Managing Distributional Ambiguity in Stochastic Optimization through a Statistical Upper Bound Framework

Abstract: Stochastic optimization is often hampered by distributional ambiguity, where critical probability distributions are poorly characterized or unknown. Addressing this challenge, we introduce a new framework that targets the minimization of a statistical upper bound for the expected value of uncertain objectives, facilitating more statistically robust decision-making. Central to our approach is the Average Percentile Upper Bound (APUB), a novel construct that simultaneously delivers a statistically rigorous upper bound for the population mean and a meaningful risk metric for the sample mean. The integration of APUB into stochastic optimization not only fortifies the process against distributional ambiguity but also reinforces key data-driven decision-making attributes, such as reliability, consistency, and comprehensibility. Notably, APUB-enriched optimization problems feature tractability, with particular advantages in two-stage stochastic optimization with random recourse. Empirical demonstrations on two-stage product mix and multi-product newsvendor benchmark problems reveal the benefit of the APUB optimization framework, in comparison with conventional techniques such as sample average approximation and distributionally robust optimization.