Eventual convexity of probability constraints with elliptical distributions

Abstract

Probability constraints are often employed to intuitively define safety of given decisions in optimization problems. They simply express that a given system of inequalities depending on a decision vector and a random vector is satisfied with high enough probability. It is known that, even if this system is convex in the decision vector, the associated probability constraint is not convex in general. In this paper, we show that some degree of convexity is still preserved, for the large class of elliptical random vectors, encompassing for example Gaussian or Student random vectors. More precisely, our main result establishes that, under mild assumptions, eventual convexity holds, i.e. the probability constraint is convex when the safety level is large enough. We also provide tools to compute a concrete convexity certificate from nominal problem data. Our results are illustrated on several examples, including the situation of polyhedral systems with random technology matrices and arbitrary covariance structure.