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.

翻译：本文重新研究了非凸随机规划中的样本平均近似方法。我们证明，对于带有期望值等式约束的问题，随着样本量的增加，应用SAA方法并不一定能保证渐近最优性。为解决该问题，我们放宽了等式约束条件。随后，在等式约束函数满足温和的光滑性和有界性条件下，我们证明了改进的SAA方法具有渐近最优性。我们的分析运用随机集理论和集中不等式来刻画抽样过程产生的近似误差。我们将该方法与分析应用于受维纳过程建模的外部扰动影响的非线性动力系统随机最优控制问题。相关随机规划问题的数值结果表明了所提方法的可靠性。在火箭动力下降问题上的计算结果显示，与确定性基准相比，我们求得的解能实现显著的不确定性降低。