Dynamic Systems Coupled with Solutions of Stochastic Nonsmooth Convex Optimization

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.