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 of such functions 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.

This work provides formulae for the ε-subdifferential of integral functions in the framework of complete σ-finite measure spaces and locally convex spaces. In this work we present here new formulae for this ε-subdifferential under the presence of continuity-type qualification conditions relying on the data involved in the integrand.We provide new formulae for the subdifferential and the ε-subdifferential of the convex integral function given by the following expressionwhere (T, Σ, µ) is a complete σ-finite measure space, and f : T × X → R is a convex normal integrand defined on a locally convex space X.General formulae have been established in [19] using a finite-dimentional reduction approach, without additional assumptions on the data represented by the integrand f . In this paper, we use natural qualifications condition, involving appropriate continuity assumption on the integrand, to give more explicit characterization of the subdifferential and the ε-subdifferential of the function I f .

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