Yang et al. (2023) recently addressed the open problem of solving Variational Inequalities (VIs) with equality and inequality constraints through a first-order gradient method. However, the proposed primal-dual method called ACVI is applicable when we can compute analytic solutions of its subproblems; thus, the general case remains an open problem. In this paper, we adopt a warm-starting technique where we solve the subproblems approximately at each iteration and initialize the variables with the approximate solution found at the previous iteration. We prove its convergence and show that the gap function of 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 and monotone, provided that the errors decrease at appropriate rates. Interestingly, we show that often in numerical experiments, this technique converges faster than its exact counterpart. Furthermore, for the cases when the inequality constraints are simple, we propose a variant of ACVI named P-ACVI and prove its convergence for the same setting. We further demonstrate the efficacy of the proposed methods through numerous experiments. We also relax the assumptions in Yang et al., yielding, to our knowledge, the first convergence result that does not rely on the assumption that the operator is $L$-Lipschitz. Our source code is provided at $\texttt{https://github.com/mpagli/Revisiting-ACVI}$.

翻译：Yang等人(2023)近期通过一阶梯度方法解决了带有等式与不等式约束的变分不等式(VI)这一开放问题。然而，他们提出的原始-对偶方法ACVI仅适用于子问题可求得解析解的情形，因此一般情形下的求解仍是开放问题。本文采用热启动技术：每次迭代中近似求解子问题，并用前一次迭代得到的近似解初始化变量。我们证明了该方法的收敛性，并表明当算子为$L$-Lipschitz且单调时，该非精确ACVI方法末次迭代的间隙函数下降率为$\mathcal{O}(\frac{1}{\sqrt{K}})$，前提是误差项以适当速率递减。有趣的是，数值实验表明该技术的收敛速度通常优于精确版本。此外，针对不等式约束为简单约束的情形，我们提出ACVI的变体P-ACVI，并在相同设定下证明其收敛性。通过大量实验验证了所提方法的有效性。我们还放宽了Yang等人中的假设条件，据我们所知，首次得到了不依赖算子$L$-Lipschitz假设的收敛结果。源代码已公开于$\texttt{https://github.com/mpagli/Revisiting-ACVI}$。