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Dual methods for probabilistic optimization problems

Author

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  • Darinka Dentcheva
  • Bogumila Lai
  • Andrzej Ruszczyński

Abstract

We consider nonlinear stochastic optimization problems with probabilistic constraints. The concept of a p-efficient point of a probability distribution is used to derive equivalent problem formulations, and necessary and sufficient optimality conditions. We analyze the dual functional and its subdifferential. Two numerical methods are developed based on approximations of the p-efficient frontier. The algorithms yield an optimal solution for problems involving r-concave probability distributions. For arbitrary distributions, the algorithms provide upper and lower bounds for the optimal value and nearly optimal solutions. The operation of the methods is illustrated on a cash matching problem with a probabilistic liquidity constraint. Copyright Springer-Verlag 2004

Suggested Citation

  • Darinka Dentcheva & Bogumila Lai & Andrzej Ruszczyński, 2004. "Dual methods for probabilistic optimization problems ," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 60(2), pages 331-346, October.
    • Handle: RePEc:spr:mathme:v:60:y:2004:i:2:p:331-346
    DOI: 10.1007/s001860400371
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    Citations

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    Cited by:

    1. Miguel A. Lejeune, 2012. "Pattern-Based Modeling and Solution of Probabilistically Constrained Optimization Problems," Operations Research, INFORMS, vol. 60(6), pages 1356-1372, December.
    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. Feng Shan & Liwei Zhang & Xiantao Xiao, 2014. "A Smoothing Function Approach to Joint Chance-Constrained Programs," Journal of Optimization Theory and Applications, Springer, vol. 163(1), pages 181-199, October.
    4. Lejeune, Miguel & Noyan, Nilay, 2010. "Mathematical programming approaches for generating p-efficient points," European Journal of Operational Research, Elsevier, vol. 207(2), pages 590-600, December.
    5. Darinka Dentcheva & Gabriela Martinez, 2012. "Augmented Lagrangian method for probabilistic optimization," Annals of Operations Research, Springer, vol. 200(1), pages 109-130, November.
    6. W. Ackooij & A. Frangioni & W. Oliveira, 2016. "Inexact stabilized Benders’ decomposition approaches with application to chance-constrained problems with finite support," Computational Optimization and Applications, Springer, vol. 65(3), pages 637-669, December.
    7. Csaba I. Fábián, 2021. "Gaining traction: on the convergence of an inner approximation scheme for probability maximization," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 29(2), pages 491-519, June.
    8. L. Jeff Hong & Yi Yang & Liwei Zhang, 2011. "Sequential Convex Approximations to Joint Chance Constrained Programs: A Monte Carlo Approach," Operations Research, INFORMS, vol. 59(3), pages 617-630, June.
    9. Miguel Lejeune, 2012. "Pattern definition of the p-efficiency concept," Annals of Operations Research, Springer, vol. 200(1), pages 23-36, November.
    10. M. C. Campi & S. Garatti, 2011. "A Sampling-and-Discarding Approach to Chance-Constrained Optimization: Feasibility and Optimality," Journal of Optimization Theory and Applications, Springer, vol. 148(2), pages 257-280, February.

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