Dynamic Systems Coupled with Solutions of Stochastic Nonsmooth Convex Optimization

In this paper, we study ordinary differential equations (ODEs) 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 that the optimal solution of the regularized SNCOP converges to the least-norm solution of the original SNCOP, which are crucial for deriving the convergence results of the discrete problems. We illustrate the theoretical results and applications for the estimation of the time-varying parameters in ODEs by numerical examples.
AB - Abstract.In this paper, we study ordinary differential equations (ODEs) 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 that the optimal solution of the regularized SNCOP converges to the least-norm solution of the original SNCOP, which are crucial for deriving the convergence results of the discrete problems. We illustrate the theoretical results and applications for the estimation of the time-varying parameters in ODEs by numerical examples.