# Subdifferential characterization of probability functions under Gaussian distribution

## TL;DR

This paper derives subdifferential formulas for Gaussian probability functions, including conditions for Lipschitz continuity and differentiability, applicable to infinite-dimensional decision variables and nonsmooth data.

## Contribution

It provides novel subdifferential characterizations of Gaussian probability functions, extending to infinite-dimensional spaces and nonsmooth inputs.

## Key findings

- Subdifferential formulas based on spheric-radial decomposition.
- Conditions for local Lipschitz continuity of probability functions.
- Criteria for differentiability of probability functions.

## 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.