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Augmented Lagrangian method for probabilistic optimization

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  • Darinka Dentcheva
  • Gabriela Martinez

Abstract

We analyze nonlinear stochastic optimization problems with probabilistic constraints described by continuously differentiable non-convex functions. We describe the tangent and the normal cone to the level sets of the underlying probability function and provide new insight into their structure. Furthermore, we formulate fist order and second order conditions of optimality for these problems based on the notion of p-efficient points. We develop an augmented Lagrangian method for the case of discrete distribution functions. The method is based on progressive inner approximation of the level set of the probability function by generation of p-efficient points. Numerical experience is provided. Copyright Springer Science+Business Media, LLC 2012

Suggested Citation

  • Darinka Dentcheva & Gabriela Martinez, 2012. "Augmented Lagrangian method for probabilistic optimization," Annals of Operations Research, Springer, vol. 200(1), pages 109-130, November.
    • Handle: RePEc:spr:annopr:v:200:y:2012:i:1:p:109-130:10.1007/s10479-011-0884-5
    DOI: 10.1007/s10479-011-0884-5
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    References listed on IDEAS

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    1. 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.
    2. 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.
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    Cited by:

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    2. Miguel A. Lejeune, 2012. "Pattern-Based Modeling and Solution of Probabilistically Constrained Optimization Problems," Operations Research, INFORMS, vol. 60(6), pages 1356-1372, December.
    3. Xiaodi Bai & Jie Sun & Xiaojin Zheng, 2021. "An Augmented Lagrangian Decomposition Method for Chance-Constrained Optimization Problems," INFORMS Journal on Computing, INFORMS, vol. 33(3), pages 1056-1069, July.
    4. 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.
    5. Hsia, Yong & Wu, Baiyi & Li, Duan, 2014. "New reformulations for probabilistically constrained quadratic programs," European Journal of Operational Research, Elsevier, vol. 233(3), pages 550-556.
    6. Wim Ackooij & Nicolas Lebbe & Jérôme Malick, 2017. "Regularized decomposition of large scale block-structured robust optimization problems," Computational Management Science, Springer, vol. 14(3), pages 393-421, July.
    7. Lukáš Adam & Martin Branda, 2016. "Nonlinear Chance Constrained Problems: Optimality Conditions, Regularization and Solvers," Journal of Optimization Theory and Applications, Springer, vol. 170(2), pages 419-436, August.
    8. Lukáš Adam & Martin Branda & Holger Heitsch & René Henrion, 2020. "Solving joint chance constrained problems using regularization and Benders’ decomposition," Annals of Operations Research, Springer, vol. 292(2), pages 683-709, September.

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