Data-Driven Adjustable Robust Optimization: Decision Making Under Uncertainty

Abstract

Optimization problems in the real-world often deal with uncertain data, such as uncertainty in the demand of a product or the price of a stock. The field of robust optimization (RO) explores decision-making under uncertainty. Despite its flexibility and computational advantages, traditional RO approaches often lead to overly conservative solutions due to assumptions made in defining uncertainty sets. Recent advancements, such as mean robust optimization (MRO) and learning for robust optimization (LRO), have leveraged data-driven techniques to improve performance. However, these methods have primarily focused on "here-and-now" decisions, neglecting the dynamic and multi-stage nature of many real-world problems. In this work, we extend LRO to multi-stage problems in the field of adjustable robust optimization (ARO), which allows decision-makers to adapt decisions over time. We propose a framework that integrates contextual optimization techniques to capture time-varying uncertainties. Specifically, on top of the stochastic augmented Lagrangian method used in LRO, we update the size and shape of our uncertainty sets by training parameters of a simple neural network, with time as an input, and condition our probabilistic guarantees of constraint satisfaction on time. Simulated examples, including a multi-stage product-inventory problem, show that our method outperforms traditional RO approaches. We discuss implementation details of our methodology and modifications to the existing open-source Python package LROPT. This work contributes to the advancement of data-driven robust optimization techniques, particularly in addressing the challenges of dynamic decision-making under uncertainty.