Empirical risk minimization for risk-neutral composite optimal control with applications to bang-bang control
<|reference_start|>Empirical risk minimization for risk-neutral composite optimal control with applications to bang-bang control: 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.<|reference_end|>
@article{milz2024empirical,
    title={Empirical risk minimization for risk-neutral composite optimal control
  with applications to bang-bang control},
    author={Johannes Milz and Daniel Walter},
    journal={arXiv preprint arXiv:2408.10384},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.10384},
    primaryClass={math.OC cs.NA math.NA}
}