  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.
