Global optimization under Lipschitzian constraints

  • Phan Thiên Thạ
  • Hoàng Tụy


We will present a new method for finding the global minimum of a Lipschitzian function under Lipschitzian constraints. The method consists in converting the given problem into one of globally minimizing a concave function subject to a convex and a reverse convex constraints. The resulting algorithm is of the same complexity as the outer approximation algorithm for a concave minimization problem.

Key words

global optimization Lipschitzian constraints, d.c. programming concave minimization outer approximation algorithm 


  1. [1]
    V. P. Bulatov, Embedding Methods in Optimization Problems. Nauka, Novosibirsk, 1977 (in Russian).Google Scholar
  2. [2]
    Yu. M. Danilin and S. A. Piyavski, On an algorithm for finding the absolute minimum. Theory of Optimal Decisions, vol. II, Inst. Cybern., Kiev, 1967 (in Russian).Google Scholar
  3. [3]
    Yu. G. Evtushenko, Numerical method for finding the global extremum of a function. Ž. Vyčisl. Mat. i Mat. Fiz.,11 (1971), 1390–1403 (in Russian).zbMATHGoogle Scholar
  4. [4]
    W. Forst, Algorithms for optimization problems of computer aided coordinate measurement techniques. 9th Symposium on Operations Research, Osnabrück, August 1984.Google Scholar
  5. [5]
    D. Q. Mayne and E. Polak, Outer approximation algorithm for nondifferentiable optimization problems. J. Optim. Theory Appl.,47 (1984), 19–30.CrossRefMathSciNetGoogle Scholar
  6. [6]
    S. A. Piyavski, Algorithm for finding the absolute minimum of a function. Theory of Optimal Decisions, vol. II, Inst. Cybern., Kiev, 1967 (in Russian).Google Scholar
  7. [7]
    P. T. Thach, The design centering problem as a d.c. programming problem. Submitted to Math. Programming Stud.Google Scholar
  8. [8]
    T. V. Thieu, B. T. Tam and V. T. Ban, An outer approximation method for globally minimizing a concave function over a compact convex set. Proc. IFIP Working Conference on Recent Advances in System Modeling and Optimization, Hanoi, January 1983.Google Scholar
  9. [9]
    H. Tuy, On outer approximation methods for solving concave minimization problems. Acta Math. Vietnam.,8 (1983), no. 2, 3–34.zbMATHMathSciNetGoogle Scholar
  10. [10]
    H. Tuy, Global minimization of a difference of two convex functions. Selected Topics in Operations Research and Mathematical Economics (eds. G. Hammer and D. Pallaschke), Springer-Verlag, Berlin, 1984, 98–118.Google Scholar
  11. [11]
    H. Tuy, A general deterministic approach to global optimization via d.c. programming. Proc. Fermat Days “Mathematics for Optimization”, Toulouse, May 1985.Google Scholar
  12. [12]
    H. Tuy, Convex programs with an additional reverse convex constraint. J. Optim. Theory Appl. (forthcoming).Google Scholar
  13. [13]
    L. M. Vidigal and S. W. Director, A design centering algorithm for nonconvex regions of acceptability. IEEE Trans. Comput. Aided Design of Integrated Circuits and Systems, vol. CAD-I (1982), 11–24.Google Scholar

Copyright information

© JJAM Publishing Committee 1987

Authors and Affiliations

  • Phan Thiên Thạ
    • 1
  • Hoàng Tụy
    • 1
  1. 1.Institute of MathematicsHanoiVietnam

Personalised recommendations