Effective Scenarios in Multistage Distributionally Robust Optimization with a Focus on Total Variation Distance

TL;DR

This paper extends the concept of critical scenarios from static to multistage distributionally robust optimization, focusing on total variation distance to identify impactful scenario paths and improve uncertainty modeling.

Contribution

It introduces the notions of effectiveness for scenario paths and realizations in multistage DRO, providing practical conditions for their identification under total variation ambiguity.

Findings

Effectiveness notions help identify impactful scenarios.

Proposed conditions are easy to check in practice.

Numerical results demonstrate practical insights into uncertainty.

Abstract

We study multistage distributionally robust optimization (DRO) to hedge against ambiguity in quantifying the underlying uncertainty of a problem. Recognizing that not all the realizations and scenario paths might have an "effect" on the optimal value, we investigate the question of how to define and identify critical scenarios for nested multistage DRO problems. Our analysis extends the work of Rahimian, Bayraksan, and Homem-de-Mello [Math. Program. 173(1--2): 393--430, 2019], which was in the context of a static/two-stage setting, to the multistage setting. To this end, we define the notions of effectiveness of scenario paths and the conditional effectiveness of realizations along a scenario path for a general class of multistage DRO problems. We then propose easy-to-check conditions to identify the effectiveness of scenario paths in the multistage setting when the distributional ambiguity is modeled via the total variation distance. Numerical results show that these notions provide useful insight on the underlying uncertainty of the problem.