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.

翻译：我们重新审视了面向非凸随机规划的样本平均逼近（SAA）方法。研究表明，将SAA方法应用于具有期望值等式约束的问题时，随着样本量增大，该方法并不必然具备渐近最优性保证。为解决此问题，我们对等式约束进行松弛处理，随后证明了在约束函数满足温和光滑性及有界性条件下，改进SAA方法的渐近最优性。我们的分析利用随机集理论及浓度不等式表征采样过程带来的近似误差。将该方法应用于受Wiener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，并通过火箭动力下降问题验证了方法的有效性。结果表明，经计算所得解能显著降低不确定性。