Convexity of Chance Constraints with Dependent Random Variables: The Use of Copulae

Abstract

We consider the convexity of chance constraints with random right-hand side. While this issue is well understood (thanks to Prékopa’s Theorem) if the mapping operating on the decision vector is componentwise concave, things become more delicate when relaxing the concavity property. In an earlier paper, the significantly weaker r-concavity concept could be exploited, in order to derive eventual convexity (starting from a certain probability level) for feasible sets defined by chance constraints. This result heavily relied on the assumption of the random vector having independent components. A generalization to arbitrary multivariate distributions is all but straightforward. The aim of this chapter is to derive the same convexity result for distributions modeled via copulae. In this way, correlated components are admitted, but a certain correlation structure is imposed through the choice of the copula. We identify a class of copulae admitting eventually convex chance constraints.