Dynamic Systems Coupled with Solutions of Stochastic Nonsmooth Convex Optimization

TL;DR

This paper investigates the coupling of ordinary differential equations with solutions to stochastic nonsmooth convex optimization problems, establishing existence, convergence, and practical applications through theoretical and numerical analysis.

Contribution

It introduces a regularization and approximation framework for coupled ODE and SNCOP problems, proving solution existence and convergence to least-norm solutions.

Findings

Existence of solutions to coupled ODE and SNCOP problems.

Convergence of regularized solutions to least-norm solutions.

Numerical validation in parameter estimation for ODEs.

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.