Asymptotic Consistency for Nonconvex Risk-Averse Stochastic Optimization with Infinite-Dimensional Decision Spaces
研究了无限维空间中非凸风险厌恶随机优化问题的经验近似解作为统计估计量的渐近一致性，利用隐式范数紧性证明了相关结论，适用于最优控制、科学机器学习和统计估计等领域。
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.