  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.
