Learning-based robust optimization: Procedures and statistical guarantees

Robust optimization (RO) is a common approach to tractably obtain safeguarding solutions for optimization problems with uncertain constraints. In this paper, we study a statistical framework to integrate data into RO based on learning a prediction set using (combinations of) geometric shapes that are compatible with established RO tools and on a simple data-splitting validation step that achieves finite-sample nonparametric statistical guarantees on feasibility. We demonstrate how our required sample size to achieve feasibility at a given confidence level is independent of the dimensions of both the decision space and the probability space governing the stochasticity, and we discuss some approaches to improve the objective performances while maintaining these dimension-free statistical feasibility guarantees.