Global Optimization Techniques for Solving the General Quadratic Integer Programming Problem

  • Nguyen Van Thoai
Article

Abstract

We consider the problem of minimizing a general quadratic function over a polytope in the n-dimensional space with integrality restrictions on all of the variables. (This class of problems contains, e.g., the quadratic 0-1 program as a special case.) A finite branch and bound algorithm is established, in which the branching procedure is the so-called “integral rectangular partition”, and the bound estimation is performed by solving a concave programming problem with a special structure. Three methods for solving this special concave program are proposed.

quadratic integer programming branch and bound algorithms concave minimization global optimization 

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References

  1. 1.
    D. Avis and K. Fukuda, "A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra," Discrete and Computational Geometry, pp. 295-313, 1992.Google Scholar
  2. 2.
    F. Barahona, "A solvable case for quadratic 0-1 programming," Discrete Applied Mathematics, vol. 13, pp. 23-26, 1986.Google Scholar
  3. 3.
    F. Barahona, M. Junger, and F. Reinelt, "Experiments in quadratic 0-1 programming," Mathematical Programming, vol. 44., pp. 127–137, 1989.Google Scholar
  4. 4.
    H.P. Benson, S.S. Erenguc, and R. Horst, "A note on adapting methods for continuos global optimization to the discrete case," Annals of Operations Research, vol. 25, pp. 243-252, 1990.Google Scholar
  5. 5.
    R.E. Burkard, "Locations with spatial interaction: The quadratic assinment problem," in Discrete Location Theory, P.B. Mirchandani and R.L. Francis (Eds.), John Wiley, 1991.Google Scholar
  6. 6.
    P.C. Chen, P. Hansen, and B. Jaumard, "On-line and off-line vertex enumeration by adjacency lists," Operations Research Letters, vol. 10, pp. 403-409, 1991.Google Scholar
  7. 7.
    R.J. Dakin, "A tree search algorithm for mixed integer programming problems," Computer Journal, vol. 8, pp. 250-255, 1965.Google Scholar
  8. 8.
    J.E. Falk and K.L. Hoffman, "A Successive underestimation method for concave minimization problems," Mathematics of Operations Research, vol. 1, pp. 251-259, 1976.Google Scholar
  9. 9.
    P. Gritzmann and V. Klee, "On the 0-1 maximization of positive definite quadratic forms," in Operations Research Proceedings, K.P. Kistner et al. (Eds.), Springer: Berlin, 1989, pp. 222-227.Google Scholar
  10. 10.
    P. Hansen, B. Jaumard, and V. Mathon, “Constrained nonlinear 0-1 programming,” Preprint G 89-38, Gerard-Montreal, 1989.Google Scholar
  11. 11.
    R. Horst, N.V. Thoai, and H. Tuy, "Outer approximation by polyhedral convex sets," Operations Research Spektrum, vol. 9, pp. 153-159, 1987.Google Scholar
  12. 12.
    R. Horst, N.V. Thoai, and J. de Vives, "On finding new vertices and redundant constraints in cutting plane algorithms for global optimization," Operations Research Letters, vol. 7, pp. 85-99, 1998.Google Scholar
  13. 13.
    R. Horst and P.T. Thach, "A decomposition method for quadratic minimization problems with integer variables," in Advances in Optimization and Parallel Computing, P.M. Pardalos (Ed.), Elsevier Science Publishers: Amsterdam, 1992, pp. 143-163.Google Scholar
  14. 14.
    R. Horst and H. Tuy, Global Optimization: Deterministic Approaches, 2nd revised edition, Springer-Verlag: Berlin, 1993.Google Scholar
  15. 15.
    R. Horst, P.M. Pardalos, and N.V. Thoai, Introduction to Global Optimization, Kluwer Academic Publishers: Dordrecht, 1995.Google Scholar
  16. 16.
    A. Kamath and N. Karmarkar, "A continuous approach to compute upper bounds in quadratic maximization problems with integer constraints," in Recent Advances in Global Optimization, C. Floudas and P. Pardalos (Eds.), Princeton University Press: Princeton, 1992, pp. 125-140.Google Scholar
  17. 17.
    J.E. Kelley, "The cutting-plane method for solving convex programs," Journal SIAM, vol. 8, pp. 703-712, 1960.Google Scholar
  18. 18.
    H. Konno, "Maximization of a convex quadratic function over a hypercube," J. of the Operations Research Society of Japan, vol. 23, pp. 171-189, 1980.Google Scholar
  19. 19.
    P.M. Pardalos and J.B. Rosen, "Constrained global optimization: Algorithms and applications," Lecture Notes in Computer Science, Springer-Verlag: Berlin, 1987, vol. 268.Google Scholar
  20. 20.
    P.M. Pardalos and G. Rodgers, "Parallel branch and bound algorithms for quadratic 0-1 programming on a hypercube architecture," Annals of Operations Research, vol. 22, pp. 271–292, 1990.Google Scholar
  21. 21.
    P.M. Pardalos, F. Rendl, and H. Wolkowicz, "The Quadratic assignment problem: A survey and recent developments," DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 16, pp. 1-39, 1994.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

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

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