Annals of Operations Research

, Volume 64, Issue 1, pp 39–65 | Cite as

Parallel decomposition of large-scale stochastic nonlinear programs

  • John R. Birge
  • Charles H. Rosa


Many practical decision problems involve both nonlinear relationships and uncertainties. The resulting stochastic nonlinear programs become quite difficult to solve as the number of possible scenarios increases. In this paper, we provide a decomposition method for problems in which nonlinear constraints appear within periods. We also show how the method extends to lower bounding refinements of the set of scenarios when the random data are independent from period to period. We then apply the method to a stochastic model of the U.S. economy based on the Global 2100 method developed by Manne and Richels.


Decomposition economics environment parallel computation stochastic programming 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J.F. Benders, Partitioning procedures for solving mixed-variables programming problems, Numerical Mathematics 4, 1962, 238–252.CrossRefGoogle Scholar
  2. [2]
    J.R. Birge, Decomposition and partitioning methods for multi-stage stochastic linear programs, Operations Research 33, 1985, 989–1007.Google Scholar
  3. [3]
    J.R. Birge and C.H. Rosa, Modeling investment uncertainty in the costs of global CO2 emission policy, European Journal of Operations Research, to be published.Google Scholar
  4. [4]
    J.R. Birge and R. J-B Wets, Designing approximation schemes for stochastic optimization problems, in particular, for stochastic programs with recourse, Mathematical Programming Study 27, 1986, 54–102.Google Scholar
  5. [5]
    A. Brooke, D. Kendrick and A. Meeraus,GAMS: A User's Guide, The Scientific Press, San Francisco, CA, 1992.Google Scholar
  6. [6]
    G. B. Dantzig and A. Madansky, On the solution of two-stage linear programs under uncertainty, inProceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, J. Neyman, ed., University of California Press, Berkeley, 1961, pp. 165–176.Google Scholar
  7. [7]
    R. Fourer, M.G. Gay and B.W. Kernighan,AMPL: A Modeling Language for Mathematical Programming, The Scientific Press, San Francisco, CA, 1993.Google Scholar
  8. [8]
    A.S. Manne, ETA-MACRO: a user's guide, EPRI Interim Report, Project 1014, 1981.Google Scholar
  9. [9]
    A.S. Manne and R.G. Richels, Global CO2 emission reductions — the impacts of rising energy costs, The Energy Journal 12, 1991, 87–107.Google Scholar
  10. [10]
    B.A. Murtagh and M.A. Saunders, MINOS 5.1 user's guide, report SOL-83-20R, Systems Optimization Library, Stanford, CA, 1987.Google Scholar
  11. [11]
    M.C. Noël and Y. Smeers, Nested decomposition of multistage nonlinear programs with recourse, Mathematical Programming 37, 1987, 131–152.Google Scholar
  12. [12]
    R.P. O'Neill, Nested decomposition of multistage convex programs, SIAM Journal on Control and Optimization 14, 1976, 409–418.CrossRefGoogle Scholar
  13. [13]
    M.V.E. Pereira and L.M.V.G. Pinto, Multi-stage stochastic optimization applied to energy planning, Mathematical Programming 52, 1991, 359–375.CrossRefMathSciNetGoogle Scholar
  14. [14]
    C.H. Rosa, Modeling investment uncertainty in the costs of global CO2 emission policy, Department of Industrial and Operations Engineering, University of Michigan, Ph.D. Thesis, Sept. 1993.Google Scholar
  15. [15]
    R. Van Slyke and R.J-B Wets, L-shaped linear programs with application to optimal control and stochastic programming, SIAM Journal on Applied Mathematics 17, 1969, 638–663.CrossRefGoogle Scholar
  16. [16]
    S.A. Vejtasa and B.L. Schumann, Technology data for carbon dioxide emission model: Global 2100, SFA Pacific Inc., Mountain View, CA, 1989.Google Scholar
  17. [17]
    S.W. Wallace and T.C. Yan, Bounding multi-stage stochastic programs from above, Mathematical Programming 61, 1993, 111–129.CrossRefGoogle Scholar

Copyright information

© J.C. Baltzer AG, Science Publishers 1996

Authors and Affiliations

  • John R. Birge
    • 1
  • Charles H. Rosa
    • 2
  1. 1.Department of Industrial and Operations EngineeringUniversity of MichiganAnn ArborUSA
  2. 2.International Institute for Applied Systems AnalysisLaxenburgAustria

Personalised recommendations