This paper proposes a novel approach to construct data-driven online solutions to optimization problems (P) subject to a class of distributionally uncertain dynamical systems. The introduced framework allows for the simultaneous learning of distributional system uncertainty via a parameterized, control-dependent ambiguity set using a finite historical data set, and its use to make online decisions with probabilistic regret function bounds. Leveraging the merits of Machine Learning, the main technical approach relies on the theory of Distributional Robust Optimization (DRO), to hedge against uncertainty and provide less conservative results than standard Robust Optimization approaches. Starting from recent results that describe ambiguity sets via parameterized, and control-dependent empirical distributions as well as ambiguity radii, we first present a tractable reformulation of the corresponding optimization problem while maintaining the probabilistic guarantees. We then specialize these problems to the cases of 1) optimal one-stage control of distributionally uncertain nonlinear systems, and 2) resource allocation under distributional uncertainty. A novelty of this work is that it extends DRO to online optimization problems subject to a distributionally uncertain dynamical system constraint, handled via a control-dependent ambiguity set that leads to online-tractable optimization with probabilistic guarantees on regret bounds. Further, we introduce an online version of Nesterov's accelerated-gradient algorithm, and analyze its performance to solve this class of problems via dissipativity theory.

翻译：本文提出了一种新颖方法，用于构建受一类分布不确定动态系统约束的优化问题(P)的数据驱动在线解。该框架能够通过有限历史数据集，利用参数化且与控制相关的模糊集同时学习系统分布不确定性，并据此做出具有概率性遗憾函数界的在线决策。借助机器学习的优势，主要技术方法基于分布鲁棒优化理论，以对冲不确定性并提供比标准鲁棒优化方法更少保守性的结果。基于近期通过参数化且与控制相关的经验分布及模糊半径描述模糊集的研究成果，我们首先给出了相应优化问题的可处理重构形式，同时保持概率性保证。随后我们将这些问题具体应用于：1）分布不确定非线性系统的最优单级控制，以及2）分布不确定性下的资源分配。本工作的创新点在于将分布鲁棒优化扩展至受分布不确定动态系统约束的在线优化问题，通过采用与控制相关的模糊集进行处理，实现了具有遗憾界概率保证的在线可处理优化。此外，我们引入了涅斯捷罗夫加速梯度算法的在线版本，并通过耗散性理论分析了其解决此类问题的性能。