Learning Decision-Focused Uncertainty Sets in Robust Optimization

TL;DR

This paper introduces a flexible, data-driven method for learning uncertainty sets in robust optimization, improving worst-case and average-case performance while ensuring constraint satisfaction through a novel constrained learning approach.

Contribution

It presents a new technique that automatically learns uncertainty sets by minimizing expected performance with probabilistic guarantees, using a stochastic augmented Lagrangian method.

Findings

Outperforms traditional robust optimization methods in out-of-sample tests.

Guarantees constraint satisfaction with finite-sample probabilistic bounds.

Demonstrates flexibility in learning various uncertainty set shapes.

Abstract

We propose a data-driven technique to automatically learn contextual uncertainty sets in robust optimization, resulting in excellent worst-case and average-case performance while also guaranteeing constraint satisfaction. Our method reshapes the uncertainty sets by minimizing the expected performance across a contextual family of problems, subject to conditional-value-at-risk constraints. Our approach is very flexible, and can learn a wide variety of uncertainty sets while preserving tractability. We solve the constrained learning problem using a stochastic augmented Lagrangian method that relies on differentiating the solutions of the robust optimization problems with respect to the parameters of the uncertainty set. Due to the nonsmooth and nonconvex nature of the augmented Lagrangian function, we apply the nonsmooth conservative implicit function theorem to establish convergence to a critical point, which is a feasible solution of the constrained problem under mild assumptions. Using empirical process theory, we show finite-sample probabilistic guarantees of constraint satisfaction for the resulting solutions. Numerical experiments show that our method outperforms traditional approaches in robust and distributionally robust optimization in terms of out-of-sample performance and constraint satisfaction guarantees.