Empirical risk minimization for risk-neutral composite optimal control with applications to bang-bang control

TL;DR

This paper develops a Monte Carlo-based method for risk-neutral composite optimal control problems, focusing on nonsmooth, potentially nonconvex objectives, with applications to bang-bang control and theoretical guarantees.

Contribution

It introduces a sample-based approximation approach with consistency and sample size estimates for nonsmooth, nonconvex risk-neutral control problems, and applies a conditional gradient method for bang-bang controls.

Findings

The method achieves asymptotic consistency.

Sample size estimates are derived for practical implementation.

Numerical results demonstrate effectiveness in bang-bang control scenarios.

Abstract

Nonsmooth composite optimization problems under uncertainty are prevalent in various scientific and engineering applications. We consider risk-neutral composite optimal control problems, where the objective function is the sum of a potentially nonconvex expectation function and a nonsmooth convex function. To approximate the risk-neutral optimization problems, we use a Monte Carlo sample-based approach, study its asymptotic consistency, and derive nonasymptotic sample size estimates. Our analyses leverage problem structure commonly encountered in PDE-constrained optimization problems, including compact embeddings and growth conditions. We apply our findings to bang-bang-type optimal control problems and propose the use of a conditional gradient method to solve them effectively. We present numerical illustrations.