Derivatives of Probability Functions: Unions of Polyhedra and Elliptical Distributions

Abstract

In many practical applications models exhibiting chance constraints play a role. Since, in practice one is also typically interesting in numerically solving the underlying optimization problems, an interest naturally arises in understanding analytical properties, such as differentiability, of probability functions. However in order to build nonlinear programming methods, not only knowledge of differentiability, but also explicit formul AE for gradients are important. Unfortunately, differentiability of probability functions cannot be taken for granted. In this paper, motivated by applications from energy management, wherein we face a variety of non-linear transforms of underlying Elliptical distributions, we investigate probability functions acting on decision dependent union of polyhedra. Union of polyhedra naturally occur as soon as one approaches the components of difference-of-convex (DC) functions with their respective cutting plane models. In this work, we will establish that the probability functions are locally Lipschitzian and exhibit explicit formul AE for the Clarke sub-gradients, under very mild conditions. We also highlight, on a numerical example, that the formul AE can be put to use in practice.