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

Published: 01 Feb 2023, 19:21, Last Modified: 02 Mar 2023, 03:30ICLR 2023 notable top 25%Readers: Everyone
Keywords: constrained variational inequality, interior point, admm
TL;DR: We derive a first-order method for solving constrained variational inequality problem when given general constraints, by combining interior-point methods and ADMM.
Abstract: 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.
Anonymous Url: I certify that there is no URL (e.g., github page) that could be used to find authors’ identity.
No Acknowledgement Section: I certify that there is no acknowledgement section in this submission for double blind review.
Code Of Ethics: I acknowledge that I and all co-authors of this work have read and commit to adhering to the ICLR Code of Ethics
Submission Guidelines: Yes
Please Choose The Closest Area That Your Submission Falls Into: Optimization (eg, convex and non-convex optimization)
12 Replies