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Lipschitz and differentiability properties of quasi-concave and singular normal distribution functions

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  • René Henrion
  • Werner Römisch

Abstract

The paper provides a condition for differentiability as well as an equivalent criterion for Lipschitz continuity of singular normal distributions. Such distributions are of interest, for instance, in stochastic optimization problems with probabilistic constraints, where a comparatively small (nondegenerate-) normally distributed random vector induces a large number of linear inequality constraints (e.g. networks with stochastic demands). The criterion for Lipschitz continuity is established for the class of quasi-concave distributions which the singular normal distribution belongs to. Copyright Springer Science+Business Media, LLC 2010

Suggested Citation

  • René Henrion & Werner Römisch, 2010. "Lipschitz and differentiability properties of quasi-concave and singular normal distribution functions," Annals of Operations Research, Springer, vol. 177(1), pages 115-125, June.
    • Handle: RePEc:spr:annopr:v:177:y:2010:i:1:p:115-125:10.1007/s10479-009-0598-0
    DOI: 10.1007/s10479-009-0598-0
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    References listed on IDEAS

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    1. Werner Römisch & Rüdiger Schultz, 1993. "Stability of Solutions for Stochastic Programs with Complete Recourse," Mathematics of Operations Research, INFORMS, vol. 18(3), pages 590-609, August.
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    Cited by:

    1. René Henrion & Andris Möller, 2012. "A Gradient Formula for Linear Chance Constraints Under Gaussian Distribution," Mathematics of Operations Research, INFORMS, vol. 37(3), pages 475-488, August.
    2. Wim Ackooij & Pedro Pérez-Aros, 2020. "Gradient Formulae for Nonlinear Probabilistic Constraints with Non-convex Quadratic Forms," Journal of Optimization Theory and Applications, Springer, vol. 185(1), pages 239-269, April.

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