Data-driven Multistage Distributionally Robust Linear Optimization with Nested Distance

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.