# A Lyapunov analysis of Korpelevich's extragradient method with fast and flexible extensions

arXiv: https://arxiv.org/abs/2502.00119 · HTML: https://mlsanity.com/paper/2502.00119/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Jan 2025.

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

- [Parameter-Free Non-Ergodic Extragradient Algorithms for Solving Monotone Variational Inequalities](https://mlsanity.com/paper/2604.07662/) (2026)

## Abstract (arXiv)

We develop a Lyapunov-based analysis of Korpelevich's extragradient method and show that it achieves an o(1/k) last-iterate convergence rate of the constructed Lyapunov function. This Lyapunov function simultaneously upper bounds several standard measures of optimality, which allows our analysis to sharpen existing last-iterate convergence guarantees for these measures. Moreover, the same analysis enables the design of a class of flexible extensions of the extragradient method in which extragradient steps are adaptively blended with user-specified directions via a Lyapunov-guided line-search procedure. These extensions retain global convergence under practical assumptions and can attain superlinear rates when the directions are chosen appropriately. Numerical experiments confirm the simplicity and efficiency of the proposed framework.