Properties and Calculation of Singular Normal Distributions

Authors René Henrion, Tamas Szantai

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René Henrion
Tamas Szantai

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René Henrion and Tamas Szantai. Properties and Calculation of Singular Normal Distributions. In Algorithms for Optimization with Incomplete Information. Dagstuhl Seminar Proceedings, Volume 5031, pp. 1-2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)


The need for calculating and characterizing singular normal distributions arises in a natural way when considering chance constraints of the type P(Az <= b(x) >= p, where A is a rectangular matrix having more rows than columns, b is some function and z is a random vector having some nondegenerate multivariate normal distribution. Such situation is typical, for instance, in stochastic networks, where a comparatively small random vector may induce a possibly large number of linear inequality constraints. Passing to the transformed random variable q:=Az, the constraint can be equivalently rewritten as F(b(x))>= p, where F is the distribution function of q. In contrast to the original random vector z, the transformed vector q has a singular normal distribution. The talk demonstrates how to get back from here to (a sum of) regular normal distributions under a full rank regularity condition. This allows for an efficient calculation of singular normal distributions and provides a numerical method which outperforms competing procedures in moderate dimensions. Computational results for test examples are provided for the sake of comparison. In general, if the mentioned regularity condition is violated, then the singular normal distribution function F may even lack continuity. The talk provides an equivalent criterion for Lipschitz continuity of F and characterizes differentiability and subdifferentiability of F.
  • singular normal distribution
  • chance constraints
  • normal probability of polyhedra


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