Data-driven distributionally robust optimization with Wasserstein metric, moment conditions and robust constraints

Abstract

We consider optimization problems where the information on the uncertain parameters reduces to a finite data sample. Using the Wasserstein metric, a ball in the space of probability distributions centered at the empirical distribution is constructed. The goal is to solve a minimization problem subject to the worst-case distribution within this Wasserstein ball. Moreover, we consider moment constraints in order to add a priori information about the random phenomena. In addition, we not only consider moment constraints but also take into account robust classical constraints. These constraints serve to hedge decisions against realizations of random variables for which we do not have distributional information other than their support set. With these assumptions we need to solve a data-driven distributionally robust optimization problem with several types of constraints. We show that strong duality holds under mild assumptions, and the distributionally robust optimization problems overWasserstein balls with moment constraints and robust classical constraints can in fact be reformulated as tractable finite programs. Finally, a taxonomy of the tractable finite programs is shown under di erent assumptions about the objective function, the constraints and the support set of the random variables.