# Optimality Conditions for Constrained Minimax Optimization

arXiv: https://arxiv.org/abs/2004.09730 · HTML: https://mlsanity.com/paper/2004.09730/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Apr 2020.

## Built on by (most cited first, 6 of 6)

- [Closed-Loop Data Transcription to an LDR via Minimaxing Rate Reduction](https://mlsanity.com/paper/2111.06636/) (2021)
- [Provably Efficient Generalized Lagrangian Policy Optimization for Safe Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2306.00212/) (2023)
- [Convex-Concave Min-Max Stackelberg Games](https://mlsanity.com/paper/2110.05192/) (2021)
- [A first-order augmented Lagrangian method for constrained minimax optimization](https://mlsanity.com/paper/2301.02060/) (2023)
- [Solving bilevel optimization via sequential minimax optimization](https://mlsanity.com/paper/2511.07398/) (2025)
- [A first-order method for nonconvex-strongly-concave constrained minimax optimization](https://mlsanity.com/paper/2512.22909/) (2025)

## Abstract (arXiv)

Minimax optimization problems arises from both modern machine learning including generative adversarial networks, adversarial training and multi-agent reinforcement learning, as well as from tradition research areas such as saddle point problems, numerical partial differential equations and optimality conditions of equality constrained optimization. For the unconstrained continuous nonconvex-nonconcave situation, Jin, Netrapalli and Jordan (2019) carefully considered the very basic question: what is a proper definition of local optima of a minimax optimization problem, and proposed a proper definition of local optimality called local minimax. We shall extend the definition of local minimax point to constrained nonconvex-nonconcave minimax optimization problems. By analyzing Jacobian uniqueness conditions for the lower-level maximization problem and the strong regularity of Karush-Kuhn-Tucker conditions of the maximization problem, we provide both necessary optimality conditions and sufficient optimality conditions for the local minimax points of constrained minimax optimization problems.