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

arXiv: https://arxiv.org/abs/2510.01168 · HTML: https://mlsanity.com/paper/2510.01168/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 34 references that are in the map.
- First posted on arXiv 1 Oct 2025.

## Builds on (most cited first, 19 of 19)

- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Certifying Some Distributional Robustness with Principled Adversarial Training](https://mlsanity.com/paper/1710.10571/) (2017)
- [Deep Decentralized Multi-task Multi-Agent Reinforcement Learning under Partial Observability](https://mlsanity.com/paper/1703.06182/) (2017)
- [Solving a Class of Non-Convex Min-Max Games Using Iterative First Order Methods](https://mlsanity.com/paper/1902.08297/) (2019)
- [Distributionally Robust Optimization: A Review](https://mlsanity.com/paper/1908.05659/) (2019)
- [Faster Single-loop Algorithms for Minimax Optimization without Strong Concavity](https://mlsanity.com/paper/2112.05604/) (2021)
- [First-order Convergence Theory for Weakly-Convex-Weakly-Concave Min-max Problems](https://mlsanity.com/paper/1810.10207/) (2018)
- [Accelerated Single-Call Methods for Constrained Min-Max Optimization](https://mlsanity.com/paper/2210.03096/) (2022)
- [Solving Nonconvex-Nonconcave Min-Max Problems exhibiting Weak Minty Solutions](https://mlsanity.com/paper/2201.12247/) (2022)
- [Distributionally Robust Optimization and Robust Statistics](https://mlsanity.com/paper/2401.14655/) (2024)
- [Escaping limit cycles: Global convergence for constrained nonconvex-nonconcave minimax problems](https://mlsanity.com/paper/2302.09831/) (2023)
- [Nonsmooth Nonconvex-Nonconcave Minimax Optimization: Primal-Dual Balancing and Iteration Complexity Analysis](https://mlsanity.com/paper/2209.10825/) (2022)
- [Universal Gradient Descent Ascent Method for Nonconvex-Nonconcave Minimax Optimization](https://mlsanity.com/paper/2212.12978/) (2022)
- [Zeroth-Order Alternating Gradient Descent Ascent Algorithms for a Class of Nonconvex-Nonconcave Minimax Problems](https://mlsanity.com/paper/2211.13668/) (2022)
- [Enhanced Adaptive Gradient Algorithms for Nonconvex-PL Minimax Optimization](https://mlsanity.com/paper/2303.03984/) (2023)
- [Nonsmooth optimization using Taylor-like models: error bounds, convergence, and termination criteria](https://mlsanity.com/paper/1610.03446/) (2016)
- [Convergence rate analysis of a sequential convex programming method with line search for a class of constrained difference-of-convex optimization problems](https://mlsanity.com/paper/2001.06998/) (2020)
- [Doubly Smoothed Optimistic Gradients: A Universal Approach for Smooth Minimax Problems](https://mlsanity.com/paper/2506.07397/) (2025)
- [A first-order method for nonconvex-nonconcave minimax problems under a local Kurdyka-Lojasiewicz condition](https://mlsanity.com/paper/2507.01932/) (2025)

## 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.