Optimized Bonferroni approximations of distributionally robust joint chance constraints

  • Weijun Xie
  • Shabbir Ahmed
  • Ruiwei JiangEmail author
Full Length Paper Series B


A distributionally robust joint chance constraint involves a set of uncertain linear inequalities which can be violated up to a given probability threshold \(\epsilon \), over a given family of probability distributions of the uncertain parameters. A conservative approximation of a joint chance constraint, often referred to as a Bonferroni approximation, uses the union bound to approximate the joint chance constraint by a system of single chance constraints, one for each original uncertain constraint, for a fixed choice of violation probabilities of the single chance constraints such that their sum does not exceed \(\epsilon \). It has been shown that, under various settings, a distributionally robust single chance constraint admits a deterministic convex reformulation. Thus the Bonferroni approximation approach can be used to build convex approximations of distributionally robust joint chance constraints. In this paper we consider an optimized version of Bonferroni approximation where the violation probabilities of the individual single chance constraints are design variables rather than fixed a priori. We show that such an optimized Bonferroni approximation of a distributionally robust joint chance constraint is exact when the uncertainties are separable across the individual inequalities, i.e., each uncertain constraint involves a different set of uncertain parameters and corresponding distribution families. Unfortunately, the optimized Bonferroni approximation leads to NP-hard optimization problems even in settings where the usual Bonferroni approximation is tractable. When the distribution family is specified by moments or by marginal distributions, we derive various sufficient conditions under which the optimized Bonferroni approximation is convex and tractable. We also show that for moment based distribution families and binary decision variables, the optimized Bonferroni approximation can be reformulated as a mixed integer second-order conic set. Finally, we demonstrate how our results can be used to derive a convex reformulation of a distributionally robust joint chance constraint with a specific non-separable distribution family.

Mathematics Subject Classification

90C15 90C47 90C11 



This research has been supported in part by the National Science Foundation Awards #1633196 and #1662774.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2019

Authors and Affiliations

  1. 1.Virginia TechBlacksburgUSA
  2. 2.Georgia Institute of TechnologyAtlantaUSA
  3. 3.University of MichiganAnn ArborUSA

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