IDEAS home Printed from https://ideas.repec.org/a/inm/oropre/v64y2016i2p528-541.html
   My bibliography  Save this article

Rectangular Sets of Probability Measures

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/opre.2015.1466
    Download Restriction: no
    File URL: https://libkey.io/10.1287/opre.2015.1466?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Epstein, Larry G. & Schneider, Martin, 2003. "Recursive multiple-priors," Journal of Economic Theory, Elsevier, vol. 113(1), pages 1-31, November.
    2. Pichler, Alois & Shapiro, Alexander, 2015. "Minimal representation of insurance prices," Insurance: Mathematics and Economics, Elsevier, vol. 62(C), pages 184-193.
    3. Wolfram Wiesemann & Daniel Kuhn & Berç Rustem, 2013. "Robust Markov Decision Processes," Mathematics of Operations Research, INFORMS, vol. 38(1), pages 153-183, February.
    4. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    5. Dan A. Iancu & Marek Petrik & Dharmashankar Subramanian, 2015. "Tight Approximations of Dynamic Risk Measures," Mathematics of Operations Research, INFORMS, vol. 40(3), pages 655-682, March.
    6. Garud N. Iyengar, 2005. "Robust Dynamic Programming," Mathematics of Operations Research, INFORMS, vol. 30(2), pages 257-280, May.
    7. Georg Ch Pflug & Werner Römisch, 2007. "Modeling, Measuring and Managing Risk," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 6478, December.
    8. Arnab Nilim & Laurent El Ghaoui, 2005. "Robust Control of Markov Decision Processes with Uncertain Transition Matrices," Operations Research, INFORMS, vol. 53(5), pages 780-798, October.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. 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.
    3. Alois Pichler, 2017. "A quantitative comparison of risk measures," Annals of Operations Research, Springer, vol. 254(1), pages 251-275, July.
    4. 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.
    5. 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).
    6. 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.
    7. Igor Cialenco & Gabriela Kov'av{c}ov'a, 2024. "Vector-valued robust stochastic control," Papers 2407.00266, arXiv.org.
    8. Angelos Georghiou & Angelos Tsoukalas & Wolfram Wiesemann, 2019. "Robust Dual Dynamic Programming," Operations Research, INFORMS, vol. 67(3), pages 813-830, May.
    9. 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.
    10. 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.
    11. Ruslan Mirmominov & Johannes Wiesel, 2024. "A dynamic programming principle for multiperiod control problems with bicausal constraints," Papers 2410.23927, arXiv.org.
    12. 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.
    13. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. Maximilian Blesch & Philipp Eisenhauer, 2021. "Robust decision-making under risk and ambiguity," Papers 2104.12573, arXiv.org, revised Oct 2021.
    3. Dan A. Iancu & Marek Petrik & Dharmashankar Subramanian, 2015. "Tight Approximations of Dynamic Risk Measures," Mathematics of Operations Research, INFORMS, vol. 40(3), pages 655-682, March.
    4. Saghafian, Soroush, 2018. "Ambiguous partially observable Markov decision processes: Structural results and applications," Journal of Economic Theory, Elsevier, vol. 178(C), pages 1-35.
    5. Nicole Bauerle & Alexander Glauner, 2020. "Distributionally Robust Markov Decision Processes and their Connection to Risk Measures," Papers 2007.13103, arXiv.org.
    6. Michael Jong Kim & Andrew E.B. Lim, 2016. "Robust Multiarmed Bandit Problems," Management Science, INFORMS, vol. 62(1), pages 264-285, January.
    7. Rasouli, Mohammad & Saghafian, Soroush, 2018. "Robust Partially Observable Markov Decision Processes," Working Paper Series rwp18-027, Harvard University, John F. Kennedy School of Government.
    8. Varagapriya, V & Singh, Vikas Vikram & Lisser, Abdel, 2024. "Rank-1 transition uncertainties in constrained Markov decision processes," European Journal of Operational Research, Elsevier, vol. 318(1), pages 167-178.
    9. Shie Mannor & Ofir Mebel & Huan Xu, 2016. "Robust MDPs with k -Rectangular Uncertainty," Mathematics of Operations Research, INFORMS, vol. 41(4), pages 1484-1509, November.
    10. 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.
    11. Arthur Flajolet & Sébastien Blandin & Patrick Jaillet, 2018. "Robust Adaptive Routing Under Uncertainty," Operations Research, INFORMS, vol. 66(1), pages 210-229, January.
    12. 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.
    13. Bren, Austin & Saghafian, Soroush, 2018. "Data-Driven Percentile Optimization for Multi-Class Queueing Systems with Model Ambiguity: Theory and Application," Working Paper Series rwp18-008, Harvard University, John F. Kennedy School of Government.
    14. Michael Jong Kim, 2016. "Robust Control of Partially Observable Failing Systems," Operations Research, INFORMS, vol. 64(4), pages 999-1014, August.
    15. Bakker, Hannah & Dunke, Fabian & Nickel, Stefan, 2020. "A structuring review on multi-stage optimization under uncertainty: Aligning concepts from theory and practice," Omega, Elsevier, vol. 96(C).
    16. Maximilian Blesch & Philipp Eisenhauer, 2023. "Robust Decision-Making under Risk and Ambiguity," Rationality and Competition Discussion Paper Series 463, CRC TRR 190 Rationality and Competition.
    17. Eli Gutin & Daniel Kuhn & Wolfram Wiesemann, 2015. "Interdiction Games on Markovian PERT Networks," Management Science, INFORMS, vol. 61(5), pages 999-1017, May.
    18. Mahmutoğulları, Ali İrfan & Çavuş, Özlem & Aktürk, M. Selim, 2018. "Bounds on risk-averse mixed-integer multi-stage stochastic programming problems with mean-CVaR," European Journal of Operational Research, Elsevier, vol. 266(2), pages 595-608.
    19. Boloori, Alireza & Saghafian, Soroush & Chakkera, Harini A. A. & Cook, Curtiss B., 2017. "Data-Driven Management of Post-transplant Medications: An APOMDP Approach," Working Paper Series rwp17-036, Harvard University, John F. Kennedy School of Government.
    20. V Varagapriya & Vikas Vikram Singh & Abdel Lisser, 2023. "Joint chance-constrained Markov decision processes," Annals of Operations Research, Springer, vol. 322(2), pages 1013-1035, March.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:inm:oropre:v:64:y:2016:i:2:p:528-541. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.