  Distributionally robust stochastic optimization (DRSO) is an approach to
optimization under uncertainty in which, instead of assuming that there is a
known true underlying probability distribution, one hedges against a chosen set
of distributions. In this paper we first point out that the set of
distributions should be chosen to be appropriate for the application at hand,
and that some of the choices that have been popular until recently are, for
many applications, not good choices. We next consider sets of distributions
that are within a chosen Wasserstein distance from a nominal distribution. Such
a choice of sets has two advantages: (1) The resulting distributions hedged
against are more reasonable than those resulting from other popular choices of
sets. (2) The problem of determining the worst-case expectation over the
resulting set of distributions has desirable tractability properties. We derive
a strong duality reformulation of the corresponding DRSO problem and construct
approximate worst-case distributions explicitly via the first-order optimality
conditions of the dual problem.
  Our contributions are four-fold. (i) We identify necessary and sufficient
conditions for the existence of a worst-case distribution, which are naturally
related to the growth rate of the objective function. (ii) We show that the
worst-case distributions resulting from an appropriate Wasserstein distance
have a concise structure and a clear interpretation. (iii) Using this
structure, we show that data-driven DRSO problems can be approximated to any
accuracy by robust optimization problems, and thereby many DRSO problems become
tractable by using tools from robust optimization. (iv) Our strong duality
result holds in a very general setting. As examples, we show that it can be
applied to infinite-dimensional process control and intensity estimation for
point processes.
