Computational Management Science

, Volume 14, Issue 1, pp 67–80 | Cite as

SDDP for multistage stochastic programs: preprocessing via scenario reduction

  • Jitka Dupačová
  • Václav Kozmík
Original Paper


Even with recent enhancements, computation times for large-scale multistage problems with risk-averse objective functions can be very long. Therefore, preprocessing via scenario reduction could be considered as a way to significantly improve the overall performance. Stage-wise backward reduction of single scenarios applied to a fixed branching structure of the tree is a promising tool for efficient algorithms like stochastic dual dynamic programming. We provide computational results which show an acceptable precision of the results for the reduced problem and a substantial decrease of the total computation time.


Multistage stochastic programs Stochastic dual dynamic programming Multiperiod CVaR Scenario reduction 

Mathematics Subject Classification

65C05 90C15 91G60 



Jitka Dupačová has initiated this project and we have worked together till the very final form of the article, unfortunately, she passed away during the publication process. I would like to dedicate this paper to Jitka, for her restless guidance, knowledge, patience and care. The research was partly supported by the project of the Czech Science Foundation P/402/12/G097 ’DYME/Dynamic Models in Economics’.


  1. Bally V, Pages G (2003) Quantization algorithm for solving multidimensional discrete-time optimal stopping problems. Bernoulli 9:1003–1049CrossRefGoogle Scholar
  2. Bayraksan G, Morton DP (2009) Assessing solution quality in stochastic programs via sampling. In: Oskoorouchi M, Gray P, Greenberg H (eds) Tutorials in operations research. Informs, Hannover, pp 102–122, ISBN 978-1-877640-24-7Google Scholar
  3. Dupačová J, Gröwe-Kuska N, Römisch W (2003) Scenario reduction in stochastic programming: an approach using probability metrics. Math Prog 95:493–511CrossRefGoogle Scholar
  4. Dupačová J, Kozmík V (2015) Structure of risk-averse multistage stochastic programs. OR Spectr 37:559–582CrossRefGoogle Scholar
  5. Eichhorn A, Römisch W (2005) Polyhedral risk measures in stochastic programming. SIAM J Optim 16:69–95CrossRefGoogle Scholar
  6. Heitsch H, Römisch W, Strugarek C (2006) Stability of multistage stochastic programs. SIAM J Optim 17:511–525CrossRefGoogle Scholar
  7. Heitsch H, Römisch W (2009) Scenario tree reduction for multistage stochastic programs. Comput Manag Sci 6:117–133CrossRefGoogle Scholar
  8. Heitsch H, Römisch W (2009) Scenario tree modeling for multistage stochastic programs. Math Progr 118:371–406CrossRefGoogle Scholar
  9. Infanger G, Morton DP (1996) Cut sharing for multistage stochastic linear programs with interstage dependency. Math Progr 75:241–256Google Scholar
  10. Kozmík V, Morton D (2015) Evaluating policies in risk-averse multi-stage stochastic programming. Math Progr 152:275–300CrossRefGoogle Scholar
  11. Löhndorf N, Wozabal D, Minner S (2013) Optimizing trading decisions for hydro storage systems using approximate dual dynamic programming. Oper Res 61:810–823CrossRefGoogle Scholar
  12. Oliveira WL, Sagastizábal C, Penna DDJ, Maceira MEP, Damázio JM (2010) Optimal scenario tree reduction for stochastic streamflows in power generation planning problems. Optim Methods Softw 25:917–936CrossRefGoogle Scholar
  13. Pereira MVF, Pinto LMVG (1991) Multi-stage stochastic optimization applied to energy planning. Math Progr 52:359–375CrossRefGoogle Scholar
  14. Pflug GCh, Pichler A (2012) A distance for multistage stochastic optimization models. SIAM J Optim 22:1–23CrossRefGoogle Scholar
  15. Pflug GCh, Pichler A (2011) Approximations for probability distributions and stochastic optimization problems. In: Bertocchi M, Consigli G, Dempster MAH (eds) Stochastic optimization methods in finance and energy. Springer, New York, pp 343–388, ISBN 978-1-4419-9585-8Google Scholar
  16. Philpott AB, de Matos VL (2012) Dynamic sampling algorithms for multi-stage stochastic programs with risk aversion. Eur J Oper Res 218:470–483CrossRefGoogle Scholar
  17. Rockafellar RT, Uryasev S (2002) Conditional value at risk for general loss distributions. J Bank Financ 26:1443–1471CrossRefGoogle Scholar
  18. Römisch W (2003) Stability of stochastic programming problems, Chapter 8. In: Ruszczyński A, Shapiro A (eds) Handbook on stochastic programming. Elsevier, Amsterdam, pp 483–554CrossRefGoogle Scholar
  19. Römisch W (2009) Scenario reduction techniques in stochastic programming. In: Watanabe O, Zeugmann T (eds) Stochastic algorithms: foundations and applications, vol 5792., Lecture notes in computer science. Springer, Sapporo, pp 1–14Google Scholar
  20. Shapiro A (2011) Analysis of stochastic dual dynamic programming method. Eur J Oper Res 209:63–72CrossRefGoogle Scholar
  21. Shapiro A, Dentcheva D, Ruszczyński A (2009) Lectures on stochastic programming: modeling and theory. SIAM Society for Industrial and Applied Mathematics, Philadelphia, ISBN 978-1107025127Google Scholar
  22. Wozabal D (2012) A framework for optimization under ambiguity. Ann Oper Res 193:21–47CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Probability and Mathematical StatisticsCharles University in PraguePragueCzech Republic

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