Skip to main content
Log in

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

  • Published:
Annals of Operations Research Aims and scope Submit manuscript


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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  • Bank, B., Guddat, J., Klatte, D., Kummer, B., & Tammer, K. (1982). Non-linear parametric optimization. Berlin: Akademie-Verlag.

    Google Scholar 

  • Barndorff-Nielsen, O. E. (1978). Information and exponential families in statistical theory. Chichester: Wiley.

    Google Scholar 

  • Borell, C. (1975). Convex sets in d-space. Periodica Mathematica Hungarica, 6, 111–136.

    Article  Google Scholar 

  • Bukszár, J., Henrion, R., Hujter, M., & Szántai, T. (2004). Polyhedral inclusion-exclusion. Weierstrass Institute Berlin, Preprint No. 913.

  • Gassmann, H. I., Deák, I., & Szántai, T. (2002). Computing multivariate normal probabilities: A new look. Journal of Computational and Graphical Statistics, 11, 920–949.

    Google Scholar 

  • Genz, A. (1992). Numerical computation of multivariate normal probabilities. Journal of Computational and Graphical Statistics, 1, 141–149.

    Article  Google Scholar 

  • Henrion, R., & Römisch, W. (1999). Metric regularity and quantitative stability in stochastic programs with probabilistic constraints. Mathematical Programming, 84, 55–88.

    Google Scholar 

  • Naiman, D. Q., & Wynn, H. P. (1997). Abstract tubes, improved inclusion-exclusion identities and inequalities and importance sampling. Annals of Statistics, 25, 1954–1983.

    Article  Google Scholar 

  • Prékopa, A. (1995). Stochastic programming. Dordrecht: Kluwer.

    Google Scholar 

  • Römisch, W., & Schultz, R. (1993). Stability of solutions for stochastic programs with complete recourse. Mathematics of Operations Research, 18, 590–609.

    Article  Google Scholar 

  • Szántai, T. (2000). Improved bounds and simulation procedures on the value of the multivariate normal probability distribution function. Annals of Operations Research, 100, 85–101.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to René Henrion.

Additional information

This work was supported by the DFG Research Center Matheon Mathematics for key technologies in Berlin.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Henrion, R., Römisch, W. Lipschitz and differentiability properties of quasi-concave and singular normal distribution functions. Ann Oper Res 177, 115–125 (2010).

Download citation

  • Published:

  • Issue Date:

  • DOI: