  Distributionally robust chance-constrained programs (DR-CCP) over Wasserstein
ambiguity sets exhibit attractive out-of-sample performance and admit
big-$M$-based mixed-integer programming (MIP) reformulations with conic
constraints. However, the resulting formulations often suffer from scalability
issues as sample size increases. To address this shortcoming, we derive
stronger formulations that scale well with respect to the sample size. Our
focus is on ambiguity sets under the so-called left-hand side (LHS)
uncertainty, where the uncertain parameters affect the coefficients of the
decision variables in the linear inequalities defining the safety sets. The
interaction between the uncertain parameters and the variable coefficients in
the safety set definition causes challenges in strengthening the original
big-$M$ formulations. By exploiting the connection between nominal
chance-constrained programs and DR-CCP, we obtain strong formulations with
significant enhancements. In particular, through this connection, we derive a
linear number of valid inequalities, which can be immediately added to the
formulations to obtain improved formulations in the original space of
variables. In addition, we suggest a quantile-based strengthening procedure
that allows us to reduce the big-$M$ coefficients drastically. Furthermore,
based on this procedure, we propose an exponential class of inequalities that
can be separated efficiently within a branch-and-cut framework. The
quantile-based strengthening procedure can be expensive. Therefore, for the
special case of covering and packing type problems, we identify an efficient
scheme to carry out this procedure. We demonstrate the computational efficacy
of our proposed formulations on two classes of problems, namely stochastic
portfolio optimization and resource planning.
