On Asymptotic Analysis of Perturbed Sweeping Processes with Application to Optimization

Zhaoyue Xia, Jun Du, Chunxiao Jiang, H. Vincent Poor, Yong Ren

Published 2024-07-26Version 1

Convergence analysis of constrained optimization methods from the dynamical systems viewpoint has attracted considerable attention because it provides a geometric demonstration towards the shadowing trajectory of a numerical scheme. In this work, we establish a tight connection between a continuous-time nonsmooth dynamical system called a perturbed sweeping process (PSP) and a proximal stochastic approximation scheme. Theoretical results are obtained by analyzing the asymptotic pseudo trajectory of a PSP. We show that under mild assumptions a proximal stochastic approximation scheme converges to an internally chain transitive invariant set of the corresponding PSP. Furthermore, given the existence of a Lyapunov function $V$ with respect to a set $\Lambda$, convergence to $\Lambda$ can be established if $V(\Lambda)$ has an empty interior. Based on these theoretical results, we are able to provide a useful framework for convergence analysis of proximal gradient methods. Illustrative examples are provided to determine the convergence of proximal variants of gradient methods (including accelerated gradient methods). Finally, numerical simulations are conducted to confirm the validity of theoretical analysis.