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Asymptotic formulas for the derivatives of probability functions and their Monte Carlo estimations

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  • Garnier, Josselin
  • Omrane, Abdennebi
  • Rouchdy, Youssef

Abstract

One of the key problems in chance constrained programming for nonlinear optimization problems is the evaluation of derivatives of joint probability functions of the form . Here is the vector of physical parameters, is a random vector describing the uncertainty of the model, is the constraints mapping, and is the vector of constraint levels. In this paper specific Monte Carlo tools for the estimations of the gradient and Hessian of P(x) are proposed when the input random vector [Lambda] has a multivariate normal distribution and small variances. Using the small variance hypothesis, approximate expressions for the first- and second-order derivatives are obtained, whose Monte Carlo estimations have low computational costs. The number of calls of the constraints mapping g for the proposed estimators of the gradient and Hessian of P(x) is only 1+2Nx+2N[Lambda]. These tools are implemented in penalized optimization routines adapted to stochastic optimization, and are shown to reduce the computational cost of chance constrained programming substantially.

Suggested Citation

  • Garnier, Josselin & Omrane, Abdennebi & Rouchdy, Youssef, 2009. "Asymptotic formulas for the derivatives of probability functions and their Monte Carlo estimations," European Journal of Operational Research, Elsevier, vol. 198(3), pages 848-858, November.
    • Handle: RePEc:eee:ejores:v:198:y:2009:i:3:p:848-858
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    References listed on IDEAS

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    1. Mark J. Schervish, 1984. "Multivariate Normal Probabilities with Error Bound," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 33(1), pages 81-94, March.
    2. Suvrajeet Sen & Julia L. Higle, 1999. "An Introductory Tutorial on Stochastic Linear Programming Models," Interfaces, INFORMS, vol. 29(2), pages 33-61, April.
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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.
    3. Fengqiao Luo & Jeffrey Larson, 2024. "An Empirical Quantile Estimation Approach for Chance-Constrained Nonlinear Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 203(1), pages 767-809, October.

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