A Characterization of the Subdifferential of Singular Gaussian Distribution Functions

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• 作者：W. van Ackooij ; M. Minoux
• 关键词：Chance constrained programming ; Probabilistic constraints ; Stochastic programming ; Gradients ; MSC 90C15 ; MSC 90B15
• 刊名：Set-Valued and Variational Analysis
• 出版年：2015
• 出版时间：September 2015
• 年：2015
• 卷：23
• 期：3
• 页码：465-483
• 全文大小：483 KB
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• 作者单位：W. van Ackooij (1)
  M. Minoux (2)
  
  1. OSIRIS, EDF R&D, 1 avenue du Général de Gaulle, F-92141, Clamart Cedex, Paris, France
  2. LIP6, Université Pierre et Marie Curie, F-75000, Paris, France
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Geometry
  
• 出版者：Springer Netherlands
• ISSN：1877-0541

文摘

For optimization problems involving probabilistic constraints it is important to be able to compute gradients of these constraints efficiently. For distribution functions of Gaussian random variables with positive definite covariance matrices, computing gradients can be done efficiently. Indeed, a formula, linking components of the partial derivative and other regular Gaussian distribution functions exists and has been frequently employed by Prékopa. Recently, Henrion and M?ller provided not only sufficient conditions under which singular Gaussian distribution functions are differentiable, but also a formula akin to the regular case. The sufficient conditions, rule out a set of points having zero (Lebesgue) measure. We show here that in these points, the sub-differential (in the sense of Clarke) of a Gaussian distribution function can be fully characterized. We moreover provide sufficient and necessary conditions under which these distribution functions are differentiable.