Distributionally robust optimization under ambiguity across two layers

Abstract

With general interest in distributionally robust optimization under multilayered ambiguity, we study a stylized model based on the inner product w · X · Y. Here, w is a deterministic, nonnegative d dimensional vector representing a strategy, while X and Y are d dimensional real-valued random vectors representing losses and economic factors, respectively, both subject to ambiguity. We first treat the ambiguity associated with X and Y separately, and then jointly, with each ambiguity set characterized by a Wasserstein ball. In both settings, we derive explicit expressions for the worst-case expectation of w · X · Y. Numerical studies further illustrate how to determine the radii of the Wasserstein balls needed to achieve a prespecified coverage probability.