Data-Driven Distributionally Robust Optimization over Time

Stochastic optimization (SO) is a classical approach for optimization under uncertainty that typically requires knowledge about the probability distribution of uncertain parameters. Because the latter is often unknown, distributionally robust optimization (DRO) provides a strong alternative that determines the best guaranteed solution over a set of distributions (ambiguity set). In this work, we present an approach for DRO over time that uses online learning and scenario observations arriving as a data stream to learn more about the uncertainty. Our robust solutions adapt over time and reduce the cost of protection with shrinking ambiguity. For various kinds of ambiguity sets, the robust solutions converge to the SO solution. Our algorithm achieves the optimization and learning goals without solving the DRO problem exactly at any step. We also provide a regret bound for the quality of the online strategy that converges at a rate of [Formula: see text], where T is the number of iterations. Furthermore, we illustrate the effectiveness of our procedure by numerical experiments on mixed-integer optimization instances from popular benchmark libraries and give practical examples stemming from telecommunications and routing. Our algorithm is able to solve the DRO over time problem significantly faster than standard reformulations. Funding: This work was supported by Deutsche Forschungsgemeinschaft (DFG): Projects B06 and B10 in CRC TRR 154 and Project-ID 416229255 - SFB 1411 and Federal Ministry for Economic Affairs and Energy, Germany [Grant 03EI1036A]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/ijoo.2023.0091 .