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Convexity of Chance Constraints with Dependent Random Variables: The Use of Copulae

In: Stochastic Optimization Methods in Finance and Energy

Author

Listed:
  • René Henrion

    (Weierstrass Institute for Applied Analysis and Stochastics)

  • Cyrille Strugarek

    (Credit Portfolio Management, Calyon Credit Agricole CIB)

Abstract

We consider the convexity of chance constraints with random right-hand side. While this issue is well understood (thanks to Prékopa’s Theorem) if the mapping operating on the decision vector is componentwise concave, things become more delicate when relaxing the concavity property. In an earlier paper, the significantly weaker r-concavity concept could be exploited, in order to derive eventual convexity (starting from a certain probability level) for feasible sets defined by chance constraints. This result heavily relied on the assumption of the random vector having independent components. A generalization to arbitrary multivariate distributions is all but straightforward. The aim of this chapter is to derive the same convexity result for distributions modeled via copulae. In this way, correlated components are admitted, but a certain correlation structure is imposed through the choice of the copula. We identify a class of copulae admitting eventually convex chance constraints.

Suggested Citation

  • René Henrion & Cyrille Strugarek, 2011. "Convexity of Chance Constraints with Dependent Random Variables: The Use of Copulae," International Series in Operations Research & Management Science, in: Marida Bertocchi & Giorgio Consigli & Michael A. H. Dempster (ed.), Stochastic Optimization Methods in Finance and Energy, edition 1, chapter 0, pages 427-439, Springer.
    • Handle: RePEc:spr:isochp:978-1-4419-9586-5_17
    DOI: 10.1007/978-1-4419-9586-5_17
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    Citations

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    Cited by:

    1. Hoang Nam Nguyen & Abdel Lisser & Vikas Vikram Singh, 2022. "Random Games Under Elliptically Distributed Dependent Joint Chance Constraints," Journal of Optimization Theory and Applications, Springer, vol. 195(1), pages 249-264, October.
    2. Rashed Khanjani-Shiraz & Salman Khodayifar & Panos M. Pardalos, 2021. "Copula theory approach to stochastic geometric programming," Journal of Global Optimization, Springer, vol. 81(2), pages 435-468, October.
    3. Martin Branda & Štěpán Hájek, 2017. "Flow-based formulations for operational fixed interval scheduling problems with random delays," Computational Management Science, Springer, vol. 14(1), pages 161-177, January.
    4. Michel Minoux & Riadh Zorgati, 2019. "Sharp upper and lower bounds for maximum likelihood solutions to random Gaussian bilateral inequality systems," Journal of Global Optimization, Springer, vol. 75(3), pages 735-766, November.
    5. Peng, Shen & Maggioni, Francesca & Lisser, Abdel, 2022. "Bounds for probabilistic programming with application to a blend planning problem," European Journal of Operational Research, Elsevier, vol. 297(3), pages 964-976.
    6. Butyn, Emerson & Karas, Elizabeth W. & de Oliveira, Welington, 2022. "A derivative-free trust-region algorithm with copula-based models for probability maximization problems," European Journal of Operational Research, Elsevier, vol. 298(1), pages 59-75.
    7. 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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