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Optimal High-Order Methods for Solving Monotone Variational Inequalities

arXiv · AI, language, vision and robotics · article · Sep 20, 2026 · UTC

We study second- and higher-order methods for solving smooth monotone variational inequalities (MVI). Monteiro and Svaiter (SIAM J. Optim., 2012) showed that a second-order method, NPE, converges at a rate of $\mathcal{O}(T^{-1.5})$. For convex-concave minimax optimization, a subclass of MVI problems, Chen, Liu, Luo, and Zhang (COLT 2025) recently improved this rate to $\tilde{\mathcal{O}}( T^{-1.75})$. However, the result has a substantial gap compared to the lower bound of $Ω(T^{-2.5})$ established by Chen et al. (2026). In this paper, we propose a novel second-order method that achieves the

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