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Rectangular Sets of Probability Measures

Author

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  • Alexander Shapiro

    (School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332)

Abstract

In this paper we consider the notion of rectangularity of a set of probability measures from a somewhat different point of view. We define rectangularity as a property of dynamic decomposition of a distributionally robust stochastic optimization problem and show how it relates to the modern theory of coherent risk measures. Consequently, we discuss robust formulations of multistage stochastic optimization problems in frameworks of stochastic programming, stochastic optimal control, and Markov decision processes.

Suggested Citation

  • Alexander Shapiro, 2016. "Rectangular Sets of Probability Measures," Operations Research, INFORMS, vol. 64(2), pages 528-541, April.
    • Handle: RePEc:inm:oropre:v:64:y:2016:i:2:p:528-541
    DOI: 10.1287/opre.2015.1466
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    References listed on IDEAS

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    1. Epstein, Larry G. & Schneider, Martin, 2003. "Recursive multiple-priors," Journal of Economic Theory, Elsevier, vol. 113(1), pages 1-31, November.
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    Citations

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    Cited by:

    1. Hadi El-Amine & Ebru K. Bish & Douglas R. Bish, 2018. "Robust Postdonation Blood Screening Under Prevalence Rate Uncertainty," Operations Research, INFORMS, vol. 66(1), pages 1-17, 1-2.
    2. Andrew J. Keith & Darryl K. Ahner, 2021. "A survey of decision making and optimization under uncertainty," Annals of Operations Research, Springer, vol. 300(2), pages 319-353, May.
    3. Angelos Georghiou & Angelos Tsoukalas & Wolfram Wiesemann, 2019. "Robust Dual Dynamic Programming," Operations Research, INFORMS, vol. 67(3), pages 813-830, May.
    4. Chen, Zengjing & Epstein, Larry G., 2022. "A central limit theorem for sets of probability measures," Stochastic Processes and their Applications, Elsevier, vol. 152(C), pages 424-451.
    5. Xin, Linwei & Goldberg, David A., 2021. "Time (in)consistency of multistage distributionally robust inventory models with moment constraints," European Journal of Operational Research, Elsevier, vol. 289(3), pages 1127-1141.
    6. Alois Pichler, 2017. "A quantitative comparison of risk measures," Annals of Operations Research, Springer, vol. 254(1), pages 251-275, July.
    7. Shapiro, Alexander, 2021. "Tutorial on risk neutral, distributionally robust and risk averse multistage stochastic programming," European Journal of Operational Research, Elsevier, vol. 288(1), pages 1-13.
    8. Alois Pichler & Rui Peng Liu & Alexander Shapiro, 2022. "Risk-Averse Stochastic Programming: Time Consistency and Optimal Stopping," Operations Research, INFORMS, vol. 70(4), pages 2439-2455, July.
    9. Henri Gérard & Michel Lara & Jean-Philippe Chancelier, 2020. "Equivalence between time consistency and nested formula," Annals of Operations Research, Springer, vol. 292(2), pages 627-647, September.
    10. Igor Cialenco & Gabriela Kov'av{c}ov'a, 2024. "Vector-valued robust stochastic control," Papers 2407.00266, arXiv.org.
    11. Roger J. A. Laeven & John G. M. Schoenmakers & Nikolaus Schweizer & Mitja Stadje, 2025. "Robust Multiple Stopping—A Duality Approach," Mathematics of Operations Research, INFORMS, vol. 50(2), pages 1250-1276, May.
    12. Larry G. Epstein & Shaolin Ji, 2022. "Optimal Learning Under Robustness and Time-Consistency," Operations Research, INFORMS, vol. 70(3), pages 1317-1329, May.
    13. Ruslan Mirmominov & Johannes Wiesel, 2024. "A dynamic programming principle for multiperiod control problems with bicausal constraints," Papers 2410.23927, arXiv.org.
    14. Park, Jangho & Bayraksan, Güzin, 2023. "A multistage distributionally robust optimization approach to water allocation under climate uncertainty," European Journal of Operational Research, Elsevier, vol. 306(2), pages 849-871.
    15. Linwei Xin & David Alan Goldberg, 2022. "Distributionally Robust Inventory Control When Demand Is a Martingale," Mathematics of Operations Research, INFORMS, vol. 47(3), pages 2387-2414, August.
    16. Chen, Zengjing & Epstein, Larry G. & Zhang, Guodong, 2023. "A central limit theorem, loss aversion and multi-armed bandits," Journal of Economic Theory, Elsevier, vol. 209(C).
    17. Henri G'erard & Michel de Lara & Jean-Philippe Chancelier, 2017. "Equivalence Between Time Consistency and Nested Formula," Papers 1711.08633, arXiv.org, revised May 2019.

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