Generalized differentiation of probability functions acting on an infinite system of constraints

Abstract

In decision-making problems under uncertainty, probability constraints are a valuable tool for expressing safety of decisions. They result from taking the probability measure of a given set of random inequalities depending on the decision vector. When uncertainty results from two different sources, unequally known, it becomes intuitively appealing to consider the probability of the worst case with respect to the part of the uncertainty vector for which little information was available. This and other models lead to probability functions acting on infinite systems of constraints. In this paper we study generalized (sub)differentiation of such probability functions. We also develop explicit formulae for the subdifferentials that should prove useful in first-order methods or in formulating optimality conditions.