  We consider optimization problems with uncertain constraints that need to be
satisfied probabilistically. When data are available, a common method to obtain
feasible solutions for such problems is to impose sampled constraints,
following the so-called scenario optimization approach. However, when the data
size is small, the sampled constraints may not statistically support a
feasibility guarantee on the obtained solution. This paper studies how to
leverage parametric information and the power of Monte Carlo simulation to
obtain feasible solutions for small-data situations. Our approach makes use of
a distributionally robust optimization (DRO) formulation that translates the
data size requirement into a Monte Carlo sample size requirement drawn from
what we call a generating distribution. We show that, while the optimal choice
of this generating distribution is the one eliciting the data or the baseline
distribution in a nonparametric divergence-based DRO, it is not necessarily so
in the parametric case. Correspondingly, we develop procedures to obtain
generating distributions that improve upon these basic choices. We support our
findings with several numerical examples.
