Data-driven Multistage Distributionally Robust Linear Optimization with Nested Distance

TL;DR

This paper develops recursive and dynamic programming methods for multistage distributionally robust linear optimization using nested distance, enabling efficient computation of robust policies under uncertainty.

Contribution

It introduces recursive reformulations and tractable cases for multistage distributionally robust optimization with nested distance, bridging static and dynamic modeling frameworks.

Findings

Robust risk evaluation can be expressed recursively.

Dynamic programming reformulations are derived under stagewise independence.

Identifies cases where value functions are efficiently computable.

Abstract

We study multistage distributionally robust linear optimization, where the uncertainty set is defined as a ball of distribution centered at a scenario tree using the nested distance. The resulting minimax problem is notoriously difficult to solve due to its inherent non-convexity. In this paper, we demonstrate that, under mild conditions, the robust risk evaluation of a given policy can be expressed in an equivalent recursive form. Furthermore, assuming stagewise independence, we derive equivalent dynamic programming reformulations to find an optimal robust policy that is time-consistent and well-defined on unseen sample paths. Our reformulations reconcile two modeling frameworks: the multistage-static formulation (with nested distance) and the multistage-dynamic formulation (with one-period Wasserstein distance). Moreover, we identify tractable cases when the value functions can be computed efficiently using convex optimization techniques.