Decomposition approach for the global minimization of biconcave functions over polytopes

  • R. Horst
  • N. V. Thoai
Contributed Papers


A decomposition approach is proposed for minimizing biconcave functions over polytopes. Important special cases include concave minimization, bilinear and indefinite quadratic programming for which new algorithms result. The approach introduces a new polyhedral partition and combines branch-and-bound techniques, outer approximation, and projection of polytopes in a suitable way.

Key Words

Global optimization biconcave programming concave minimization bilinear and quadratic programming branch-and-bound algorithms outer approximations. 


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  1. 1.
    Al-Khayyal, F. A.,Constrained Bilinear Programming and Jointly Related Problems, Report J-83-3, Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia, 1983.Google Scholar
  2. 2.
    Horst, R., andTuy, H.,On the Convergence of Global Methods in Multiextremal Optimization, Journal of Optimization Theory and Applications, Vol. 54, pp. 253–271, 1987.Google Scholar
  3. 3.
    Horst, R., andTuy, H.,Global Optimization: Deterministic Approaches 2nd Revised Edition, Springer, Berlin, Germany, 1993.Google Scholar
  4. 4.
    Thoai, N. V., andTuy, H.,Convergent Algorithms for Minimizing a Concave Function, Mathematics of Operations Research, Vol. 5, pp. 556–566, 1980.Google Scholar
  5. 5.
    Horst, R., andTuy, H.,Convergence and Restart in Branch-and-Bound Algorithms for Global Optimization: Application to Concave Minimization and DC-Optimization Problems, Mathematical Programming, Vol. 41, pp. 161–183, 1988.Google Scholar
  6. 6.
    Horst, R.,An Algorithm for Nonconvex Programming Problems, Mathematical Programming, Vol. 10, pp. 312–321, 1976.Google Scholar
  7. 7.
    Benson, H. P.,Concave Minimization, Handbook of Global Optimization, Edited by R. Horst and P. Pardalos, Kluwer Academic Publishers, Dordrecht, Netherlands, pp. 43–148, 1995.Google Scholar
  8. 8.
    Horst, R., Thoai, N. V., andDe Vries, J.,On Finding New Vertices and Redundant Constraints in Cutting-Plane Algorithms for Global Optimization, Operations Research Letters, Vol. 7, pp. 85–90, 1988.Google Scholar
  9. 9.
    Chen, P. C., Hansen, P., andJaumard, B.,On-Line and Off-Line Vertex Enumeration by Adjacency Lists, Operations Research Letters, Vol. 10, pp. 403–409, 1991.Google Scholar
  10. 10.
    Tuy, H.,Concave Minimization under Linear Constraints with Special Structure, Optimization, Vol. 16, pp. 335–352, 1985.Google Scholar
  11. 11.
    Horst, R., andThoai, N. V.,Constraints Decomposition Algorithms in Global Optimization, Journal of Global Optimization, Vol. 5, pp. 333–348, 1994.Google Scholar
  12. 12.
    Thoai, N. V.,A Global Optimization Approach for Solving Convex Multiplicative Programming Problems, Journal of Global Optimization, Vol. 1, pp. 341–357, 1991.Google Scholar
  13. 13.
    Thoai, N. V.,Canonical DC-Programming Techniques for Solving a Convex Program with an Additional Constraint of Multiplicative Type, Computing, Vol. 50, pp. 241–253, 1993.Google Scholar
  14. 14.
    Horst, R., andThoai, N. V.,A New Algorithm for Solving the General Quadratic Problem, Computational Optimization and Applications Vol. 5, pp. 39–48, 1990.Google Scholar
  15. 15.
    Horst, R., andThoai, N. V.,Modification Implementation and Comparison of Three Algorithms for Globally Solving Linearly-Constrained Concave Minimization Problems, Computing, Vol. 42, pp. 271–289, 1989.Google Scholar
  16. 16.
    Horst, R., Thoai, N. V., andBenson, H. P.,Concave Minimization via Conical Partitions and Polyhedral Outer Approximation, Mathematical Programming, Vol. 50, pp. 259–274, 1991.Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • R. Horst
    • 1
  • N. V. Thoai
    • 2
  1. 1.Fachbereich IV, Department of MathematicsUniversity of TrierTrierGermany
  2. 2.Institute of MathematicsHanoiVietnam

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