# Data-driven Multistage Distributionally Robust Linear Optimization with Nested Distance

arXiv: https://arxiv.org/abs/2407.16346 · HTML: https://mlsanity.com/paper/2407.16346/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Jul 2024.

## Built on by (most cited first, 1 of 1)

- [Scalable Bi-causal Optimal Transport via KL Relaxation and Policy Gradients](https://mlsanity.com/paper/2605.17271/) (2026)

## Abstract (arXiv)

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.