Differentiability of Probability Functions Involving Star-Shaped Valued Set-Valued Maps

In energy, mechanical engineering, and economics, the use of probability—or chance—constraints provides an intuitive way to deal with uncertainty. The careful study of analytical properties of the involved objects provides a stepping stone to improved numerical treatment. Probability functions consist of “measuring” that a given random vector belongs to a parameter dependent set. The analytical properties of the thus obtained probability function depend on the nature of the random vector and analytical properties of the parameter dependent set. In this paper we are concerned with the investigation of first-order information of probability functions wherein the parameter dependent sets are star-shaped. Under mild conditions we establish local Lipschitz continuity and provide formulae for the subdifferentials. We instantiate these formulae in concrete settings.