On Approximations of Data-Driven Chance Constrained Programs over Wasserstein Balls

TL;DR

This paper analyzes three approximation methods for distributionally robust chance constrained programs over Wasserstein balls, highlighting their properties, limitations, and the potential for poor performance in data-driven contexts.

Contribution

It provides a theoretical characterization of the CVaR approximation, offers a new transportation interpretation, and compares the three methods' effectiveness and limitations.

Findings

CVaR approximation is a tight convex approximation.

The three methods can perform arbitrarily poorly in data-driven settings.

The three approximations are generally incomparable.

Abstract

Distributionally robust chance constrained programs minimize a deterministic cost function subject to the satisfaction of one or more safety conditions with high probability, given that the probability distribution of the uncertain problem parameters affecting the safety condition(s) is only known to belong to some ambiguity set. We study three popular approximation schemes for distributionally robust chance constrained programs over Wasserstein balls, where the ambiguity set contains all probability distributions within a certain Wasserstein distance to a reference distribution. The first approximation replaces the chance constraint with a bound on the conditional value-at-risk, the second approximation decouples different safety conditions via Bonferroni's inequality, and the third approximation restricts the expected violation of the safety condition(s) so that the chance constraint is satisfied. We show that the conditional value-at-risk approximation can be characterized as a tight convex approximation, which complements earlier findings on classical (non-robust) chance constraints, and we offer a novel interpretation in terms of transportation savings. We also show that the three approximations can perform arbitrarily poorly in data-driven settings, and that they are generally incomparable with each other.