Efficient Serial and Parallel Implementations of the Cutting Angle Method

  • Gleb Beliakov
  • Kai Ming Ting
  • Manzur Murshed
  • Alex Rubinov
  • Marcello Bertoli
Part of the Applied Optimization book series (APOP, volume 82)


We examine efficient computer implementation of one method of deterministic global optimization, the cutting angle method. In this method the objective function is approximated from below with piecewise linear auxiliary functions. The sequence of global minima of these auxiliary functions converges to the global minimum of the objective function. Computing the minima of the auxiliary function is a combinatorial problem, and we show that it can be effectively parallelized. We discuss the improvements made to the serial implementation of the cutting angle method, and ways of distributing computations across multiple processors on parallel and cluster computers.


global optimization cutting angle method parallel computing 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    M. Andramonov, A. Rubinov and B. Glover, Cutting angle methods in global optimization, Applied Mathematics Letters, vol. 12 (1999), pp. 95–100.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    A. Bagirov, Derivative-free methods for unconstrained nonsmooth optimization and its numerical analysis, Journal Investigacao Operational, vol. 19 (1999), pp. 75–93.Google Scholar
  3. [3]
    A. Bagirov and A. Rubinov, Global minimization of increasing positively homogeneous function over the unit simplex, Annals of Operations Research, vol. 98 (2000), pp. 171–187.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    A. Bagirov and A. Rubinov, Modified versions of the cutting angle method, in N. Hadjisavvas and P. M. Pardalos, eds., Convex analysis and global optimization, Kluwer Academic Publishers, Dordrecht, 2001, pp. 245–268.CrossRefGoogle Scholar
  5. [5]
    A. M. Bagirov and A. M. Rubinov, Cutting angle method and a local search, Journal of Global Optimization, to appear.Google Scholar
  6. [6]
    L. M. Batten and G. Beliakov, Fast algorithm for the cutting angle method of global optimization, Journal of Global Optimization, to appear.Google Scholar
  7. [7]
    I.M. Bomze and E. de Klerk, Solving standard quadratic optimization problems via linear, semidefinite and copositive programming. Journal of Global Optimization, to appear.Google Scholar
  8. [8]
    T. H. Cormen, C. E. Leiserson and R. L. Rivest, Introduction to algorithms, MIT Press, McGraw-Hill, Cambridge, Mass. New York, 1990.zbMATHGoogle Scholar
  9. [9]
    C. A. Floudas, Deterministic global optimization: theory,methods,and applications, Kluwer Academic Publishers, Dordrecht, 2000.Google Scholar
  10. [10]
    R. Horst and H. Tuy, Global optimization: deterministic approaches, Springer-Verlag, Berlin; New York, 1993.Google Scholar
  11. [11]
    R. Horst and P. M. Pardalos, Handbook of global optimization, Kluwer Academic Publishers, Dordrecht; Boston, 1995.zbMATHGoogle Scholar
  12. [12]
    J. L. Kiepeis, M. G. Ierapetritou and C. A. Floudas, Protein Folding and Peptide Docking - a Molecular Modeling and Global Optimization Approach, Computers and Chemical Engineering, vol. 22 (1998), pp. S 3–S 10.Google Scholar
  13. [13]
    F. T. Leighton, Introduction to parallel algorithms and architectures: arrays,trees,hypercubes, M. Kaufmann Publishers, San Mateo, Calif., 1992.Google Scholar
  14. [14]
    X. Liu, Finding global minima with a computable filled function, Journal of Global Optimization vol. 19(2001), pp. 151–161.zbMATHCrossRefGoogle Scholar
  15. [15]
    H. S. Morse,Practical parallel computing, AP Professional, Boston, 1994.Google Scholar
  16. [16]
    R. V. Pappu, R. K. Hart and J. W. Ponder, Analysis and application of potential energy smoothing and search methods for global optimization, Journal of Physical Chemistry B, vol. 102 (1998), pp. 9725–9742.CrossRefGoogle Scholar
  17. [17]
    P. M. Pardalos and C. A. Floudas, Optimization in computational chemistry and molecular biology: local and global approaches, Kluwer Academic Publishers, Boston, 2000.zbMATHGoogle Scholar
  18. [18]
    J. Pintér, Global optimization in action: continuous and Lipschitz optimization–algorithms, implementations, and applications, Kluwer Academic Publishers, Dordrecht; Boston, 1996.zbMATHGoogle Scholar
  19. [19]
    A. M. Rubinov, Abstract convexity and global optimization,Kluwer Academic Publishers, Dordrecht; Boston, 2000.zbMATHGoogle Scholar
  20. [20]
    T. Takaoka, Theory of Trinomial Heaps, in D.-Z. Du, P. Eades, V. Estivill-Castro, X. Lin and A. Sharma, eds., Computing and Combinatorics, Springer, Sydney, 2000, pp. 362–372.CrossRefGoogle Scholar
  21. [21]
    A. Törn and A. Zhilinskas, Global optimization, Springer-Verlag, Berlin; New York, 1989.zbMATHGoogle Scholar
  22. [22]
    B. Wilkinson and C. M. Allen, Parallel programming: techniques and applications using networked workstations and parallel computers, Prentice Hall, Upper Saddle River, N.J., 1999.Google Scholar

Copyright information

© Kluwer Academic Publishers B.V. 2003

Authors and Affiliations

  • Gleb Beliakov
    • 1
  • Kai Ming Ting
    • 2
  • Manzur Murshed
    • 2
  • Alex Rubinov
    • 3
  • Marcello Bertoli
    • 3
  1. 1.School of Computing and MathematicsDeakin UniversityClaytonAustralia
  2. 2.Gippsland School of Computing and Information TechnologyMonash UniversityVictoriaAustralia
  3. 3.School of Information technology and Mathematical SciencesUniversity of BallaratBallaratAustralia

Personalised recommendations