  Probability functions appear in constraints of many optimization problems in
practice and have become quite popular. Understanding their first-order
properties has proven useful, not only theoretically but also in implementable
algorithms, giving rise to competitive algorithms in several situations.
Probability functions are built up from a random vector belonging to some
parameter-dependent subset of the range of a given random vector. In this
paper, we investigate first order information of probability functions
specified through a convex-valued set-valued application. We provide conditions
under which the resulting probability function is indeed locally Lipschitzian
as well as subgradient formulae. The resulting formulae are made concrete in a
classic optimization setting and put to work in an illustrative example coming
from an energy application.
