  In the optimization under uncertainty, decision-makers first select a
wait-and-see policy before any realization of uncertainty and then place a
here-and-now decision after the uncertainty has been observed. Two-stage
stochastic programming is a popular modeling paradigm for the optimization
under uncertainty that the decision-makers first specifies a probability
distribution, and then seek the best decisions to jointly optimize the
deterministic wait-and-see and expected here-and-now costs. In practice, such a
probability distribution may not be fully available but is probably observable
through an empirical dataset. Therefore, this paper studies distributionally
robust two-stage stochastic program (DRTSP) which jointly optimizes the
deterministic wait-and-see and worst-case expected here-and-now costs, and the
probability distribution comes from a family of distributions which are
centered at the empirical distribution using $\infty-$Wasserstein metric. There
have been successful developments on deriving tractable approximations of the
worst-case expected here-and-now cost in DRTSP. Unfortunately, limited results
on exact tractable reformulations of DRTSP. This paper fills this gap by
providing sufficient conditions under which the worst-case expected
here-and-now cost in DRTSP can be efficiently computed via a tractable convex
program. By exploring the properties of binary variables, the developed
reformulation techniques are extended to DRTSP with binary random parameters.
The main tractable reformulations in this paper are projected into the original
decision space and thus can be interpreted as conventional two-stage stochastic
programs under discrete support with extra penalty terms enforcing the
robustness. These tractable results are further demonstrated to be sharp
through complexity analysis.
