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.

翻译：非光滑复合优化问题在不确定性条件下广泛存在于科学与工程应用中。本文研究风险中性复合最优控制问题，其目标函数由可能非凸的期望函数与非光滑凸函数之和构成。为逼近风险中性优化问题，我们采用基于蒙特卡洛采样的方法，分析其渐近一致性，并推导非渐近样本量估计。我们的分析利用了偏微分方程约束优化问题中常见的结构特征，包括紧嵌入性与增长条件。我们将所得结论应用于Bang-Bang型最优控制问题，并提出采用条件梯度法进行高效求解。文中给出了数值算例。