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On Solving Multistage Stochastic Programs with Coherent Risk Measures

Author

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  • 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)

Abstract

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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    1. Shapiro, Alexander & Tekaya, Wajdi & da Costa, Joari Paulo & Soares, Murilo Pereira, 2013. "Risk neutral and risk averse Stochastic Dual Dynamic Programming method," European Journal of Operational Research, Elsevier, vol. 224(2), pages 375-391.
    2. Shapiro, Alexander, 2011. "Analysis of stochastic dual dynamic programming method," European Journal of Operational Research, Elsevier, vol. 209(1), pages 63-72, February.
    3. Riedel, Frank, 2004. "Dynamic coherent risk measures," Stochastic Processes and their Applications, Elsevier, vol. 112(2), pages 185-200, August.
    4. David R. Cariño & Terry Kent & David H. Myers & Celine Stacy & Mike Sylvanus & Andrew L. Turner & Kouji Watanabe & William T. Ziemba, 1994. "The Russell-Yasuda Kasai Model: An Asset/Liability Model for a Japanese Insurance Company Using Multistage Stochastic Programming," Interfaces, INFORMS, vol. 24(1), pages 29-49, February.
    5. Philpott, A.B. & de Matos, V.L., 2012. "Dynamic sampling algorithms for multi-stage stochastic programs with risk aversion," European Journal of Operational Research, Elsevier, vol. 218(2), pages 470-483.
    6. Hans Föllmer & Alexander Schied, 2002. "Convex measures of risk and trading constraints," Finance and Stochastics, Springer, vol. 6(4), pages 429-447.
    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.
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