  We address high-dimensional zero-one random parameters in two-stage convex
conic optimization problems. Such parameters typically represent failures of
network elements and constitute rare, high-impact random events in several
applications. Given a sparse training dataset of the parameters, we motivate
and study a distributionally robust formulation of the problem using a
Wasserstein ambiguity set centered at the empirical distribution. We present a
simple, tractable, and conservative approximation of this problem that can be
efficiently computed and iteratively improved. Our method relies on a
reformulation that optimizes over the convex hull of a mixed-integer conic
programming representable set, followed by an approximation of this convex hull
using lift-and-project techniques. We illustrate the practical viability and
strong out-of-sample performance of our method on nonlinear optimal power flow
and multi-commodity network design problems that are affected by random
contingencies, and we report improvements of up to 20\% over existing sample
average approximation and two-stage robust optimization methods.
