论文标题
非convex风险避免风险随机优化的渐近一致性，无限尺寸决策空间
Asymptotic Consistency for Nonconvex Risk-Averse Stochastic Optimization with Infinite Dimensional Decision Spaces
论文作者
论文摘要
随机优化问题的经验近似值的最佳值和解决方案可以看作是其真实值的统计估计值。从这个角度来看，重要的是要了解这些估计量的渐近行为，因为样本量变为无穷大。该研究领域在随机编程方面具有悠久的传统。但是，文献缺乏一致性分析，即从无限的维度空间中获取决策变量的问题，这些空间是在最佳控制，科学机器学习和统计估计中产生的。通过利用这些应用中发现的典型问题结构，这些结构引起了解决方案集的隐藏规范紧凑性，我们证明了在无限尺寸空间中提出的非convex风险避免风险的随机优化问题的一致性结果。该证明是基于变异收敛理论的几个关键结果。对于文献中出现的几个重要问题类别，证明了理论结果。
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.
