Numerical Techniques in Applied Multistage Stochastic Programming

  • Karl Frauendorfer
  • Gido Haarbrücker
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 59)


This contribution deals with the apparent difficulties when solving an optimization problem with random influences by the use of multistage stochastic linear programming. It names specific numerical solution techniques which are suitable for coping with the curse of dimensionality, with increasing scenario trees and associated large-scale LPs. The focus lies on classic decomposition methods which are natural candidates to apply parallelization techniques. In addition, an alternative approach is sketched where the replacement of optimization runs by optimality checks leads to an efficient handling of consecutive discretization steps given some structural requirements arc fulfilled.


Multistage stochastic linear programming discretization large-scale linear program numerical techniques 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Birge, J. R. and Wallace, S. W.: A separable piecewise linear upper bound for stochastic linear programs, SIAM J. Control Optim. 26 (1988), 725–739.CrossRefMathSciNetzbMATHGoogle Scholar
  2. [2]
    Birge, J. R., and Wets R. J.-B.: Sublinear upper bounds for stochastic programs with recourse, Math. Programming 43 (1989), 131–149.CrossRefMathSciNetzbMATHGoogle Scholar
  3. [3]
    Wallace, S. W.: Solving stochastic programs with network recourse. Networks 16 (1986), 295–317.MathSciNetzbMATHGoogle Scholar
  4. [4]
    Birge, J. R. and Qi, L.: Continuous approximation schemes for stochastic progams, Ann. Oper. Res. 56 (1995), 15–38.CrossRefMathSciNetzbMATHGoogle Scholar
  5. [5]
    Birge, J. R. and Louveaux, F.: Introduction to Stochastic Programming, Springer-Verlag, New York, 1997.zbMATHGoogle Scholar
  6. [6]
    Kall, P. and Wallace, S. W.: Stochastic Programming, Wiley, Chichester, 1994.zbMATHGoogle Scholar
  7. [7]
    Wets, R. J.-B.: Stochastic programming: Solution techniques and approximation schemes, In: A. Bachem, M. Grötschel and B. Korte (eds), Mathematical Programming: The State-of-the-art 1982, Springer-Verlag, Berlin, 1983, pp. 566–603.Google Scholar
  8. [8]
    Rockafellar, R. T. and Wets, R. J.-B.: Nonanticipativity and L 1-martingales in stochastic optimization problems, Math. Programming Study 6 (1976), 170–187.MathSciNetGoogle Scholar
  9. [9]
    Wets, R. J.-B.: Stochastic programs with fixed recourse: The equivalent deterministic program, SIAM Rev. 16 (1974), 309–339.MathSciNetzbMATHGoogle Scholar
  10. [10]
    Wets, R. J.-B.: Stochastic programs with recourse: A basic theorem for multistage problems, Z. Wahrschein. verw. Geb. 21 (1972), 201–206.MathSciNetzbMATHGoogle Scholar
  11. [11]
    Birge, J. R. and Qi, L.: Computing block-angular Karmarkar projections with applications to stochastic programmin, Management Sci. 34 (1988), 1472–1479.MathSciNetzbMATHGoogle Scholar
  12. [12]
    Jessup, E. R., Yang, D. and Zenios, S. A.: Parallel factorization of structured matrices arising in stochastic programming, SIAM J. Optim. 4 (1994), 833–846.CrossRefMathSciNetzbMATHGoogle Scholar
  13. [13]
    Czyzyk, J., Fourer, R. and Mehrotra, S.: Using a massively parallel processor to solve large sparse linear programs by an interior point method, SIAM J. Sci. Comput. 19 (1998), 553–565.CrossRefMathSciNetzbMATHGoogle Scholar
  14. [14]
    Benders, J.: Partitioning methods for solving mixed variables programming problems, Numer. Math. 4 (1962), 238–252.CrossRefMathSciNetzbMATHGoogle Scholar
  15. [15]
    Dantzig, G. B. and Wolfe, P.: The decomposition principle for linear programs, Oper. Res. 8 (1960), 101–111.zbMATHGoogle Scholar
  16. [16]
    Van Slyke, R. M. and Wets, R. J.-B.: L-shaped linear programs with application to optimal control and stochastic programming, SIAM J. Appl. Math. 17 (1969), 638–663.MathSciNetzbMATHGoogle Scholar
  17. [17]
    Ruszczyński, A.: A regularized decomposition method for minimizing a sum of polyhedral functions, Math. Programming 35 (1986), 309–333.MathSciNetzbMATHGoogle Scholar
  18. [18]
    Birge, J. R.: Using sequential approximations in the L-shaped and generalized programming algorithms for stochastic linear programs, Technical Report 83-12, Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI, 1983.Google Scholar
  19. [19]
