Journal of Global Optimization

, Volume 3, Issue 3, pp 377–391 | Cite as

Combined branch-and-bound and cutting plane methods for solving a class of nonlinear programming problems

  • LÊ D. Muu
  • W. Oettli


We propose unified branch-and-bound and cutting plane algorithms for global minimization of a functionf(x, y) over a certain closed set. By formulating the problem in terms of two groups of variables and two groups of constraints we obtain new relaxation bounding and adaptive branching operations. The branching operation takes place in y-space only and uses the iteration points obtained through the bounding operation. The cutting is performed in parallel with the branch-and-bound procedure. The method can be applied implementably for a certain class of nonconvex programming problems.

Key words

Branch-and-bound cutting plane decomposition convex-concave function global optimization 


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  1. 1.
    Al-Khayyal, F. A. and Falk, J. E. (1983), Jointly constrained biconvex programming,Mathematics of Operations Research 8, 273–286.Google Scholar
  2. 2.
    Benders, J. F. (1962), Partitioning procedures for solving mixed-variables programming problems,Numerische Mathematik 4, 238–252.Google Scholar
  3. 3.
    Benson, H. P. (1982), On the convergence of two branch-and-bound algorithms for nonconvex programming problems,Journal of Optimization Theory and Applications 36, 129–134.Google Scholar
  4. 4.
    Blum, E. and Oettli, W. (1975),Mathematische Optimierung, Springer, Berlin.Google Scholar
  5. 5.
    Cheney, E. W. and Goldstein, A. A. (1959), Newton's method for convex programming and Tchebycheff approximation,Numerische Mathematik 1, 253–268.Google Scholar
  6. 6.
    Dantzig, G. B. and Wolfe, P. (1960), Decomposition principle for linear programs,Operations Research 8, 101–111.Google Scholar
  7. 7.
    Falk, J. E. and Soland, R. M. (1969), An algorithm for separable nonconvex programming problems,Management Science 15, 550–569.Google Scholar
  8. 8.
    Fukushima, M. (1983), An outer approximation algorithm for solving general convex programs,Operations Research 31, 101–113.Google Scholar
  9. 9.
    Fukushima, M. (1984), On the convergence of a class of outer approximation algorithms for convex programs,Journal of Computational and Applied Mathematics 10, 147–156.Google Scholar
  10. 10.
    Fukushima, M. (1984), A descent algorithm for nonsmooth convex optimization,Mathematical Programming 30, 163–175.Google Scholar
  11. 11.
    Geoffrion, A. (1970), Elements of large-scale mathematical programming, Part I: Concepts,Management Science 16, 652–675.Google Scholar
  12. 12.
    Horst, R. (1976), An algorithm for nonconvex programming problems,Mathematical Programming 10, 312–321.Google Scholar
  13. 13.
    Horst, R., de Vries, J., and Thoai, N. V. (1988), On finding new vertices and redundant constraints in cutting plane algorithms for global optimization,Operations Research Letters 7, 85–90.Google Scholar
  14. 14.
    Horst, R. (1988), Deterministic global optimization with partition sets whose feasibility is not known. Application to concave minimization, reverse convex constraints, DC-programming, and Lipschitzian optimization,Journal of Optimization Theory and Applications 58, 11–37.Google Scholar
  15. 15.
    Horst, R. and Tuy, H. (1987), On the convergence of global methods in multiextremal optimization,Journal of Optimization Theory and Applications 54, 253–271.Google Scholar
  16. 16.
    Horst, R. and Tuy, H. (1990),Global Optimization, Springer, Berlin; second edition 1992.Google Scholar
  17. 17.
    Kelley, J. E. (1960), The cutting-plane method for solving convex programs,Journal of the SIAM 8, 703–712.Google Scholar
  18. 18.
    Kiwiel, K. C. (1985),Methods of Descent for Nondifferentiable Optimization, Lecture Notes in Mathematics1133, Springer, Berlin.Google Scholar
  19. 19.
    Mayne, D. Q. and Polak, E. (1984), Outer approximation algorithm for nondifferentiable optimization problems,Journal of Optimization Theory and Applications 42, 19–30.Google Scholar
  20. 20.
    McCormick, G. P. (1976), Computability of global solutions to factorable nonconvex programs: Part I—Convex underestimating problems,Mathematical Programming 10, 147–175.Google Scholar
  21. 21.
    Muu, L. D. (1985), A convergent algorithm for solving linear programs with an additional reverse convex constraint,Kybernetika (Prague) 21, 428–435.Google Scholar
  22. 22.
    Muu, L. D. and Oettli, W. (1991), An algorithm for indefinite quadratic programming with convex constraints,Operations Research Letters 10, 323–327.Google Scholar
  23. 23.
    Muu, L. D. and Oettli, W. (1991), A method for minimizing a convex-concave function over a convex set,Journal of Optimization Theory and Applications 70, 377–384.Google Scholar
  24. 24.
    Pardalos, P. M. (1987), Generation of large-scale quadratic programs for use as global optimization test problems,ACM Transactions on Mathematical Software 13, 133–137.Google Scholar
  25. 25.
    Pardalos, P. M. and Rosen, J. B. (1987),Constrained Global Optimization: Algorithms and Applications. Lecture Notes in Computer Science268.Google Scholar
  26. 26.
    Rosen, J. B. and Pardalos, P. M. (1986), Global minimization of large-scale constrained concave quadratic problems by separable programming,Mathematical Programming 34, 163–174.Google Scholar
  27. 27.
    Soland, R. M. (1971), An algorithm for separable nonconvex programming problems II: Nonconvex constraints,Management Science 17, 759–773.Google Scholar
  28. 28.
    Topkis, D. M. (1970), Cutting-plane methods without nested constraint sets,Operations Research 18, 404–413.Google Scholar
  29. 29.
    Thoai, N. V. and Tuy, H. (1980), Convergent algorithms for minimizing a concave function,Mathematics of Operations Research 5, 556–566.Google Scholar
  30. 30.
    Tuy, H. (1987), Global minimization of a difference of two convex functions,Mathematical Programming Study 30, 150–182.Google Scholar
  31. 31.
    Tuy, H. and Horst, R. (1988), Convergence and restart in branch-and-bound algorithms for global optimization. Application to concave minimization and d.c. optimization problems,Mathematical Programming 41, 161–183.Google Scholar
  32. 32.
    Tuy, H., Thieu, T. V., and Thai, N. Q. (1985), A conical algorithm for globally minimizing a concave function over a closed convex set,Mathematics of Operations Research 10, 498–514.Google Scholar
  33. 33.
    Veinott, A. F. (1967), The supporting hyperplane method for unimodal programming,Operations Research 15, 147–152.Google Scholar

Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • LÊ D. Muu
    • 1
  • W. Oettli
    • 1
  1. 1.UniversitÄt MannheimMannheimGermany

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