11
  3. 2023Enhanced Adaptive Gradient Algorithms for Nonconvex-PL Minimax Optimizationcited by 6
  4. 2016Nonsmooth optimization using Taylor-like models: error bounds, convergence, and termination criteriacited by 5
  5. 2020Convergence rate analysis of a sequential convex programming method with line search for a class of constrained difference-of-convex optimization problemscited by 5
  6. 2025Doubly Smoothed Optimistic Gradients: A Universal Approach for Smooth Minimax Problemscited by 2
  7. 2025A first-order method for nonconvex-nonconcave minimax problems under a local Kurdyka-Lojasiewicz conditioncited by 1

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