Robust Stochastic Optimization with Rare-Event Modeling

Abstract

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.