Wasserstein distributionally robust optimization and its tractable regularization formulations

We study a variety of Wasserstein distributionally robust optimization (WDRO) problems where the distributions in the ambiguity set are chosen by constraining their Wasserstein discrepancies to the empirical distribution. Using the notion of weak Lipschitz property, we derive lower and upper bounds of the corresponding worst-case loss quantity and propose sufficient conditions under which this quantity coincides with its regularization scheme counterpart. Our constructive methodology and elementary analysis also directly characterize the closed-form of the approximate worst-case distribution. Extensive applications show that our theoretical results are applicable to various problems, including regression, classification and risk measure problems.