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Global Minimization of Separable Concave Functions under Linear Constraints with Totally Unimodular Matrices

  • Reiner Horst
  • Nguyen Van Thoai
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 7)

Abstract

Two types of new finite branch and bound algorithms are proposed for global minimization of a separable concave function under linear constraints with a totally unimodular matrix and additional box constraints. The key idea for establishing these algorithms is based upon the fact that the underlying problem can be viewed as an integer global optimization problem. For the case that a fixed number of the components of the objective function is nonlinear, an upper bound for the running time is given, which is polynomial in the data of the box constraints.

Keywords

Global optimization integer global optimization separable programming concave minimization branch and bound unimodularity 

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References

  1. 1.
    Ahuja, R.K., Magnanti, T.L., and Orlin, J.B., “Network Flows,” Prentice-Hall, New Jersey, 1993.zbMATHGoogle Scholar
  2. 2.
    Benson, H.P., “Concave Minimization: Theory, Applications and Algorithms”, in: Handbook of Global Optimization, Edited by R. Horst and P.M. Pardalos, Kluwer Academic Publishers, Dordrecht, 1995, 43 - 148.Google Scholar
  3. 3.
    Benson, H.P., Erenguc, S.S., and Horst, R., “A Note on Adapting Methods for Continuous Global Optimization to the Discrete Case”, Annals of Operations Research 25, 243–252, 1990.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Falk, I.E., and Soland, R.M., “An Algorithm for Separable Nonconvex Programming Problems”, Management Science 15, 550–569, 1969.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Guisewite, G.M., “Network Problems”, in: Handbook of Global Optimization, Edited by R. Horst and P.M. Pardalos, Kluwer Academic Publishers, Dordrecht, 1995, 609–648.Google Scholar
  6. 6.
    Guisewite, G.M., and Pardalos, P.M., “Minimum Concave-Cost Network Flow Problems: Applications, Complexity, and Algorithms”, Annals of Operations Research 25, 75–100, 1990.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Guisewite, G.M., and Pardalos, P.M., “Algorithms for the Single-Source Uncapacitated Minimum Concave-Cost Network Flow Problem”, Journal of Global Optimization 3, 245–266, 1991.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Guisewite, G.M., and Pardalos, P.M., “A Polynomial Time Solvable Concave Network Flow Problem”, Networks 23, 143–147, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Horst, R., and Thoai, N.V., “A Decomposition Approach for the Global Minimization of Biconcave Functions over Polytopes”, Research Report Nr. 93–23, University of Trier, 1993, forthcoming in Journal of Optimization Theory and Applications.Google Scholar
  10. 10.
    Horst, R., and Thoai, N.V., “An Integer Concave Minimization Approach for the Minimum Concave Cost Capacitated Flow Problem on Networks”, Research Report Nr. 94–13, University of Trier, 1994.Google Scholar
  11. 11.
    Horst, R, Pardalos, P.M., and Thoai, N.V., “Introduction to Gloabal Optimization”, Kluwer Academic Publishers, 1995.Google Scholar
  12. 12.
    Horst, R., and Tuy, H., “Global Optimization: Deterministic Approaches”, 2nd revised edition, Springer-Verlag, Berlin, 1993.Google Scholar
  13. 13.
    Klinz, B., and Tuy, H., “Minimum Concave Cost Network Flow Problem with a Single Nonlinear Arc Cost” in: Network Optimization Problems, Edited by D.Z. Du and P.M. Pardalos, World Scientific, Singapore, 1993.Google Scholar
  14. 14.
    Li, J., and Pardalos, P.M., “Integer Separable Programming Problems with a Unimodular Constraint Matrix”, Technical Report, University of Florida, 1995.Google Scholar
  15. 15.
    Pardalos, P.M, and Rosen, J.B., “Constrained Global Optimization: Algorithms and Applications”, Lecture Notes in Computer Science, 268, Springer-Verlag, Berlin, 1987.Google Scholar
  16. 16.
    Soland, R.M., “Optimal Facility Location with Concave Costs”, Operations Research 22, 373–382, 1974.Google Scholar
  17. 17.
    Thoai, N.V ., “Global Optimization Techniques for Solving the General Quadratic Integer Programming Problem”, University of Trier, Research Report, Nr. 94–09, 1994.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Reiner Horst
    • 1
  • Nguyen Van Thoai
    • 1
  1. 1.Department of MathematicsUniversity of TrierTrierGermany

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