  Distributionally robust stochastic optimization (DRSO) is a framework for
decision-making problems under certainty, which finds solutions that perform
well for a chosen set of probability distributions. Many different approaches
for specifying a set of distributions have been proposed. The choice matters,
because it affects the results, and the relative performance of different
choices depend on the characteristics of the problems. In this paper, we
consider problems in which different random variables exhibit some form of
dependence, but the exact values of the parameters that represent the
dependence are not known. We consider various sets of distributions that
incorporate the dependence structure, and we study the corresponding DRSO
problems.
  In the first part of the paper, we consider problems with linear dependence
between random variables. We consider sets of distributions that are within a
specified Wasserstein distance of a nominal distribution, and that satisfy a
second-order moment constraint. We obtain a tractable dual reformulation of the
corresponding DRSO problem. This approach is compared with the traditional
moment-based DRSO and Wasserstein-based DRSO with no moment constraints.
Numerical experiments suggest that our new formulation has superior
out-of-sample performance.
  In the second part of the paper, we consider problems with various types of
rank dependence between random variables, including rank dependence measured by
Spearman's footrule distance between empirical rankings, comonotonic
distributions, box uncertainty for individual observations, and Wasserstein
distance between copulas associated with continuous distributions. We also
obtain a dual reformulation of the DRSO problem. A desirable byproduct of the
formulation is that it also avoids an issue associated with the one-sided
moment constraints in moment-based DRSO problems.
