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On probabilistic constraints induced by rectangular sets and multivariate normal distributions

Author

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  • Wim Van Ackooij
  • René Henrion
  • Andris Möller
  • Riadh Zorgati

Abstract

In this paper, we consider optimization problems under probabilistic constraints which are defined by two-sided inequalities for the underlying normally distributed random vector. As a main step for an algorithmic solution of such problems, we prove a derivative formula for (normal) probabilities of rectangles as functions of their lower or upper bounds. This formula allows to reduce the calculus of such derivatives to the calculus of (normal) probabilities of rectangles themselves thus generalizing a similar well-known statement for multivariate normal distribution functions. As an application, we consider a problem from water reservoir management. One of the outcomes of the problem solution is that the (still frequently encountered) use of simple individual probabilistic constraints can completely fail. By contrast, the (more difficult) use of joint probabilistic constraints, which heavily depends on the derivative formula mentioned before, yields very reasonable and robust solutions over the whole time horizon considered. Copyright Springer-Verlag 2010

Suggested Citation

  • Wim Van Ackooij & René Henrion & Andris Möller & Riadh Zorgati, 2010. "On probabilistic constraints induced by rectangular sets and multivariate normal distributions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 71(3), pages 535-549, June.
    • Handle: RePEc:spr:mathme:v:71:y:2010:i:3:p:535-549
    DOI: 10.1007/s00186-010-0316-3
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    Citations

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    Cited by:

    1. Holger Berthold & Holger Heitsch & René Henrion & Jan Schwientek, 2022. "On the algorithmic solution of optimization problems subject to probabilistic/robust (probust) constraints," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 96(1), pages 1-37, August.
    2. 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.
    3. I. Bremer & R. Henrion & A. Möller, 2015. "Probabilistic constraints via SQP solver: application to a renewable energy management problem," Computational Management Science, Springer, vol. 12(3), pages 435-459, July.
    4. Holger Heitsch & René Henrion & Thomas Kleinert & Martin Schmidt, 2022. "On convex lower-level black-box constraints in bilevel optimization with an application to gas market models with chance constraints," Journal of Global Optimization, Springer, vol. 84(3), pages 651-685, November.
    5. Kawas, Ban & Thiele, Aurélie, 2011. "Short sales in Log-robust portfolio management," European Journal of Operational Research, Elsevier, vol. 215(3), pages 651-661, December.
    6. Wim Ackooij, 2014. "Decomposition approaches for block-structured chance-constrained programs with application to hydro-thermal unit commitment," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 80(3), pages 227-253, December.
    7. Wim Ackooij, 2017. "A comparison of four approaches from stochastic programming for large-scale unit-commitment," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 5(1), pages 119-147, March.
    8. Yongzhen Li & Jia Shu & Miao Song & Jiawei Zhang & Huan Zheng, 2017. "Multisourcing Supply Network Design: Two-Stage Chance-Constrained Model, Tractable Approximations, and Computational Results," INFORMS Journal on Computing, INFORMS, vol. 29(2), pages 287-300, May.
    9. Wim Ackooij & Welington Oliveira, 2014. "Level bundle methods for constrained convex optimization with various oracles," Computational Optimization and Applications, Springer, vol. 57(3), pages 555-597, April.
    10. Michel Minoux & Riadh Zorgati, 2017. "Global probability maximization for a Gaussian bilateral inequality in polynomial time," Journal of Global Optimization, Springer, vol. 68(4), pages 879-898, August.

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