We develop an interior-point approach to solve constrained variational inequality (cVI) problems. Inspired by the efficacy of the alternating direction method of multipliers (ADMM) method in the single-objective context, we generalize ADMM to derive a first-order method for cVIs, that we refer to as ADMM-based interior-point method for constrained VIs (ACVI). We provide convergence guarantees for ACVI in two general classes of problems: (i) when the operator is $\xi$-monotone, and (ii) when it is monotone, some constraints are active and the game is not purely rotational. When the operator is, in addition, L-Lipschitz for the latter case, we match known lower bounds on rates for the gap function of $\mathcal{O}(1/\sqrt{K})$ and $\mathcal{O}(1/K)$ for the last and average iterate, respectively. To the best of our knowledge, this is the first presentation of a first-order interior-point method for the general cVI problem that has a global convergence guarantee. Moreover, unlike previous work in this setting, ACVI provides a means to solve cVIs when the constraints are nontrivial. Empirical analyses demonstrate clear advantages of ACVI over common first-order methods. In particular, (i) cyclical behavior is notably reduced as our methods approach the solution from the analytic center, and (ii) unlike projection-based methods that zigzag when near a constraint, ACVI efficiently handles the constraints.

翻译：我们提出了一种内点方法用于求解约束变分不等式问题。受交替方向乘子法在单目标问题中有效性的启发，我们将ADMM推广到约束变分不等式的一阶求解方法，称为基于ADMM的约束变分不等式内点法（ACVI）。我们为ACVI在两类一般性问题中提供了收敛性保证：（i）当算子为$\xi$-单调时；（ii）当算子为单调、部分约束活跃且博弈不完全是旋转型时。对于后一种情况，若算子进一步满足L-Lipschitz条件，我们匹配了已知的间隙函数收敛速率下界，即最后迭代点的$\mathcal{O}(1/\sqrt{K})$和平均迭代点的$\mathcal{O}(1/K)$。据我们所知，这是首个具有全局收敛保证的通用约束变分不等式一阶内点法。此外，与先前在该领域的研究不同，ACVI提供了一种在约束非平凡时求解约束变分不等式的方法。实验分析表明ACVI相较于常见一阶方法具有明显优势。具体而言：（i）当我们的方法从解析中心逼近解时，循环行为显著减少；（ii）与靠近约束时会出现锯齿现象的投影方法不同，ACVI能高效处理约束。