Skip to main content
Log in

Nested decomposition of multistage nonlinear programs with recourse

  • Published:
Mathematical Programming Submit manuscript

Abstract

Nested decomposition is extended to the case of arborescent nonlinear programs. Duals of extensive forms of nonlinear multistage stochastic programs constitute a particular class of those problems; the method is tested on a set of problems of that type.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  • D. Ament, J.K. Ho, E. Loute and M. Remmelswaal, “LIFT: A nested decomposition algorithm for solving lower block triangular linear programs,” in: G.B. Dantzig, M.A.H. Dempster and M.J. Kallio, eds.,Large Scale Programming (IIASA, Laxenburg, 1981) pp. 383–408.

    Google Scholar 

  • J.F. Benders, “Partitioning procedures for solving mixed variables programming problems,”Numerische Mathematik 1 (1982) 238–252.

    MathSciNet  Google Scholar 

  • J.R. Birge, “Decomposition and partitioning methods for multi-stage stochastic linear programs,”Operations Research 33 (1985) 989–1007.

    MATH  MathSciNet  Google Scholar 

  • G.B. Dantzig and A. Madansky, “On the solution of two-stage linear programs under uncertainty,” in:Proceedings, 4th Berkeley Symposium on Mathematical Statistics and Probability (UC Press, Berkeley, 1961).

    Google Scholar 

  • G.B. Dantzig and P. Wolfe, “The decomposition principle for linear programs,”Operations Research 8 (1960) 101–111.

    Article  MATH  Google Scholar 

  • G.B. Dantzig,Linear Programming and Extensions (Princeton University Press, Princeton, 1963).

    MATH  Google Scholar 

  • X. de Groote, M.-C. Noël and Y. Smeers, “Some test problems for stochastic nonlinear multistage programs,” to appear in: Y. Ermoliev and R. Wets, eds.,Numerical Techniques for Stochastic Optimization Problems (IIASA, Laxenburg, 1986).

    Google Scholar 

  • J.K. Ho and A.S. Manne, “Nested decomposition for dynamic models,”Mathematical Programming 6 (1974) 121–140.

    Article  MATH  MathSciNet  Google Scholar 

  • J.K. Ho, Nested decomposition of large scale linear programs with the staircase structure,” Ph.D. Dissertation, Stanford University (Stanford, 1974).

  • J.K. Ho and E. Loute, “Computational experience with advanced implementation of decomposition algorithms for linear programming”,Mathematical Programming 27 (1983) 283–290.

    MATH  Google Scholar 

  • M. Kallio and E.L. Porteus, “Decomposition of arborescent linear programs,”Mathematical Programming 13(1977) 348–356.

    Article  MATH  MathSciNet  Google Scholar 

  • A.S. Manne, “ETA-MACRO: A model of energy-economy interactions,” Research Project 1014, Department of Operations Research, Stanford University, Stanford, California 94305 (1977).

    Google Scholar 

  • A.S. Manne, M.A. Beltramo, T.F. Rutherford, A.N. Svoronos and T.F. Wilson, “ETA-MACRO: A progress report,” Research Project 1014, Department of Operations Research, Stanford University, Stanford, California 94305 (1983).

    Google Scholar 

  • A.B. Murtagh and M.A. Saunders, “Minos: A large-scale nonlinear programming system,” User's guide. Technical Report 77-9, Stanford University, Department of Operations Research (1977).

  • M.-C. Noël and Y. Smeers, “On the use of nested decomposition for solving non-linear multistage stochastic programs,” in: F. Archetti, D. Di Pillo and M. Lucertini, eds.,Stochastic Programming (Springer-Verlag, Berlin, 1985) pp. 235–245.

    Chapter  Google Scholar 

  • P. Olsen, “Multistage stochastic programming with recourse: the equivalent deterministic problem,”SIAM Journal on Control and Optimization 14 (876) 495–517.

  • R.P. O'Neill, “Nested decomposition of multistage convex programs,”SIAM Journal on Control and Optimization 14 (1976) 409–418.

    Article  MATH  MathSciNet  Google Scholar 

  • R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, 1970).

    MATH  Google Scholar 

  • R.M. Van Slyke and R. Wets, “L-shaped linear programs with applications to optimal control and stochastic programming,”SIAM Journal on Applied Mathematics 17 (1969) 638–663.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Noël, MC., Smeers, Y. Nested decomposition of multistage nonlinear programs with recourse. Mathematical Programming 37, 131–152 (1987). https://doi.org/10.1007/BF02591691

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02591691

Key words

Navigation