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：我们开发了一种内点方法来解决限制的变差不平等问题。在单一目标背景下,由于乘数交替方向方法(ADMM)方法(ADMM(ADMM)方法(ADMM)方法(ADMM)方法(ADMM)方法(ADVI)方法(ADVI)方法(ADMM)方法(ADVI)方法(CVI),我们称之为ADMM基于ADM的内点方法(ADVI)方法(CVI)方法(CVI),我们为AC在两种一般性问题中分别提供了趋同保证:(i)当操作者为美元和平均偏差时,以及(ii)当操作者为单调时,一些限制是活动,游戏并非纯粹循环。此外,当操作者(L-LIPS)方法(L-LIS)方法(L-LIF)方法(L-LI(L-LI)方法(LI)方法(AV)方法(UDR)方法(AV)方法(Arental 方法(AV)方法(AV) 方法(Arentrentral) 方法(I) 方法(LU) 方法(Aral 方法(I) 方法(I) 方法(I) 方法(Arentrent) 方法(I) 方法(I) 方法(I) 方法(I) 方法(L) 方法(LU) 方法(I) ) 方法(I) ) 方法(Arent) 方法(I) 方法(I) 方法(LU) 方法(LU) ) 方法(A) )与方法(LU)相比, ) ) ) 方法(A 方法(I) 方法(I) 方法(I) 方法(I) 方法(LU) 方法(IV) 方法(I) 方法(I) 方法(I) 方法(I) 方法(I) 方法(A) 方法(I) ) 方法(I) ) ) 方法(A) 方法(I) 方法(I) 方法(I) 方法(</s>
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