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A framework for optimization under ambiguity

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  • David Wozabal

Abstract

In this paper, single stage stochastic programs with ambiguous distributions for the involved random variables are considered. Though the true distribution is unknown, existence of a reference measure $\hat {P}$ enables the construction of non-parametric ambiguity sets as Kantorovich balls around $\hat{P}$ . The original stochastic optimization problems are robustified by a worst case approach with respect to these ambiguity sets. The resulting problems are infinite optimization problems and can therefore not be solved computationally by straightforward methods. To nevertheless solve the robustified problems numerically, equivalent formulations as finite dimensional non-convex, semi definite saddle point problems are proposed. Finally an application from portfolio selection is studied for which methods to solve the robust counterpart problems explicitly are proposed and numerical results for sample problems are computed. Copyright Springer Science+Business Media, LLC 2012

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  • David Wozabal, 2012. "A framework for optimization under ambiguity," Annals of Operations Research, Springer, vol. 193(1), pages 21-47, March.
    • Handle: RePEc:spr:annopr:v:193:y:2012:i:1:p:21-47:10.1007/s10479-010-0812-0
    DOI: 10.1007/s10479-010-0812-0
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    Cited by:

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    4. Max Nendel & Alessandro Sgarabottolo, 2022. "A parametric approach to the estimation of convex risk functionals based on Wasserstein distance," Papers 2210.14340, arXiv.org.
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    6. Hu, Jian & Bansal, Manish & Mehrotra, Sanjay, 2018. "Robust decision making using a general utility set," European Journal of Operational Research, Elsevier, vol. 269(2), pages 699-714.
    7. Hachmi Ben Ameur & Mouna Boujelbène & J. L. Prigent & Emna Triki, 2020. "Optimal Portfolio Positioning on Multiple Assets Under Ambiguity," Computational Economics, Springer;Society for Computational Economics, vol. 56(1), pages 21-57, June.
    8. Corina Birghila & Tim J. Boonen & Mario Ghossoub, 2020. "Optimal Insurance under Maxmin Expected Utility," Papers 2010.07383, arXiv.org.
    9. Ran Ji & Miguel A. Lejeune, 2021. "Data-Driven Optimization of Reward-Risk Ratio Measures," INFORMS Journal on Computing, INFORMS, vol. 33(3), pages 1120-1137, July.
    10. Jose Blanchet & Lin Chen & Xun Yu Zhou, 2022. "Distributionally Robust Mean-Variance Portfolio Selection with Wasserstein Distances," Management Science, INFORMS, vol. 68(9), pages 6382-6410, September.
    11. Corina Birghila & Tim J. Boonen & Mario Ghossoub, 2023. "Optimal insurance under maxmin expected utility," Finance and Stochastics, Springer, vol. 27(2), pages 467-501, April.
    12. Jose Blanchet & Karthyek Murthy, 2019. "Quantifying Distributional Model Risk via Optimal Transport," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 565-600, May.
    13. Ameur, H. Ben & Prigent, J.L., 2013. "Optimal portfolio positioning under ambiguity," Economic Modelling, Elsevier, vol. 34(C), pages 89-97.
    14. Bansal, Manish & Mehrotra, Sanjay, 2019. "On solving two-stage distributionally robust disjunctive programs with a general ambiguity set," European Journal of Operational Research, Elsevier, vol. 279(2), pages 296-307.
    15. Debora Daniela Escobar & Georg Ch. Pflug, 2020. "The distortion principle for insurance pricing: properties, identification and robustness," Annals of Operations Research, Springer, vol. 292(2), pages 771-794, September.
    16. Jitka Dupačová & Václav Kozmík, 2017. "SDDP for multistage stochastic programs: preprocessing via scenario reduction," Computational Management Science, Springer, vol. 14(1), pages 67-80, January.
    17. Grechuk, Bogdan & Zabarankin, Michael, 2018. "Direct data-based decision making under uncertainty," European Journal of Operational Research, Elsevier, vol. 267(1), pages 200-211.
    18. Luo, Fengqiao & Mehrotra, Sanjay, 2019. "Decomposition algorithm for distributionally robust optimization using Wasserstein metric with an application to a class of regression models," European Journal of Operational Research, Elsevier, vol. 278(1), pages 20-35.
    19. Daniela Escobar & Georg Pflug, 2018. "The distortion principle for insurance pricing: properties, identification and robustness," Papers 1809.06592, arXiv.org.
    20. Manish Bansal & Yingqiu Zhang, 2021. "Scenario-based cuts for structured two-stage stochastic and distributionally robust p-order conic mixed integer programs," Journal of Global Optimization, Springer, vol. 81(2), pages 391-433, October.

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