# Optimality Conditions and Numerical Algorithms for A Class of Linearly Constrained Minimax Optimization Problems

arXiv: https://arxiv.org/abs/2204.09185 · HTML: https://mlsanity.com/paper/2204.09185/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Apr 2022.

## Built on by (most cited first, 4 of 4)

- [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)
- [Iterative Minimax Games with Coupled Linear Constraints](https://mlsanity.com/paper/2212.04672/) (2022)
- [A first-order method for nonconvex-strongly-concave constrained minimax optimization](https://mlsanity.com/paper/2512.22909/) (2025)

## Abstract (arXiv)

It is well known that there have been many numerical algorithms for solving nonsmooth minimax problems, numerical algorithms for nonsmooth minimax problems with joint linear constraints are very rare. This paper aims to discuss optimality conditions and develop practical numerical algorithms for minimax problems with joint linear constraints. First of all, we use the properties of proximal mapping and KKT system to establish optimality conditions. Secondly, we propose a framework of alternating coordinate algorithm for the minimax problem and analyze its convergence properties. Thirdly, we develop a proximal gradient multi-step ascent decent method (PGmsAD) as a numerical algorithm and demonstrate that the method can find an ε-stationary point for this kind of nonsmooth nonconvex-nonconcave problem in cal O(ε-2logε-1) iterations. Finally, we apply PGmsAD to generalized absolute value equations, generalized linear projection equations and linear regression problems and report the efficiency of PGmsAD on large-scale optimization.