# Min-Max Optimization Is Strictly Easier Than Variational Inequalities

arXiv: https://arxiv.org/abs/2511.03052 · HTML: https://mlsanity.com/paper/2511.03052/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 30 references that are in the map.
- First posted on arXiv 4 Nov 2025.

## Builds on (most cited first, 9 of 9)

- [On Gradient Descent Ascent for Nonconvex-Concave Minimax Problems](https://mlsanity.com/paper/1906.00331/) (2019)
- [Accelerated Algorithms for Smooth Convex-Concave Minimax Problems with O(1/k2) Rate on Squared Gradient Norm](https://mlsanity.com/paper/2102.07922/) (2021)
- [Improved Algorithms for Convex-Concave Minimax Optimization](https://mlsanity.com/paper/2006.06359/) (2020)
- [Near-optimal Local Convergence of Alternating Gradient Descent-Ascent for Minimax Optimization](https://mlsanity.com/paper/2102.09468/) (2021)
- [The First Optimal Algorithm for Smooth and Strongly-Convex-Strongly-Concave Minimax Optimization](https://mlsanity.com/paper/2205.05653/) (2022)
- [Accelerating Smooth Games by Manipulating Spectral Shapes](https://mlsanity.com/paper/2001.00602/) (2020)
- [Negative Stepsizes Make Gradient-Descent-Ascent Converge](https://mlsanity.com/paper/2505.01423/) (2025)
- [Fundamental Benefit of Alternating Updates in Minimax Optimization](https://mlsanity.com/paper/2402.10475/) (2024)
- [Two-timescale Extragradient for Finding Local Minimax Points](https://mlsanity.com/paper/2305.16242/) (2023)

## Built on by (most cited first, 1 of 1)

- [Negative Momentum for Convex-Concave Optimization](https://mlsanity.com/paper/2604.17145/) (2026)

## Abstract (arXiv)

Classically, a mainstream approach for solving a convex-concave min-max problem is to instead solve the variational inequality problem arising from its first-order optimality conditions. Is it possible to solve min-max problems faster by bypassing this reduction? This paper initiates this investigation. We show that the answer is yes in the textbook setting of unconstrained quadratic objectives: the optimal convergence rate for first-order algorithms is strictly better for min-max problems than for the corresponding variational inequalities. The key reason that min-max algorithms can be faster is that they can exploit the asymmetry of the min and max variables--a property that is lost in the reduction to variational inequalities. Central to our analyses are sharp characterizations of optimal convergence rates in terms of extremal polynomials which we compute using Green's functions and conformal mappings.