(Sub-)Differentiability of Probability Functions with Elliptical Distributions

Wim van Ackooij (1) , I. Aleksovska (2) , M. Munoz-Zuniga (2)

1 EDF R&D OSIRIS - Optimisation, Simulation, Risque et Statistiques pour les Marchés de l’Energie
2 IFPEN - IFP Energies nouvelles

In this paper we investigate probability functions acting on nonlinear systems wherein the random vector can follow an elliptically symmetric distribution. We provide first and second order differentiability results as well as readily implementable formulæ. We also demonstrate that these formulæ can be readily employed within standard non-linear programming software through a set of illustrative numerical experiments.

Mots clés: Stochastic optimisation; Probabilistic constraints; Chance constraints; Gradients of probability functions

Domaines: Probabilités [math.PR]

Wim van Ackooij, I. Aleksovska, M. Munoz-Zuniga. (Sub-)Differentiability of Probability Functions with Elliptical Distributions. Set-Valued and Variational Analysis, 2018, 26 (4), pp.887-910. ⟨10.1007/s11228-017-0454-3⟩. ⟨hal-02013571⟩