  We study optimal transport-based distributionally robust optimization
problems where a fictitious adversary, often envisioned as nature, can choose
the distribution of the uncertain problem parameters by reshaping a prescribed
reference distribution at a finite transportation cost. In this framework, we
show that robustification is intimately related to various forms of variation
and Lipschitz regularization even if the transportation cost function fails to
be (some power of) a metric. We also derive conditions for the existence and
the computability of a Nash equilibrium between the decision-maker and nature,
and we demonstrate numerically that nature's Nash strategy can be viewed as a
distribution that is supported on remarkably deceptive adversarial samples.
Finally, we identify practically relevant classes of optimal transport-based
distributionally robust optimization problems that can be addressed with
efficient gradient descent algorithms even if the loss function or the
transportation cost function are nonconvex (but not both at the same time).
