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On Solving Multistage Stochastic Programs with Coherent Risk Measures


  • Andy Philpott

    () (Electric Power Optimization Centre, Department of Engineering Science, University of Auckland, Auckland, 1142, New Zealand)

  • Vitor de Matos

    () (Plan4 Engenharia SS, Florianópolis-SC, 88034-160, Brazil)

  • Erlon Finardi

    () (Laboratório de Planejamento em Sistemas de Energia Elétrica, Universidade Federal de Santa Catarina, Florianópolis-SC, 88040-900, Brazil)


We consider a class of multistage stochastic linear programs in which at each stage a coherent risk measure of future costs is to be minimized. A general computational approach based on dynamic programming is derived that can be shown to converge to an optimal policy. By computing an inner approximation to future cost functions, we can evaluate an upper bound on the cost of an optimal policy, and an outer approximation delivers a lower bound. The approach we describe is particularly useful in sampling-based algorithms, and a numerical example is provided to show the efficacy of the methodology when used in conjunction with stochastic dual dynamic programming.

Suggested Citation

  • Andy Philpott & Vitor de Matos & Erlon Finardi, 2013. "On Solving Multistage Stochastic Programs with Coherent Risk Measures," Operations Research, INFORMS, vol. 61(4), pages 957-970, August.
  • Handle: RePEc:inm:oropre:v:61:y:2013:i:4:p:957-970
    DOI: 10.1287/opre.2013.1175

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    References listed on IDEAS

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    Cited by:

    1. Powell, Warren B., 2019. "A unified framework for stochastic optimization," European Journal of Operational Research, Elsevier, vol. 275(3), pages 795-821.
    2. Daniel R. Jiang & Warren B. Powell, 2018. "Risk-Averse Approximate Dynamic Programming with Quantile-Based Risk Measures," Mathematics of Operations Research, INFORMS, vol. 43(2), pages 554-579, May.
    3. Weini Zhang & Hamed Rahimian & Güzin Bayraksan, 2016. "Decomposition Algorithms for Risk-Averse Multistage Stochastic Programs with Application to Water Allocation under Uncertainty," INFORMS Journal on Computing, INFORMS, vol. 28(3), pages 385-404, August.
    4. 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.
    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. Reus, Lorenzo & Pagnoncelli, Bernardo & Armstrong, Margaret, 2019. "Better management of production incidents in mining using multistage stochastic optimization," Resources Policy, Elsevier, vol. 63(C), pages 1-1.
    7. Moret, Fabio & Pinson, Pierre & Papakonstantinou, Athanasios, 2020. "Heterogeneous risk preferences in community-based electricity markets," European Journal of Operational Research, Elsevier, vol. 287(1), pages 36-48.
    8. Volker Krätschmer & Marcel Ladkau & Roger J. A. Laeven & John G. M. Schoenmakers & Mitja Stadje, 2018. "Optimal Stopping Under Uncertainty in Drift and Jump Intensity," Mathematics of Operations Research, INFORMS, vol. 43(4), pages 1177-1209, November.
    9. Schur, Rouven & Gönsch, Jochen & Hassler, Michael, 2019. "Time-consistent, risk-averse dynamic pricing," European Journal of Operational Research, Elsevier, vol. 277(2), pages 587-603.
    10. Bernardo K. Pagnoncelli & Adriana Piazza, 2017. "The optimal harvesting problem under price uncertainty: the risk averse case," Annals of Operations Research, Springer, vol. 258(2), pages 479-502, November.
    11. Alois Pichler, 2017. "A quantitative comparison of risk measures," Annals of Operations Research, Springer, vol. 254(1), pages 251-275, July.
    12. Alonso-Ayuso, Antonio & Escudero, Laureano F. & Guignard, Monique & Weintraub, Andres, 2018. "Risk management for forestry planning under uncertainty in demand and prices," European Journal of Operational Research, Elsevier, vol. 267(3), pages 1051-1074.
    13. Alois Pichler & Ruben Schlotter, 2020. "Quantification of Risk in Classical Models of Finance," Papers 2004.04397,, revised Jul 2020.
    14. Borgonovo, E. & Cappelli, V. & Maccheroni, F. & Marinacci, M., 2018. "Risk analysis and decision theory: A bridge," European Journal of Operational Research, Elsevier, vol. 264(1), pages 280-293.
    15. Dominic White & Niven Winchester, 2018. "Energy- and multi-sector modelling of climate change mitigation in New Zealand: current practice and future needs," Working Papers 18_15, Motu Economic and Public Policy Research.
    16. Liu, Rui Peng & Shapiro, Alexander, 2020. "Risk neutral reformulation approach to risk averse stochastic programming," European Journal of Operational Research, Elsevier, vol. 286(1), pages 21-31.
    17. A. B. Philpott & V. L. Matos & L. Kapelevich, 2018. "Distributionally robust SDDP," Computational Management Science, Springer, vol. 15(3), pages 431-454, October.
    18. Andre Luiz Diniz & Maria Elvira P. Maceira & Cesar Luis V. Vasconcellos & Debora Dias J. Penna, 2020. "A combined SDDP/Benders decomposition approach with a risk-averse surface concept for reservoir operation in long term power generation planning," Annals of Operations Research, Springer, vol. 292(2), pages 649-681, September.


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