Revisiting the ACVI Method for Constrained Variational Inequalities

ACVI is a recently proposed first-order method for solving variational inequalities (VIs) with general constraints. Yang et al. (2022) showed that the gap function of the last iterate decreases at a rate of $\mathcal{O}(\frac{1}{\sqrt{K}})$ when the operator is $L$-Lipschitz, monotone, and at least one constraint is active. In this work, we show that the same guarantee holds when only assuming that the operator is monotone. To our knowledge, this is the first analytically derived last-iterate convergence rate for general monotone VIs, and overall the only one that does not rely on the assumption that the operator is $L$-Lipschitz. Furthermore, when the sub-problems of ACVI are solved approximately, we show that by using a standard warm-start technique the convergence rate stays the same, provided that the errors decrease at appropriate rates. We further provide empirical analyses and insights on its implementation for the latter case.