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Probability functions appear in constraints of many optimization problems inpractice and have become quite popular. Understanding their first-orderproperties has proven useful, not only theoretically but also in implementablealgorithms, giving rise to competitive algorithms in several situations.Probability functions are built up from a random vector belonging to someparameter-dependent subset of the range of a given random vector. In thispaper, we investigate first order information of probability functionsspecified through a convex-valued set-valued application. We provide conditionsunder which the resulting probability function is indeed locally Lipschitzianas well as subgradient formulae. The resulting formulae are made concrete in aclassic optimization setting and put to work in an illustrative example comingfrom an energy application.

Probability functions generated by set-valued mappings: a study of first order information