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

2024/08/19 by Milz, Johannes, Walter, Daniel · 1 citation
#35Q93 #49M41 #90C06 #90C15 #90C59 #90C60 #FOS: Mathematics #Numerical Analysis (math.NA) #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2408.10384

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.

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