Dimension Reduction of Distributionally Robust Optimization Problems

Abstract: 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 under which the images of these sets under scalar-valued aggregation functions are equal to or contained in uncertainty sets of univariate random variables defined via a univariate Wasserstein distance. This allows to rewrite or bound high-dimensional DRO problems with simpler DRO problems over the space of univariate random variables. We generalize the results to uncertainty sets defined via the Bregman-Wasserstein divergence and the max-sliced Wasserstein and Bregman-Wasserstein divergence. The max-sliced divergences allow us to jointly model distributional uncertainty around the reference random vector and uncertainty in the aggregation function. Finally, we derive explicit bounds for worst-case risk measures that belong to the class of signed Choquet integrals.