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Journal of Optimization Theory and Applications

, Volume 10, Issue 4, pp 237–260 | Cite as

Generalized Benders decomposition

  • A. M. Geoffrion
Contributed Papers

Abstract

J. F. Benders devised a clever approach for exploiting the structure of mathematical programming problems withcomplicating variables (variables which, when temporarily fixed, render the remaining optimization problem considerably more tractable). For the class of problems specifically considered by Benders, fixing the values of the complicating variables reduces the given problem to an ordinary linear program, parameterized, of course, by the value of the complicating variables vector. The algorithm he proposed for finding the optimal value of this vector employs a cutting-plane approach for building up adequate representations of (i) the extremal value of the linear program as a function of the parameterizing vector and (ii) the set of values of the parameterizing vector for which the linear program is feasible. Linear programming duality theory was employed to derive the natural families ofcuts characterizing these representations, and the parameterized linear program itself is used to generate what are usuallydeepest cuts for building up the representations.

In this paper, Benders' approach is generalized to a broader class of programs in which the parametrized subproblem need no longer be a linear program. Nonlinear convex duality theory is employed to derive the natural families of cuts corresponding to those in Benders' case. The conditions under which such a generalization is possible and appropriate are examined in detail. An illustrative specialization is made to the variable factor programming problem introduced by R. Wilson, where it offers an especially attractive approach. Preliminary computational experience is given.

Keywords

Programming Problem Duality Theory Parameterized Linear Bender Decomposition Mathematical Programming Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Geoffrion, A. M.,Elements of Large-Scale Mathematical Programming, Management Science, Vol. 16, No. 11, 1970.Google Scholar
  2. 2.
    Benders, J. F.,Partitioning Procedures for Solving Mixed-Variables Programming Problems, Numerische Mathematik, Vol. 4, 1962.Google Scholar
  3. 3.
    Wilson, R. Programming Variable Factors, Management Science, Vol. 13, No. 1, 1966.Google Scholar
  4. 4.
    Balas, E.,Duality in Discrete Programming: IV. Applications, Carnegie-Mellon University, Graduate School of Industrial Administration, Report No. 145, 1968.Google Scholar
  5. 5.
    Meyer, R.,The Validity of a Family of Optimization Methods, SIAM Journal on Control, Vol. 8, No. 1, 1970.Google Scholar
  6. 6.
    Dantzig, G. B.,Linear Programming and Extensions, Princeton University Press, Princeton, New Jersey, 1963.Google Scholar
  7. 7.
    Geoffrion, A. M.,Primal Resource-Directive Approaches for Optimizing Nonlinear Decomposable Systems, Operations Research, Vol. 18, No. 3, 1970.Google Scholar
  8. 8.
    Hogan, W.,Application of a General Convergence Theory for Outer Approximation Algorithms, University of California at Los Angeles, Western Management Science Institute, Working Paper No. 174, 1971.Google Scholar
  9. 9.
    Eaves, B. C., andZangwill, W. I.,Generalized Cutting Plane Algorithms, SIAM Journal on Control, Vol. 9, No. 4, 1971.Google Scholar
  10. 10.
    Geoffrion, A. M.,A New Global Optimization Technique for Gaseous Diffusion Plant Operation and Capital Investment, University of California at Los Angeles, Graduate School of Business Administration, Discussion Paper, 1970.Google Scholar
  11. 11.
    Geoffrion, A. M.,Duality in Nonlinear Programming: A Simplified Application-Oriented Development, SIAM Review, Vol. 13, No. 1, 1971.Google Scholar

Copyright information

© Plenum Publishing Corporation 1972

Authors and Affiliations

  • A. M. Geoffrion
    • 1
  1. 1.University of California at Los AngelesLos Angeles

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