Optimized Bonferroni approximations of distributionally robust joint chance constraints

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 nonseparable distribution family.

Automated summary

This paper investigates the Bonferroni approximation of distributionally robust joint chance constraints and proposes an optimized version of the Bonferroni approximation. It is shown that the optimized Bonferroni approximation is exact when the uncertainties are separable across the individual inequalities but leads to NP-hard optimization problems in certain cases. Sufficient conditions for the optimized Bonferroni approximation to be convex and tractable are derived. The paper also demonstrates how the results can be used to derive a convex reformulation of a distributionally robust joint chance constraint with a specific nonseparable distribution family.