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

## TL;DR

This paper proposes a distributionally robust optimization framework that leverages parametric assumptions and Monte Carlo simulation to obtain feasible solutions in small-data probabilistic constraint problems.

## Contribution

It introduces a novel DRO approach that incorporates parametric information and Monte Carlo methods to improve feasibility guarantees with limited data.

## Key findings

- Parametric assumptions can improve feasibility in small-data scenarios.
- Optimal generating distributions differ from baseline in parametric DRO.
- Numerical examples demonstrate the effectiveness of the proposed approach.

## Abstract

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.