# On Feasibility of Sample Average Approximation Solutions

## TL;DR

This paper investigates the feasibility of sample average approximation solutions in stochastic programming, showing that infeasible solutions decrease exponentially with larger sample sizes under certain conditions.

## Contribution

It extends the analysis of SAA solutions' feasibility to general stochastic programming problems without the relatively complete recourse assumption, including multistage cases.

## Key findings

- Infeasible SAA solutions decay exponentially with sample size for convex problems.
- Feasibility improves for functions with chain-constrained domains as sample size increases.
- Results extend to multistage stochastic programming scenarios.

## Abstract

When there are infinitely many scenarios, the current studies of two-stage stochastic programming problems rely on the relatively complete recourse assumption. However, such assumption can be unrealistic for many real-world problems. This motivates us to study general stochastic programming problems where the sample average approximation (SAA) solutions are not necessarily feasible. When the problems are convex and the true solutions lie in the interior of feasible solutions, we show the portion of infeasible SAA solutions decays exponentially as the sample size increases. We also study functions with chain-constrained domain, and show the portion of SAA solutions having a low degree of feasibility decays exponentially as the sample size increases. This result is then extended to multistage stochastic programming.