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

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