A Nonsmooth Dynamical Systems Perspective on Accelerated Extensions of ADMM
Abstract
Recently, there has been great interest in connections between continuous-time dynamical systems and optimization methods, notably in the context of accelerated methods for smooth and unconstrained problems. In this paper we extend this perspective to nonsmooth and constrained problems by obtaining differential inclusions associated to novel accelerated variants of the alternating direction method of multipliers (ADMM). Through a Lyapunov analysis, we derive rates of convergence for these dynamical systems in different settings that illustrate an interesting tradeoff between decaying versus constant damping strategies. We also obtain modified equations capturing fine-grained details of these methods, which have improved stability and preserve the leading order convergence rates. An extension to general nonlinear equality and inequality constraints in connection with singular perturbation theory is provided.
Keywords
Cite
@article{arxiv.1808.04048,
title = {A Nonsmooth Dynamical Systems Perspective on Accelerated Extensions of ADMM},
author = {Guilherme França and Daniel P. Robinson and René Vidal},
journal= {arXiv preprint arXiv:1808.04048},
year = {2023}
}
Comments
Last version was completely rewritten. New results for modified/perturbed equations, constraints, and singular perturbation theory. Matches the version to appear on IEEE Transactions on Automatic Control