Mathematical Programming

, Volume 51, Issue 1–3, pp 229–245 | Cite as

Normal conical algorithm for concave minimization over polytopes

  • Hoang Tuy


A new conical algorithm is developed for finding the global minimum of a concave function over a polytope. To ensure faster convergence and overcome some major drawbacks of previous conical algorithms, a normal (rather than exhaustive) subdivision process is used.

Key words

Concave minimization conical algorithm convergence condition bisection normal subdivision process 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    S. Bali, “Minimization of a concave function on a bounded convex polyhedron,” Ph.D. Dissertation, University of California (Los Angeles, A, 1973).Google Scholar
  2. [2]
    V.P. Bulatov,Metody Pogrujeniia v Zadatchak Optimizatsii (Nauka, Sibirskoe Otdelenie, Novosibirsk, 1977).Google Scholar
  3. [3]
    M. Hamami and S.E. Jacobsen, “Exhaustive nondegenerate conical processes for concave minimization on convex polytopes,”Mathematics of Operations Research 13 (1988) 479–487.Google Scholar
  4. [4]
    R. Horst, “An algorithm for nonconvex programming problems,”Mathematical Programming 10 (1976) 312–321.Google Scholar
  5. [5]
    R. Horst, “On the global minimization of concave functions: Introduction and survey,”Operations Research Spektrum 6 (1984) 195–205.Google Scholar
  6. [6]
    R. Horst and Ng.V. Thoai, “Implementation, modification and comparison of some algorithms for concave minimization problems,” Preprint, Department of Mathematics, University of Trier (1988), to appear in:Computing. Google Scholar
  7. [7]
    S.E. Jacobsen, “Convergence of a Tuy-type algorithm for concave minimization subject to linear constraints,”Applied Mathematics and Optimization 7 (1981) 1–9.Google Scholar
  8. [8]
    P.M. Pardalos and J.B. Rosen, “Methods for global concave minimization: a bibliographic survey,”SIAM Review 26 (1986) 367–379.Google Scholar
  9. [9]
    Ng.V. Thoai and H. Tuy, “Convergent algorithm for minimizing a concave function,”Mathematics of Operations Research 5 (1980) 556–566.Google Scholar
  10. [10]
    Ng.V. Thoai and J. de Vries, “Numerical experiments on concave minimization algorithms,” to appear in:Methods of Operations Research. Google Scholar
  11. [11]
    H. Tuy, “Vognutoe programmirovaniie pri lineinyk ogranitchenyakh,”Doklady AN SSSR 158 (1964) 32–35. [Translated as: “Concave programming under linear constraints,”Soviet Mathematics 5 (1964) 1437–1440.]Google Scholar
  12. [12]
    H. Tuy, T.V. Thieu and Ng.Q. Thai, “A conical algorithm for globally minimizing a concave function over a closed convex set,”Mathematics of Operations Research 10 (1985) 498–514.Google Scholar
  13. [13]
    H. Tuy, V. Khachaturov and S. Utkin, “A class of exhaustive cone splitting procedures in conical algorithms for concave minimization,”Optimization 18 (1987) 791–808.Google Scholar
  14. [14]
    H. Tuy, “A general deterministic approach to global optimization via D.C. programming,” in: J.B. Hiriari-Urruty, ed.,Fermat Days 1985: Mathematics for Optimization. Mathematics Studies Series (North-Holland, Amsterdam, 1986).Google Scholar
  15. [15]
    H. Tuy, “On polyhedral annexation method for concave minimization,” submitted for publication.Google Scholar
  16. [16]
    H. Tuy and R. Horst, “Convergence and restart in branch and bound algorithms for global optimization algorithms,”Mathematical Programming 41 (1988) 161–184.Google Scholar
  17. [17]
    P.B. Zwart, “Nonlinear Programming: counterexamples to two global optimization algorithms,”Operations Research 21 (1973) 1260–1266.Google Scholar
  18. [18]
    P.B. Zwart, “Global maximization of a convex function with linear inequality constraints,”Operations Research 22 (1974) 602–609.Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1991

Authors and Affiliations

  • Hoang Tuy
    • 1
  1. 1.Institute of MathematicsBo Ho, HanoiVietnam

Personalised recommendations