Skip to main content
SpringerLink
Log in

Scenario tree modeling for multistage stochastic programs

  • FULL LENGTH PAPER
  • Published:
Mathematical Programming Submit manuscript

Abstract

An important issue for solving multistage stochastic programs consists in the approximate representation of the (multivariate) stochastic input process in the form of a scenario tree. In this paper, we develop (stability) theory-based heuristics for generating scenario trees out of an initial set of scenarios. They are based on forward or backward algorithms for tree generation consisting of recursive scenario reduction and bundling steps. Conditions are established implying closeness of optimal values of the original process and its tree approximation, respectively, by relying on a recent stability result in Heitsch, Römisch and Strugarek (SIAM J Optim 17:511–525, 2006) for multistage stochastic programs. Numerical experience is reported for constructing multivariate scenario trees in electricity portfolio management.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Barty, K.: Contributions à la discrétisation des contraintes de mesurabilité pour les problèmes d’optimisation stochastique. Thèse de Doctorat, École Nationale des Ponts et Chaussées, (2004)

  2. Casey M. and Sen S. (2005). The scenario generation algorithm for multistage stochastic linear programming. Math. Oper. Res. 30: 615–631

    Article  MATH  MathSciNet  Google Scholar 

  3. Chiralaksanakul, A.; Morton, D.P.: Assessing policy quality in multi-stage stochastic programming. Stochastic Programming E-Print Series 12–2004 (<www.speps.org>).

  4. Corvera Poiré, X.: Model Generation and Sampling Algorithms for Dynamic Stochastic Programming. PhD Thesis, Department of Mathematics, University of Essex (1995)

  5. Dempster M.A.H. (2004). Sequential importance sampling algorithms for dynamic stochastic programming. Zap. Nauchn. Semin. POMI 312: 94–129

    MATH  Google Scholar 

  6. Dudley R.M. (1989). Real Analysis and Probability. Chapman & Hall, New York

    MATH  Google Scholar 

  7. Dunford N. and Schwartz J.T. (1988). Linear Operators, Part I: General Theory. Wiley Classics Library, New York

    MATH  Google Scholar 

  8. Dupačová J., Consigli G. and Wallace S.W. (2000). Scenarios for multistage stochastic programs. Ann. Oper. Res. 100: 25–53

    Article  MATH  MathSciNet  Google Scholar 

  9. Dupačová J., Gröwe-Kuska N. and Römisch W. (2003). Scenario reduction in stochastic programming: an approach using probability metrics. Math. Program. 95: 493–511

    Article  MATH  MathSciNet  Google Scholar 

  10. Eichhorn, A.; Römisch, W.; Wegner, I.: Mean-risk optimization of electricity portfolios using multiperiod polyhedral risk measures, IEEE St. Petersburg Power Tech (2005)

  11. Fetter H. (1977). On the continuity of conditional expectations. J. Math. Anal. Appl. 61: 227–231

    Article  MATH  MathSciNet  Google Scholar 

  12. Fleten, S.-E.; Kristoffersen, T.K.: Short-term hydropower production planning by stochastic programming. Comput. Oper. Res. (to appear)

  13. Frauendorfer K. (1996). Barycentric scenario trees in convex multistage stochastic programming. Math. Programm. Ser. B 75: 277–293

    MathSciNet  Google Scholar 

  14. Givens C.R. and Shortt R.M. (1984). A class of Wasserstein metrics for probability distributions. Michigan Math. J. 31: 231–240

    Article  MATH  MathSciNet  Google Scholar 

  15. Graf S. and Luschgy H. (2000). Foundations of Quantization for Probability. Distributions Lecture Notes in Mathematics, vol. 1730. Springer, Berlin

    Google Scholar 

  16. Gröwe-Kuska, N., Heitsch, H., Römisch, W.: Modellierung stochastischer Datenprozesse für Optimierungsmodelle der Energiewirtschaft, IT-Lösungen für die Energiewirtschaft in liberalisierten Märkten, VDI-Berichte 1647, pp. 69–78. VDI-Verlag, Düsseldorf (2001)

  17. Gröwe-Kuska, N., Heitsch, H., Römisch, W.: Scenario reduction and scenario tree construction for power management problems. In: Borghetti, A., Nucci, C.A., Paolone, M. (eds.) IEEE Bologna Power Tech Proceedings (2003)

  18. Gröwe-Kuska, N., Römisch, W.: Stochastic unit commitment in hydro-thermal power production planning. In: Wallace, S.W., Ziemba, W.T. (eds.) Chapter 30 in Applications of Stochastic Programming, MPS-SIAM Series in Optimization (2005)

  19. Heitsch H. and Römisch W. (2003). Scenario reduction algorithms in stochastic programming. Comput. Optim. Appl. 24: 187–206

    Article  MATH  MathSciNet  Google Scholar 

  20. Heitsch, H., Römisch, W.: Generation of multivariate scenario trees to model stochasticity in power management. IEEE St. Petersburg Power Tech (2005)

  21. Heitsch, H., Römisch, W.: Stability and scenario trees for multistage stochastic programs, Preprint 324, DFG Research Center Matheon. In: Dantzig, G., Infanger, G. (eds.) Mathematics for key technologies, 2006 and submitted to Stochastic Programming—The State of the Art (2006)

