  Adaptive robust optimization problems are usually solved approximately by
restricting the adaptive decisions to simple parametric decision rules.
However, the corresponding approximation error can be substantial. In this
paper we show that two-stage robust and distributionally robust linear programs
can often be reformulated exactly as conic programs that scale polynomially
with the problem dimensions. Specifically, when the ambiguity set constitutes a
2-Wasserstein ball centered at a discrete distribution, then the
distributionally robust linear program is equivalent to a copositive program
(if the problem has complete recourse) or can be approximated arbitrarily
closely by a sequence of copositive programs (if the problem has sufficiently
expensive recourse). These results directly extend to the classical robust
setting and motivate strong tractable approximations of two-stage problems
based on semidefinite approximations of the copositive cone. We also
demonstrate that the two-stage distributionally robust optimization problem is
equivalent to a tractable linear program when the ambiguity set constitutes a
1-Wasserstein ball centered at a discrete distribution and there are no support
constraints.
