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, which is both of theoretical as well as practical interest. 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.

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具有无限维决策空间的非凸风险规避随机优化的渐近一致性

随机优化问题的经验近似的最优值和解可以看作是其真实值的统计估计量。从这个角度来看，重要的是要了解这些估计量在样本量趋于无穷大时的渐近行为，这既有理论意义，也有实际意义。这一研究领域在随机规划方面有着悠久的传统。然而，文献缺乏对决策变量取自无限维空间的问题的一致性分析，这些问题出现在最优控制、科学机器学习和统计估计中。通过利用在这些应用程序中发现的典型问题结构，这些结构会产生解决方案集的隐藏范数紧致性属性，我们证明了在无限维空间中制定的非凸风险规避随机优化问题的一致性结果。该证明基于变分收敛理论的几个关键结果。理论结果证明了文献中出现的几个重要问题类别。