Optimal High-Order Methods for Solving Monotone Variational Inequalities

2026-09-20

AI Summary

本文针对光滑单调变分不等式问题，提出了一种新的二阶方法及更高阶方法，达到了最优收敛速度，解决了现有方法收敛速度低于理论下界的问题。

Abstract

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 optimal rate of $\mathcal{O}(T^{-2.5})$. Our algorithm also extends to higher-order methods: for any integer $ p \ge 1$, we obtain a $p$th-order method with a convergence rate of $\mathcal{O}(T^{-(3p-1)/2})$, matching the known lower bounds and therefore establishing optimal rates across all orders.

Problem

• Monotone Variational Inequalities
• Convergence Rate
• Higher-Order Methods

Innovation

• second-order method
• optimal rate
• higher-order methods
• monotone variational inequalities