Sample Average Approximation for Stochastic Programming with Equality Constraints

TL;DR

This paper analyzes the limitations of the sample average approximation method for stochastic programming with equality constraints and proposes a relaxed approach with proven asymptotic optimality, validated through numerical experiments.

Contribution

It introduces a relaxed SAA method for constrained stochastic programming and proves its asymptotic optimality under mild conditions.

Findings

Relaxing equality constraints improves asymptotic guarantees.

The proposed method reliably solves stochastic control problems.

Significant uncertainty reduction achieved in rocket descent simulations.

Abstract

We revisit the sample average approximation (SAA) approach for non-convex stochastic programming. We show that applying the SAA approach to problems with expected value equality constraints does not necessarily result in asymptotic optimality guarantees as the sample size increases. To address this issue, we relax the equality constraints. Then, we prove the asymptotic optimality of the modified SAA approach under mild smoothness and boundedness conditions on the equality constraint functions. Our analysis uses random set theory and concentration inequalities to characterize the approximation error from the sampling procedure. We apply our approach and analysis to the problem of stochastic optimal control for nonlinear dynamical systems under external disturbances modeled by a Wiener process. Numerical results on relevant stochastic programs show the reliability of the proposed approach. Results on a rocket-powered descent problem show that our computed solutions allow for significant uncertainty reduction compared to a deterministic baseline.