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Gradient and Hessian of Joint Probability Function with Applications on Chance-Constrained Programs

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Abstract

Joint probability function refers to the probability function that requires multiple conditions to satisfy simultaneously. It appears naturally in chance-constrained programs. In this paper, we derive closed-form expressions of the gradient and Hessian of joint probability functions and develop Monte Carlo estimators of them. We then design a Monte Carlo algorithm, based on these estimators, to solve chance-constrained programs. Our numerical study shows that the algorithm works well, especially only with the gradient estimators.

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Notes

  1. There are technical conditions for interchanging the order of differentiations (see, for instance, Marsden and Hoffman [31]). The conditions are weak and typically satisfied by practical problems. To avoid too much technicality, we implicitly assume that the order can be interchanged throughout the paper.

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Authors and Affiliations

  1. Department of Economics and Finance and Department of Management Sciences, City University of Hong Kong, Hong Kong, China

    L. Jeff Hong

  2. Department of Economics and Finance, City University of Hong Kong, Hong Kong, China

    Guang-Xin Jiang

Authors
  1. L. Jeff Hong
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  2. Guang-Xin Jiang
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Correspondence to L. Jeff Hong.

Additional information

This research was supported by the Hong Kong Research Grants Council (No. GRF 613213).

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Hong, L.J., Jiang, GX. Gradient and Hessian of Joint Probability Function with Applications on Chance-Constrained Programs. J. Oper. Res. Soc. China 5, 431–455 (2017). https://doi.org/10.1007/s40305-017-0154-6

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  • DOI: https://doi.org/10.1007/s40305-017-0154-6

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