Asymptotic behaviour of coupled random dynamical systems with multiscale aspects
D. Russell Luke, Johannes-Carl Schnebel, Mathias Staudigl, Juan Peypouquet, Siqi Qu
TL;DR
This work develops a penalized, multiscale stochastic framework for solving generalized variational inequalities under convex constraints by embedding an exterior penalty with a time-varying parameter $\beta(t)$ into a non-autonomous stochastic differential inclusion $dX(t) + \mathsf{A}(X(t))dt + \beta(t)\nabla\Psi(X(t))dt \ni \sigma(t,X(t))dW(t)$. It establishes strong solution existence, almost-sure and ergodic convergence to the solution set $\mathcal{S}=\mathrm{zer}(\mathsf{A}+\mathrm{N}_C)$, and, in the subgradient case $\mathsf{A}=\partial\Phi$, convergence of $\Phi(X(t))$ to $\min_C\Phi$ and of $\Psi(X(t))$ to zero, with finite-time rates expressed via gap functions. The paper also derives concentration bounds for the gap, connects the continuous-time dynamics to a discrete-time forward-backward scheme, and validates the approach through numerical experiments on basis pursuit, linear inverse problems, and CT imaging. Additional discussions cover invariant measures in stationary settings and potential stochastic-control extensions. Overall, the results provide a rigorous stochastic-dynamical perspective on penalty-regularized, multiscale variational inequalities with practical implications for bilevel optimization and constrained problems.
Abstract
We examine a class of stochastic differential inclusions involving multiscale effects designed to solve a class of generalized variational inequalities. This class of problems contains constrained convex non-smooth optimization problems, constrained saddle-point problems and various equilibrium problems in economics and engineering. In order to respect constraints we adopt a penalty approach, introducing an explicit time-dependency into the evolution system. The resulting dynamics are described in terms of a non-autonomous stochastic evolution equation governed by maximally monotone operators in the drift and perturbed by a Brownian motion. We study the asymptotic behavior, as well as finite time convergence rates in terms of gap functions. The condition we use to prove convergence involves a Legendre transform of the function describing the set C, a condition first used by Attouch and Czarnecki (J. Differ. Equations, Vol. 248, Issue 6, 2010) in the context of deterministic evolution equations. We also establish a large deviations principle showing that individual trajectories exhibit exponential concentration around the solution set. Finally we show how our continuous-time approach relates to penalty-regulated algorithms of forward-backward type after performing a suitable Euler-Maruyama discretisation.