A Characterization of the Subdifferential of Singular Gaussian Distribution Functions

Abstract

For optimization problems involving probabilistic constraints it is important to be able to compute gradients of these constraints efficiently. For distribution functions of Gaussian random variables with positive definite covariance matrices, computing gradients can be done efficiently. Indeed, a formula, linking components of the partial derivative and other regular Gaussian distribution functions exists and has been frequently employed by Prekopa. Recently, Henrion and Moller provided not only sufficient conditions under which singular Gaussian distribution functions are differentiable, but also a formula akin to the regular case. The sufficient conditions, rule out a set of points having zero (Lebesgue) measure. We show here that in these points, the sub-differential (in the sense of Clarke) of a Gaussian distribution function can be fully characterized. We moreover provide sufficient and necessary conditions under which these distribution functions are differentiable.