# Optimality conditions for nonsmooth nonconvex-nonconcave min-max problems and generative adversarial networks

arXiv: https://arxiv.org/abs/2203.10914 · HTML: https://mlsanity.com/paper/2203.10914/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Mar 2022.

## Built on by (most cited first, 5 of 5)

- [Two-Timescale Gradient Descent Ascent Algorithms for Nonconvex Minimax Optimization](https://mlsanity.com/paper/2408.11974/) (2024)
- [Fairness-aware Regression Robust to Adversarial Attacks](https://mlsanity.com/paper/2211.04449/) (2022)
- [An accelerated first-order regularized momentum descent ascent algorithm for stochastic nonconvex-concave minimax problems](https://mlsanity.com/paper/2310.15448/) (2023)
- [A stochastic smoothing framework for nonconvex-nonconcave minEmax problems with applications to Wasserstein distributionally robust optimization](https://mlsanity.com/paper/2502.17602/) (2025)
- [Iterative Minimax Games with Coupled Linear Constraints](https://mlsanity.com/paper/2212.04672/) (2022)

## Abstract (arXiv)

This paper considers a class of nonsmooth nonconvex-nonconcave min-max problems in machine learning and games. We first provide sufficient conditions for the existence of global minimax points and local minimax points. Next, we establish the first-order and second-order optimality conditions for local minimax points by using directional derivatives. These conditions reduce to smooth min-max problems with Fr'echet derivatives. We apply our theoretical results to generative adversarial networks (GANs) in which two neural networks contest with each other in a game. Examples are used to illustrate applications of the new theory for training GANs.