Title: Subdifferential characterization of probability functions under Gaussian distribution,
Abstract: Probability functions figure prominently in optimization problems of
engineering. They may be nonsmooth even if all input data are smooth.This fact
motivates the consideration of subdifferentials for such typically just
continuous functions. The aim of this paper is to provide subdifferential
formulae in the case of Gaussian distributions for possibly
infinite-dimensional decision variables and nonsmooth (locally Lipschitzian)
input data. These formulae are based on the spheric-radial decomposition of
Gaussian random vectors on the one hand and on a cone of directions of moderate
growth on the other. By successively adding additional hypotheses, conditions
are satisfied under which the probability function is locally Lipschitzian or
even differentiable.