Optimal values and solutions of empirical approximations of stochastic optimization problems can be viewed as statistical estimators of their true values. From this perspective, it is important to understand the asymptotic behavior of these estimators as the sample size goes to infinity. This area of study has a long tradition in stochastic programming. However, the literature is lacking consistency analysis for problems in which the decision variables are taken from an infinite dimensional space, which arise in optimal control, scientific machine learning, and statistical estimation. By exploiting the typical problem structures found in these applications that give rise to hidden norm compactness properties for solution sets, we prove consistency results for nonconvex risk-averse stochastic optimization problems formulated in infinite dimensional space. The proof is based on several crucial results from the theory of variational convergence. The theoretical results are demonstrated for several important problem classes arising in the literature.

翻译：随机优化问题的经验近似的最优值和解可视为其真实值的统计估计量。从这一角度看，理解当样本量趋于无穷时这些估计量的渐近行为至关重要。该研究领域在随机规划中具有悠久传统。然而，现有文献缺乏对决策变量取自无限维空间的问题（此类问题出现在最优控制、科学机器学习及统计估计中）的一致性分析。通过利用这些应用中典型的、为解集赋予隐式范数紧致性的问题结构，我们证明了无限维空间中非凸风险厌恶随机优化问题的一致性结果。证明基于变分收敛理论中的若干关键结论。本文针对文献中出现的几类重要问题类展示了理论结果。