Solving Constrained Variational Inequalities via a First-order Interior Point-based Method

We focus on the open problem to develop a first-order method that can solve constrained Variational Inequality (cVI) problems when given general constraints. We generalize the \textit{alternating direction method of multipliers} (ADMM) method and combine it with interior-point approaches, yielding a first-order method that we refer to as ADMM-based interior-point method for cVIs (ACVI). We provide convergence guarantees for ACVI in two general classes of problems: (i) when the operator is $\xi$-monotone, and (i) 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 our knowledge, this is the first \emph{first}-order interior-point method for the general cVI problem that has a global convergence guarantee. Empirical analyses demonstrate clear advantages of ACVI over common first-order methods. In particular, (i) cyclical behavior is notably reduced as our method approaches the solution from the analytic center, and (ii) unlike projection-based methods that zigzag when near a constraint, ACVI efficiently handles the constraints.