Dynamic Systems Coupled with Solutions of Stochastic Nonsmooth Convex Optimization
Jianfeng Luo, Xiaojun Chen
TL;DR
This work studies ODEs dynamically coupled with a stochastic nonsmooth convex optimization problem, formulating the system as $\dot{x}(t)=\mathbb{E}[f(t,x,y,\xi)]$ with $y(t)$ solving a convex program under a convex constraint $K(t,x(t))$. It develops a comprehensive convergence framework built on a regularization approach, the sample average approximation (SAA), and an implicit time-stepping scheme, proving existence of solutions and convergence (in probability) of discrete approximations to the original problem. Key results include a linear-growth bound for the optimal control, convergence of the regularized problem to the least-norm solution, and rigorous convergence of SAA and time-stepping schemes to weak (or classic) solutions of the coupled system. Theoretical findings are complemented by a numerical example demonstrating parameter-estimation in time-varying ODEs, with empirical evidence that increasing sample size and decreasing regularization yield improved accuracy.
Abstract
In this paper, we study ordinary differential equations (ODE) coupled with solutions of a stochastic nonsmooth convex optimization problem (SNCOP). We use the regularization approach, the sample average approximation and the time-stepping method to construct discrete approximation problems. We show the existence of solutions to the original problem and the discrete problems. Moreover, we show that the optimal solution of the SNCOP with a strong convex objective function admits a linear growth condition and the optimal solution of the regularized SNCOP converges to the least-norm solution of the original SNCOP, which are crucial for us to derive the convergence results of the discrete problems. We illustrate the theoretical results and applications for the estimation of the time-varying parameters in ODE by numerical examples.