Practicable Robust Stochastic Optimization under Divergence Measures

Abstract

We seek to provide practicable approximations of the two-stage robust stochastic optimization (RSO) model when its ambiguity set is constructed with an f-divergence radius. These models are known to be numerically challenging to various degrees, depending on the choice of the f-divergence function. The numerical challenges are even more pronounced under mixed-integer first-stage decisions. In this paper, we propose novel divergence functions that produce practicable robust counterparts, while maintaining versatility in modeling diverse ambiguity aversions. Our functions yield robust counterparts that have comparable numerical difficulties to their nominal problems. We also propose ways to use our divergences to mimic existing f-divergences without affecting the practicability. We implement our models in a realistic location-allocation model for humanitarian operations in Brazil. Our humanitarian model optimizes an effectiveness-equity trade-off, defined with a new utility function and a Gini mean difference coefficient. With the case study, we showcase 1) the significant improvement in practicability of the RSO counterparts with our proposed divergence functions compared to existing f-divergences, 2) the greater equity of humanitarian response that our new objective function enforces and 3) the greater robustness to variations in probability estimations of the resulting plans when ambiguity is considered.