Statistical Inference for Scenario-Based Dynamic Optimization under Uncertainty

Motivated by batch and semi-batch process operation, we study finite-horizon open-loop dynamic optimization problems with uncertain parameters. A common computational approach replaces the expected performance criterion by an average over finitely many sampled parameter realizations. We develop a statistical theory for the resulting sample-based optimal value as an estimator of the population optimal value. The analysis is based on a stability estimate showing that terminal losses depend Lipschitz continuously on the time-integrated control, which records the cumulative input delivered up to each time. This estimate yields a functional central limit theorem for the sample-based objective and a statistical limit theorem for the corresponding optimal value error. As a consequence, we obtain confidence intervals for the population optimal value. When the population optimizer is unique, the limit is Gaussian and leads to a plug-in confidence interval. When multiple optimal policies may exist, we use a subsampling confidence interval that does not require uniqueness. The methodology is illustrated on two fed-batch case studies in which feed-rate profiles are optimized under parametric uncertainty.