Mathematical Programming

, Volume 174, Issue 1–2, pp 167–194 | Cite as

Subdifferential characterization of probability functions under Gaussian distribution

  • Abderrahim HantouteEmail author
  • René Henrion
  • Pedro Pérez-Aros
Full Length Paper Series B


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.


Probability functions Probabilistic constraint Stochastic optimization Multivariate Gaussian distribution Spheric-radial decomposition Clarke subdifferential Mordukhovich subdifferential 

Mathematics Subject Classification

90C15 90C30 49J52 49J53 


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018

Authors and Affiliations

  • Abderrahim Hantoute
    • 1
    Email author
  • René Henrion
    • 2
  • Pedro Pérez-Aros
    • 3
  1. 1.Center for Mathematical ModelingUniversidad de ChileSantiagoChile
  2. 2.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany
  3. 3.Institute of Engineering SciencesUniversity of O’higgnsRancaguaChile

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