  In this paper, we propose a novel robust stochastic optimization approach
with a distinctive consideration for rare events, in which divergence measures
are used to bound the event-wise ambiguity sets. This is done by using the
Poisson distribution with uncertain expected value parameter to model rare
events and showing that the distribution possesses theoretical properties that
enable the derivation of rare-event probability bounds. We employ the proven
bijection between the Bregman divergence and the Poisson distribution to
characterize variations within the probability bounds, and demonstrate that the
explicit use of the Poisson-specific Bregman divergence results in reduced
conservatism in decision-making. Moreover, we derive a probability bound on
this conservatism reduction, which is a concept akin to the popular probability
of constraint violation in robust optimization. The bound shows that the rarer
the event, the more the rare-event information becomes powerful. Finally, we
provide tractable reformulations for our robust counterparts under rare events.
Our computational study, based on a realistic case study involving 18 years of
Brazilian flood and landslide data, reveals that our new approach generates
significantly better decisions than state-of-the-art approaches that are not
driven by rare events.
