Convexity of chance constraints for elliptical and skewed distributions with copula structures dependent on decision variables

Abstract

Chance constraints describe a set of given random inequalities depending on the decision vector satisfied with a large enough probability. They are widely used in decision making under uncertain data in many engineering problems. This paper aims to derive the convexity of chance constraints with row dependent elliptical and skewed random variables via a copula depending on decision vectors. We obtain best thresholds of the $r$-concavity for any real number $r$ and improve probability thresholds of the eventual convexity. We prove the eventual convexity with elliptical distributions and a Gumbel-Hougaard copula despite the copula's singularity near the origin. We determine the $\alpha$-decreasing densities of generalized hyperbolic distributions by estimating the modified Bessel functions. By applying the $\alpha$-decreasing property and a radial decomposition, we achieve the eventual convexity for three types of skewed distributions. Finally, we provide an example to illustrate the eventual convexity of a feasible set containing the origin.