Data-driven Multistage Distributionally Robust Linear Optimization with Nested Distance
<|reference_start|>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.<|reference_end|>
@article{gao2024data-driven,
    title={Data-driven Multistage Distributionally Robust Linear Optimization with
  Nested Distance},
    author={Rui Gao, Rohit Arora, Yizhe Huang},
    journal={arXiv preprint arXiv:2407.16346},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.16346},
    primaryClass={math.OC cs.LG math.PR stat.ML}
}