    Dantzig, G. B. and Glynn, P.: Parallel processors for planning under uncertainty, Ann. Oper. Res. 22 (1990), 1–21.CrossRefMathSciNetzbMATHGoogle Scholar
  20. [20]
    Dantzig, G. B. and Infanger, G.: Large-scale stochastic linear programs — Importance sampling and Benders decomposition, In: Computational and Applied Mathematics, I. Algorithms and Theory, Sel. Rev. Pap. IMACS 13th World Congr., Dublin/Irel. 1991, 1992, pp. 111–120.Google Scholar
  21. [21]
    Higle, J. and Sen, S.: Stochastic decomposition: An algorithm for two stage linear programs with recourse, Math. Oper. Res. 16 (1991), 650–669.MathSciNetzbMATHGoogle Scholar
  22. [22]
    Louveaux, F.: A solution method for multistage stochastic programs with recourse with application to an energy investment problem, Oper. Res. 28 (1980), 889–902.MathSciNetzbMATHGoogle Scholar
  23. [23]
    Birge, J. R.: Decomposition and partitioning methods for multistage stochastic linear programs, Oper. Res. 33 (1985), 989–1007.MathSciNetzbMATHGoogle Scholar
  24. [24]
    Wittrock, R. J.: Advances in a nested decomposition algorithm for solving staircase linear programs, Technical Report SOL 83-2, Systems Optimization Laboratory, Stanford University, Stanford, CA, 1983.Google Scholar
  25. [25]
    Gassmann, H.: MSLIP, a computer code for the multistage stochastic linear programming problem, Math. Programming 47 (1990), 407–423.CrossRefMathSciNetzbMATHGoogle Scholar
  26. [26]
    Morton, D. P.: An enhanced decomposition algorithm for multistage stochastic hydroelectric scheduling, Technical Report NPSOR-94-001, Department of Operations Research, Naval Postgraduate School, Monterey, CA, 1994.Google Scholar
  27. [27]
    Ruszczyński, A.: An augmented Lagrangian decomposition method for block diagonal linear programming problems, Oper. Res. Lett. 8 (1989), 287–294.MathSciNetGoogle Scholar
  28. [28]
    Rosa, C. and Ruszczyński, A.: On augmented Lagrangian decomposition methods for multistage stochastic programs, Working Paper WP-94-125, IIASA International Institute for Applied Systems Analysis, Laxenburg, Austria, 1994.Google Scholar
  29. [29]
    Mulvey, J. M. and Ruszczyński, A.: A new scenario decomposition method for large-scale stochastic optimization, Oper. Res. 43(3) (1995), 333–353.CrossRefGoogle Scholar
  30. [30]
    Ruszczyński, A.: Parallel decomposition of multistage stochastic programming problems, Math. Programming 58 (1993), 201–228.MathSciNetzbMATHGoogle Scholar
  31. [31]
    Gondzio, J. and Kouwenberg, R.: High performance computing for asset liability management, Preprint MS-99-004, Department of Mathematics & Statistics, The University of Edinburgh, UK, 1999.Google Scholar
  32. [32]
    Gondzio, J.: HOPDM (version 2.12) — A fast LP solver based on a primal-dual interior point method, Europ. J. Oper. Res. 85 (1955), 221–225.Google Scholar
  33. [33]
    Frauendorfer, K.: Stochastic Two-Stage Programming, Lecture Notes in Econom. Math. Systems 392, Springer-Verlag, Berlin, 1992.zbMATHGoogle Scholar
  34. [34]
    Frauendorfer, K.: Multistage stochastic programming: Error analysis for the convex case, Z. Oper. Res. 39(1) (1994), 93–122.MathSciNetzbMATHGoogle Scholar
  35. [35]
    Frauendorfer, K.: Barycentric scenario trees in convex multistage stochastic programming, Math. Programming 75(2) (1996), 277–294.MathSciNetzbMATHGoogle Scholar
  36. [36]
    Marohn, C.: Stochastische mehrstufige lineare Programmierung im Asset & Liability Management, Bank-und finanzwirtschaftliche Forschungen, Bd. 282, Paul Haupt Verlag, Bern, 1998.Google Scholar
  37. [37]
    Haarbrücker, G.: Sequentielle Optimierung verfeinerter Approximationen in der mehrstufigen stochastischen linearen Programmierung, Doctoral thesis No. 2410, University of St. Gallen, 2000.Google Scholar
  38. [38]
    Frauendorfer, K. and Haarbrücker, G.: Solving sequences of refined multistage stochastic linear programs, Submitted for appearance in Ann. Oper. Res. Google Scholar
  39. [39]
    Yang, D. and Zenios, S. A.: A scalable parallel interior point algorithm for stochastic linear programming and robust optimization, Comput. Optim. Appl. 7(1) (1997), 143–158.CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Karl Frauendorfer
    • 1
  • Gido Haarbrücker
    • 1
  1. 1.Institute for Operations ResearchUniversity of St. GallenSwitzerland

Personalised recommendations