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Minimax and risk averse multistage stochastic programming

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  • Shapiro, Alexander

Abstract

In this paper we study relations between the minimax, risk averse and nested formulations of multistage stochastic programming problems. In particular, we discuss conditions for time consistency of such formulations of stochastic problems. We also describe a connection between law invariant coherent risk measures and the corresponding sets of probability measures in their dual representation. Finally, we discuss a minimax approach with moment constraints to the classical inventory model.

Suggested Citation

  • Shapiro, Alexander, 2012. "Minimax and risk averse multistage stochastic programming," European Journal of Operational Research, Elsevier, vol. 219(3), pages 719-726.
    • Handle: RePEc:eee:ejores:v:219:y:2012:i:3:p:719-726
    DOI: 10.1016/j.ejor.2011.11.005
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    References listed on IDEAS

    as
    1. Andrzej Ruszczynski & Alexander Shapiro, 2004. "Conditional Risk Mappings," Risk and Insurance 0404002, University Library of Munich, Germany, revised 08 Oct 2005.
    2. Andrzej Ruszczynski & Alexander Shapiro, 2004. "Optimization of Convex Risk Functions," Risk and Insurance 0404001, University Library of Munich, Germany, revised 08 Oct 2005.
    3. Andrzej Ruszczyński & Alexander Shapiro, 2006. "Optimization of Convex Risk Functions," Mathematics of Operations Research, INFORMS, vol. 31(3), pages 433-452, August.
    4. 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.
    5. Riedel, Frank, 2004. "Dynamic coherent risk measures," Stochastic Processes and their Applications, Elsevier, vol. 112(2), pages 185-200, August.
    6. Harry Markowitz, 1952. "Portfolio Selection," Journal of Finance, American Finance Association, vol. 7(1), pages 77-91, March.
    7. 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.
    8. Garud N. Iyengar, 2005. "Robust Dynamic Programming," Mathematics of Operations Research, INFORMS, vol. 30(2), pages 257-280, May.
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    Citations

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    Cited by:

    1. Alexander Shapiro & Wajdi Tekaya & Murilo Pereira Soares & Joari Paulo da Costa, 2013. "Worst-Case-Expectation Approach to Optimization Under Uncertainty," Operations Research, INFORMS, vol. 61(6), pages 1435-1449, December.
    2. Powell, Warren B., 2019. "A unified framework for stochastic optimization," European Journal of Operational Research, Elsevier, vol. 275(3), pages 795-821.
    3. Sungchul Hong & Jong-June Jeon, 2023. "Uniform Pessimistic Risk and its Optimal Portfolio," Papers 2303.07158, arXiv.org, revised May 2024.
    4. Homem-de-Mello, Tito & Pagnoncelli, Bernardo K., 2016. "Risk aversion in multistage stochastic programming: A modeling and algorithmic perspective," European Journal of Operational Research, Elsevier, vol. 249(1), pages 188-199.
    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. Anderson, Edward & Zachary, Stan, 2023. "Minimax decision rules for planning under uncertainty: Drawbacks and remedies," European Journal of Operational Research, Elsevier, vol. 311(2), pages 789-800.
    7. De Lara, Michel & Leclère, Vincent, 2016. "Building up time-consistency for risk measures and dynamic optimization," European Journal of Operational Research, Elsevier, vol. 249(1), pages 177-187.
    8. 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.
    9. François Clautiaux & Boris Detienne & Henri Lefebvre, 2023. "A two-stage robust approach for minimizing the weighted number of tardy jobs with objective uncertainty," Journal of Scheduling, Springer, vol. 26(2), pages 169-191, April.
    10. Martello, Silvano & Pinto Paixão, José M., 2012. "A look at the past and present of optimization – An editorial," European Journal of Operational Research, Elsevier, vol. 219(3), pages 638-640.
    11. 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).
    12. Pichler, Alois & Shapiro, Alexander, 2015. "Minimal representation of insurance prices," Insurance: Mathematics and Economics, Elsevier, vol. 62(C), pages 184-193.
    13. Yan Deng & Shabbir Ahmed & Siqian Shen, 2018. "Parallel Scenario Decomposition of Risk-Averse 0-1 Stochastic Programs," INFORMS Journal on Computing, INFORMS, vol. 30(1), pages 90-105, February.
    14. Nicole Bauerle & Alexander Glauner, 2020. "Markov Decision Processes with Recursive Risk Measures," Papers 2010.07220, arXiv.org.
    15. Yongchao Liu & Alois Pichler & Huifu Xu, 2019. "Discrete Approximation and Quantification in Distributionally Robust Optimization," Mathematics of Operations Research, INFORMS, vol. 44(1), pages 19-37, February.

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