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Decomposition Branch-and-Bound Based Algorithm for Linear Programs with Additional Multiplicative Constraints

  • H. P. Benson
Article

Abstract

This article presents an algorithm for globally solving a linear program (P) that contains several additional multiterm multiplicative constraints. To our knowledge, this is the first algorithm proposed to date for globally solving Problem (P). The algorithm decomposes the problem to obtain a master problem of low rank. To solve the master problem, the algorithm uses a branch-and-bound scheme where Lagrange duality theory is used to obtain the lower bounds. As a result, the lower-bounding subproblems in the algorithm are ordinary linear programs. Convergence of the algorithm is shown and a solved sample problem is given.

Keywords

Global optimization multiplicative programming branch-and-bound schemes decomposition 

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References

  1. 1.
    Tuy, H. 1998Convex Analysis and Global OptimizationKluwerDordrecht, NetherlandsGoogle Scholar
  2. 2.
    Thoai, N. V. 2002Convergence and Application of a Decomposition Method Using Duality Bounds for Nonconvex Global OptimizationJournal of Optimization Theory and Applications113165193CrossRefGoogle Scholar
  3. 3.
    Kuno, T., Konno, H., Irie, A. 1999A Deterministic Approach to Linear Programs with Several Additional Multiplicative ConstraintsComputational Optimization and Applications14347366CrossRefGoogle Scholar
  4. 4.
    Konno, H., Kuno, T. 1992Linear Multiplicative ProgrammingMathematical Programming565164CrossRefGoogle Scholar
  5. 5.
    Konno, H., Kuno, T. 1995Multiplicative Programming ProblemsHorst, R.Pardalos, P. M. eds. Handbook of Global OptimizationKluwerDordrecht, Netherlands369405Google Scholar
  6. 6.
    Konno, H., Gao, C., Saitoh, I. 1998Cutting Plane/Tabu Search Algorithms for Solving Low-Rank Concave Quadratic Programming ProblemsJournal of Global Optimization13225240CrossRefGoogle Scholar
  7. 7.
    Kuno, T., Utsunomiya, T. 1997A Pseudopolynomial Primal-Dual Algorithm for Globally Solving a Production-Transportation ProblemJournal of Global Optimization11163180CrossRefGoogle Scholar
  8. 8.
    Benders, J. F. 1962Partitioning Procedures for Solving Mixed-Variables Programming ProblemsNumerische Mathematik4238252CrossRefGoogle Scholar
  9. 9.
    Floudas, C. A., Visweswaran, V. 1993A Primal-Relaxed Dual Global Optimization ApproachJournal of Optimization Theory and Applications78187225CrossRefGoogle Scholar
  10. 10.
    Horst, R., Thoai, N. V. 1996Decomposition Approach for the Global Minimization of Biconcave Functions over PolytopesJournal of Optimization Theory and Applications88561583Google Scholar
  11. 11.
    Muu, L. D., Oettli, W. 1991Method for Minimizing Convex-Concave Functions over a Convex SetJournal of Optimization Theory and Applications70377384CrossRefGoogle Scholar
  12. 12.
    Thoai, N. V. 1991A Global Optimization Approach for Solving the Convex Multiplicative Programming ProblemJournal of Global Optimization1341357CrossRefGoogle Scholar
  13. 13.
    Benson, H. P., Boger, G. M. 1997Multiplicative Programming Problems: Analysis and Efficient Point Search HeuristicJournal of Optimization Theory and Applications94487510CrossRefGoogle Scholar
  14. 14.
    Benson, H. P. 1999An Outcome Space Branch-and-Bound–Outer–Approximation Algorithm for Convex Multiplicative ProgrammingJournal of Global Optimization15315342CrossRefGoogle Scholar
  15. 15.
    Benson, H. P., Boger, G. M. 2000An Outcome Space, Cutting Plane Algorithm for Linear Multiplicative ProgrammingJournal of Optimization Theory and Applications104301322CrossRefGoogle Scholar
  16. 16.
    Benson, H. P., Boger, G. M. 2000Analysis of an Outcome Space Formulation of the Multiplicative Programming ProblemZeleny, M.Shi, Y. eds. New Frontiers of Decision Making for the Information Technology EraWorld Scientific PublishingRiver Edge, New Jersey100122Google Scholar
  17. 17.
    Dur, M. 2001Dual Bounding Procedures Lead to Convergent Branch-and-Bound AlgorithmsMathematical Programming91117125Google Scholar
  18. 18.
    Dur, M. 2002A Class of Problems Where Dual Bounds Beat Underestimation BoundsJournal of Global Optimization224957CrossRefGoogle Scholar
  19. 19.
    Dur, M., Horst, R. 1997Lagrange-Duality and Partitioning Techniques in Nonconvex Global OptimizationJournal of Optimization Theory and Applications95347369CrossRefGoogle Scholar
  20. 20.
    Nowak, I. 2000Dual Bounds and Optimality Cuts for All-Quadratic Programs with Convex ConstraintsJournal of Global Optimizatiion18337356CrossRefGoogle Scholar
  21. 21.
    Thoai, N. V. 2000Duality Bound Method for the General Quadratic Programming Problem with Quadratic ConstraintsJournal of Optimization Theory and Applications107331354CrossRefGoogle Scholar
  22. 22.
    Horst, R., Tuy, H. 1996Global Optimization: Deterministic ApproachesSpringerBerlin, GermanyGoogle Scholar
  23. 23.
    Avriel, M., Diewert, W. E., Schaible, S., Zang, I. 1988Generalized ConcavityPlenumNew York, NYGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • H. P. Benson
    • 1
  1. 1.Professor, Department of Decision and Information SciencesUniversity of FloridaGainesvilleFlorida

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