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Augmented Lagrangian Methods for Solving Optimization Problems with Stochastic-Order Constraints

Author

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  • Darinka Dentcheva

    (Stevens Institute of Technology, Department of Mathematical Sciences, Hoboken, New Jersey 07030)

  • Gabriela Martinez

    (Kern Center for Science of Health Care Delivery, Mayo Clinic, Rochester, Minnesota 55905)

  • Eli Wolfhagen

    (Stevens Institute of Technology, Department of Mathematical Sciences, Hoboken, New Jersey 07030)

Abstract

We investigate risk-averse stochastic optimization problems with a risk-shaping constraint in the form of a stochastic-order relation. Both univariate and multivariate orders are considered. We extend the notion of a linear multivariate order, adding flexibility with respect to the controlled portion of the distributions. We propose several methods for the numerical solution of these problems based on augmented Lagrangian framework and analyze their convergence. The methods construct finite-dimensional risk-neutral approximations of the optimization problem whose solutions converge to the solution of the original problem. In the case of univariate stochastic dominance, we explore augmented Lagrangian functionals based on inverse formulations of the stochastic-order constraint. The performance of the methods is compared to other extant numerical methods and shows the numerical advantage of the augmented Lagrangian framework. The proposed numerical approach is particularly successful when applied to problems with multivariate stochastic dominance constraints.

Suggested Citation

  • Darinka Dentcheva & Gabriela Martinez & Eli Wolfhagen, 2016. "Augmented Lagrangian Methods for Solving Optimization Problems with Stochastic-Order Constraints," Operations Research, INFORMS, vol. 64(6), pages 1451-1465, December.
    • Handle: RePEc:inm:oropre:v:64:y:2016:i:6:p:1451-1465
    DOI: 10.1287/opre.2016.1521
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    References listed on IDEAS

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    2. Darinka Dentcheva & Andrzej Ruszczynski, 2004. "Optimization Under First Order Stochastic Dominance Constraints," GE, Growth, Math methods 0403002, University Library of Munich, Germany, revised 07 Aug 2005.
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    Cited by:

    1. Jianming Xia, 2023. "Benchmark Beating with the Increasing Convex Order," Papers 2311.01692, arXiv.org.
    2. Liwei Zhang & Yule Zhang & Jia Wu & Xiantao Xiao, 2022. "Solving Stochastic Optimization with Expectation Constraints Efficiently by a Stochastic Augmented Lagrangian-Type Algorithm," INFORMS Journal on Computing, INFORMS, vol. 34(6), pages 2989-3006, November.

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