# A Fast Optimistic Method for Monotone Variational Inequalities

arXiv: https://arxiv.org/abs/2307.11281 · HTML: https://mlsanity.com/paper/2307.11281/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Jul 2023.

## Built on by (most cited first, 3 of 3)

- [Accelerated and Stable Convergence with Anchored Generalized Optimistic Method](https://mlsanity.com/paper/2606.21528/) (2026)
- [On Same-Sample and Independent-Sample Stochastic Extragradient for Monotone Variational Inequalities](https://mlsanity.com/paper/2608.06182/) (2026)
- [Polyak-Type Extragradient Methods for Monotone Root-Finding Problems](https://mlsanity.com/paper/2609.26581/) (2026)

## Abstract (arXiv)

We study monotone variational inequalities that can arise as optimality conditions for constrained convex optimisation or convex-concave minimax problems and propose a novel algorithm that uses only one gradient/operator evaluation and one projection onto the constraint set per iteration. The algorithm, which we call fOGDA-VI, achieves a o left( frac1k right) rate of convergence in terms of the restricted gap function as well as the natural residual for the last iterate. Moreover, we provide a convergence guarantee for the sequence of iterates to a solution of the variational inequality. These are the best theoretical convergence results for numerical methods for (only) monotone variational inequalities reported in the literature. To empirically validate our algorithm we investigate a two-player matrix game with mixed strategies of the two players. Concluding, we show promising results regarding the application of fOGDA-VI to the training of generative adversarial nets.