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等式和不等式约束的变分不等式求解这一开放性问题。然而，他们所提出的名为ACVI的原始-对偶方法仅在可计算子问题解析解时适用，因此一般情形仍有待解决。本文采用热启动技术，在每次迭代中近似求解子问题，并用上一次迭代求得的近似解初始化变量。我们证明了该方法的收敛性，并表明在算子为$L$-利普希茨连续且单调的条件下，若误差以适当速率递减，此非精确ACVI方法最后一步迭代的间隙函数以$\mathcal{O}(\frac{1}{\sqrt{K}})$速率下降。有趣的是，数值实验表明该技术通常比其精确对应方法收敛更快。此外，针对不等式约束为简单形式的情形，我们提出ACVI的变体P-ACVI，并证明了其在相同设定下的收敛性。通过大量实验进一步验证了所提方法的有效性。我们还放宽了Yang等人中的假设，据我们所知，这首次得到了不依赖算子$L$-利普希茨连续性假设的收敛性结果。源代码公布于$\texttt{https://github.com/mpagli/Revisiting-ACVI}$。