Sample Average Approximation for Stochastic Programming with Equality Constraints

Marco Pavone, Riccardo Bonalli, Thomas Lew

Publication date: 20 June 2022

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 to the problem of stochastic optimal control for nonlinear dynamical systems subject to external disturbances modeled by a Wiener process. We verify our approach on a rocket-powered descent problem and show that our computed solutions allow for significant uncertainty reduction.

Has companion code repository: https://github.com/stanfordasl/stochasticedl

Mathematics Subject Classification ID

Geometric probability and stochastic geometry (60D05) Parametric inference (62F99) Monte Carlo methods (65C05) Stochastic programming (90C15) Optimal stochastic control (93E20)

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