  This paper proposes a second-order conic programming (SOCP) approach to solve
distributionally robust two-stage stochastic linear programs over 1-Wasserstein
balls. We start from the case with distribution uncertainty only in the
objective function and exactly reformulate it as an SOCP problem. Then, we
study the case with distribution uncertainty only in constraints, and show that
such a robust program is generally NP-hard as it involves a norm maximization
problem over a polyhedron. However, it is reduced to an SOCP problem if the
extreme points of the polyhedron are given as a prior. This motivates to design
a constraint generation algorithm with provable convergence to approximately
solve the NP-hard problem. In sharp contrast to the exiting literature, the
distribution achieving the worst-case cost is given as an "empirical"
distribution by simply perturbing each sample for both cases. Finally,
experiments illustrate the advantages of the proposed model in terms of the
out-of-sample performance and the computational complexity.
