Bounds for Distributionally Robust Optimization Problems

We study distributionally robust optimization (DRO) problems with uncertainty sets consisting of high-dimensional random vectors that are close in the multivariate Wasserstein distance to a reference random vector. We give conditions when the images of these sets under scalar-valued aggregation functions are contained in and contain uncertainty sets of univariate random variables defined via a univariate Wasserstein distance. This provides lower and upper bounds for the solution to general multivariate DRO problems that are computationally tractable. Furthermore, we generalize the results to uncertainty sets characterized by Bregman-Wasserstein divergences, which allows for asymmetric deviations from the reference random vector. Moreover, for DRO problems with risk measure criterion in the class of signed Choquet integrals, we derive semi-analytic formulae for the upper and lower bounds and the distribution that attains these bounds.