This paper studies distributional robust optimization problems based on optimal transport, where a hypothetical adversary (nature) can choose a distribution of uncertain problem parameters by modifying a predetermined reference distribution under finite transport costs. Within this framework, the authors show that robustness is closely related to various forms of variation and Lipschitz regularization, even when the transport cost function is not a (partially squared) metric. Furthermore, we derive conditions for the existence and computability of a Nash equilibrium between a decision maker and nature, and numerically demonstrate that nature's Nash strategy can be considered a distribution that supports surprisingly deceptive adversarial samples. Finally, we identify a class of practically relevant optimal transport-based distributional robust optimization problems that can be solved by efficient gradient descent algorithms even when either the loss function or the transport cost function is nonconvex (except when both are nonconvex).
