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Yang et al. (2023) recently addressed the open problem of solving VariationalInequalities (VIs) with equality and inequality constraints through afirst-order gradient method. However, the proposed primal-dual method calledACVI is applicable when we can compute analytic solutions of its subproblems;thus, the general case remains an open problem. In this paper, we adopt awarm-starting technique where we solve the subproblems approximately at eachiteration and initialize the variables with the approximate solution found atthe previous iteration. We prove its convergence and show that the gap functionof the last iterate of this inexact-ACVI method decreases at a rate of$\mathcal{O}(\frac{1}{\sqrt{K}})$ when the operator is $L$-Lipschitz andmonotone, provided that the errors decrease at appropriate rates.Interestingly, we show that often in numerical experiments, this techniqueconverges faster than its exact counterpart. Furthermore, for the cases whenthe inequality constraints are simple, we propose a variant of ACVI namedP-ACVI and prove its convergence for the same setting. We further demonstratethe efficacy of the proposed methods through numerous experiments. We alsorelax the assumptions in Yang et al., yielding, to our knowledge, the firstconvergence result that does not rely on the assumption that the operator is$L$-Lipschitz. Our source code is provided at$\texttt{https://github.com/mpagli/Revisiting-ACVI}$.

A Primal-dual Approach for Solving Variational Inequalities with General-form Constraints