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

Parametric Scenario Optimization under Limited Data: A Distributionally Robust Optimization View