  22. Heitsch H., Römisch W. and Strugarek C. (2006). Stability of multistage stochastic programs. SIAM J. Optim. 17: 511–525

    Article  MATH  MathSciNet  Google Scholar 

  23. Higle, J.L., Rayco, B., Sen, S.: Stochastic scenario decomposition for multistage stochastic programs. Ann. Oper. Res. (submitted)

  24. Hochreiter R. and Pflug G. Ch. (2007). Financial scenario generation for stochastic multi-stage decision processes as facility location problems. Ann. Oper. Res. 152: 257–272

    Article  MATH  MathSciNet  Google Scholar 

  25. Høyland K. and Wallace S.W. (2001). Generating scenario trees for multi-stage decision problems. Manage. Sci. 47: 295–307

    Article  Google Scholar 

  26. Høyland K., Kaut M. and Wallace S.W. (2003). A heuristic for moment-matching scenario generation. Comput. Optim. Appl. 24: 169–185

    Article  MathSciNet  Google Scholar 

  27. Kaut, M., Wallace, S.W.: Evaluation of scenario-generation methods for stochastic programming. Stochastic Programming E-Print Series 14–2003 (<www.speps.org>).

  28. Kuhn D. (2005). Generalized Bounds for Convex Multistage Stochastic Programs. Lecture Notes in Economics and Mathematical Systems, vol. 548. Springer, Berlin

    Google Scholar 

  29. Möller, A., Römisch, W., Weber, K.: Airline network revenue management by multistage stochastic programming. Comput. Manage. Sci. (to appear)

  30. Pennanen T. (2005). Epi-convergent discretizations of multistage stochastic programs. Math. Oper. Res. 30: 245–256

    Article  MATH  MathSciNet  Google Scholar 

  31. Pennanen, T.: Epi-convergent discretizations of multistage stochastic programs via integration quadratures. Stochastic Programming E-Print Series 19–2004 (<www.speps.org>). Math. Programm. (to appear)

  32. Pflug G.Ch. (2001). Scenario tree generation for multiperiod financial optimization by optimal discretization. Math. Program. 89: 251–271

    Article  MATH  MathSciNet  Google Scholar 

  33. Rachev S.T. (1991). Probability Metrics and the Stability of Stochastic Models. Wiley, New York

    MATH  Google Scholar 

  34. Rachev S.T. and Römisch W. (2002). Quantitative stability in stochastic programming: The method of probability metrics. Math. Oper. Res. 27: 792–818

    Article  MATH  MathSciNet  Google Scholar 

  35. Rachev S.T. and Rüschendorf L. (1998). Mass Transportation Problems, vols. I and II. Springer, Berlin

    Google Scholar 

  36. Rachev S.T. and Schief A. (1992). On L p -minimal metrics. Prob. Math. Stat. 13: 311–320

    MATH  MathSciNet  Google Scholar 

  37. Robinson S.M. (1981). Some continuity properties of polyhedral multifunctions. Math. Program. Study 14: 206–214

    MATH  Google Scholar 

  38. Rockafellar R.T. and Wets R.J.-B. (1998). Variational Analysis. Springer, Berlin

    MATH  Google Scholar 

  39. Römisch W. (1979). Kennwertmethoden für stochastische Volterrasche Integralgleichungen. Wiss. Zeitschr. Humb.-Univ. Berlin, Math.-Nat. R. XXVIII: 523–533

    Google Scholar 

  40. Römisch W. (1981). On the approximate solution of random operator equations. Wiss. Zeitschr. Humb.-Univ. Berlin, Math.-Nat. R. XXX: 455–462

    Google Scholar 

  41. Römisch W. (2003). Stability of stochastic programming problems. In: Ruszczyński, A. and Shapiro, A. (eds) Stochastic Programming, Handbooks in Operations Research and Management Science, vol 10, pp 483–554. Elsevier, Amsterdam

    Google Scholar 

  42. Römisch W. and Schultz R. (1991). Stability analysis for stochastic programs. Ann. Oper. Res. 30: 241–266

    Article  MATH  MathSciNet  Google Scholar 

  43. Ruszczyński A. and Shapiro A. (eds) (2003) Stochastic Programming, Handbooks in Operations Research and Management Science, vol 10. Elsevier, Amsterdam

    Google Scholar 

  44. Shapiro A. (2003). Inference of statistical bounds for multistage stochastic programming problems. Math. Meth. Oper. Res. 58: 57–68

    Article  MATH  Google Scholar 

  45. Shapiro A. (2006). On complexity of multistage stochastic programs. Oper. Res. Lett. 34: 1–8

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Humboldt-University Berlin, 10099, Berlin, Germany

    Holger Heitsch & Werner Römisch

Authors
  1. Holger Heitsch
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Werner Römisch
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Werner Römisch.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Heitsch, H., Römisch, W. Scenario tree modeling for multistage stochastic programs. Math. Program. 118, 371–406 (2009). https://doi.org/10.1007/s10107-007-0197-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-007-0197-2

Keywords

Mathematics Subject Classification (2000)

Search

Navigation