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Yang et al. (2023) recently showed how to use first-order gradient methods tosolve general variational inequalities (VIs) under a limiting assumption thatanalytic solutions of specific subproblems are available. In this paper, wecircumvent this assumption via a warm-starting technique where we solvesubproblems approximately and initialize variables with the approximatesolution found at the previous iteration. We prove the convergence of thismethod and show that the gap function of the last iterate of the methoddecreases at a rate of $O(\frac{1}{\sqrt{K}})$ when the operator is$L$-Lipschitz and monotone. In numerical experiments, we show that thistechnique can converge much faster than its exact counterpart. Furthermore, forthe cases when the inequality constraints are simple, we introduce analternative variant of ACVI and establish its convergence under the sameconditions. Finally, we relax the smoothness assumptions in Yang et al.,yielding, to our knowledge, the first convergence result for VIs with generalconstraints that does not rely on the assumption that the operator is$L$-Lipschitz.

A Primal-Dual Approach to Solving Variational Inequalities with General Constraints