On Simplicial Longest Edge Bisection in Lipschitz Global Optimization

  • Juan F. R. Herrera
  • Leocadio G. Casado
  • Eligius M. T. Hendrix
  • Inmaculada García
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8580)


Simplicial subsets are popular in branch-and-bound methods for Global Optimization. Longest Edge Bisection is a convenient way to divide a simplex. When the number of dimensions is greater than two, irregular simplices (not all edges have the same length) may appear with more than one longest edge. In these cases, the first longest edge is usually selected. We study the impact of other selection rule of the longest edge to be bisected next on the development of a branch-and-bound algorithm to solve multidimensional Lipschitz Global Optimization instances. Experiments show a significant reduction in the number of evaluated simplices for most of the test problems.


Longest Edge Bisection Branching rule Branch-and-bound Lipschitz optimization 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Christodoulos, A.F., Pardalos, P.M. (eds.): State of the Art in Global Optimization: Computational Methods and Applications. Nonconvex Optimization and Its Applications, vol. 7. Kluwer Academic Publishers (1996)Google Scholar
  2. 2.
    Mitten, L.G.: Branch and bound methods: general formulation and properties. Oper. Res. 18(1), 24–34 (1970)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Ibaraki, T.: Theoretical comparisons of search strategies in branch and bound algorithms. Int. J. Comput. Inf. Sci. 5(4), 315–344 (1976)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Aparicio, G., Casado, L.G., Hendrix, E.M.T., García, I., Toth, B.G.: On computational aspects of a regular n-simplex bisection. In: 2013 Eighth International Conference on P2P, Parallel, Grid, Cloud and Internet Computing (3PGCIC), pp. 513–518 (October 2013)Google Scholar
  5. 5.
    Paulavičius, R., Žilinskas, J.: Global optimization using the branch-and-bound algorithm with a combination of Lipschitz bounds over simplices. Technol. Econ. Dev. Econ. 15(2), 310–325 (2009)CrossRefGoogle Scholar
  6. 6.
    Horst, R., Tuy, H.: Global Optimization (Deterministic Approaches). Springer, Berlin (1990)CrossRefzbMATHGoogle Scholar
  7. 7.
    Todd, M.J.: The computation of fixed points and applications. Lecture Notes in Economics and Mathematical Systems, vol. 24. Springer (1976)Google Scholar
  8. 8.
    Herrera, J.F.R., Casado, L.G., Paulavičius, R., Žilinskas, J., Hendrix, E.M.T.: On a hybrid MPI-Pthread approach for simplicial branch-and-bound. In: 2013 IEEE 27th International Parallel and Distributed Processing Symposium Workshops PhD Forum (IPDPSW), pp. 1764–1770 (May 2013)Google Scholar
  9. 9.
    Herrera, J.F.R., Casado, L.G., Hendrix, E.M.T., Paulavičius, R., Žilinskas, J.: Dynamic and hierarchical load-balancing techniques applied to parallel branch-and-bound methods. In: 2013 Eighth International Conference on P2P, Parallel, Grid, Cloud and Internet Computing (3PGCIC), pp. 497–502 (October 2013)Google Scholar
  10. 10.
    Paulavičius, R., Žilinskas, J., Grothey, A.: Investigation of selection strategies in branch and bound algorithm with simplicial partitions and combination of Lipschitz bounds. Optim. Lett. 4(2), 173–183 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Hannukainen, A., Korotov, S.: On numerical regularity of the face-to-face longest-edge bisection algorithm for tetrahedral partitions. Sci. Comput. Program. 90, Part A, 34–41 (2014)Google Scholar
  12. 12.
    Paulavičius, R., Žilinskas, J.: Simplicial global optimization. Springer Briefs in Optimization. Springer, New York (2014)CrossRefzbMATHGoogle Scholar
  13. 13.
    Hansen, P., Jaumard, B.: Lipschitz optimization. In: Horst, R., Pardalos, P. (eds.) Handbook of Global Optimization. Nonconvex Optimization and Its Applications, vol. 2, pp. 407–493. Springer US (1995)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Juan F. R. Herrera
    • 1
  • Leocadio G. Casado
    • 1
  • Eligius M. T. Hendrix
    • 2
  • Inmaculada García
    • 2
  1. 1.Universidad de Almería (ceiA3)AlmeríaSpain
  2. 2.Universidad de MálagaMálagaSpain

Personalised recommendations