Global Optimization Simplex Bisection Revisited Based on Considerations by Reiner Horst

  • Eligius M. T. Hendrix
  • Leocadio G. Casado
  • Paula Amaral
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7335)

Abstract

In this paper, the use of non-optimality spheres in a simplicial branch and bound (B&B) algorithm is investigated. In this context, some considerations regarding the use of bisection on the longest edge in relation with ideas of Reiner Horst are reminded. Three arguments highlight the merits of bisection of simplicial subsets in B&B schemes.

Keywords

Global Optimization simplicial partition branch and bound bisection 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baritompa, W.P.: Customizing methods for global optimization, a geometric viewpoint. Journal of Global Optimization 3, 193–212 (1993)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Breiman, L., Cutler, A.: A deterministic algorithm for global optimization. Mathematical Programming 58, 179–199 (1993)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Casado, L.G., García, I., Tóth, B.G., Hendrix, E.M.T.: On determining the cover of a simplex by spheres centered at its vertices. Journal of Global Optimization 50, 654–655 (2011)CrossRefGoogle Scholar
  4. 4.
    Casado, L.G., Hendrix, E.M.T., García, I.: Infeasibility spheres for finding robust solutions of blending problems with quadratic constraints. Journal of Global Optimization 39, 557–593 (2007)CrossRefGoogle Scholar
  5. 5.
    Danilin, Y., Piyavski, S.A.: An algorithm for finding the absolute minimum. Theory of Optimal Decisions 2, 25–37 (1967) (in Russian)Google Scholar
  6. 6.
    Evtushenko, Y., Posypkin, M.: Coverings for global optimization of partial-integer nonlinear problems. Doklady Mathematics 83, 1–4 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Horst, R.: On generalized bisection of n-simplices. Mathematics of Computation 66(218), 691–698 (1997)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Horst, R.: Bisection by global optimization revisited. Journal of Optimization Theory and Applications 144, 501–510 (2010)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Horst, R., Pardalos, P.M., Thoai, N.V.: Introduction to Global Optimization, Nonconvex Optimization and its Applications, vol. 3. Kluwer Academic Publishers, Dordrecht (1995)Google Scholar
  10. 10.
    Horst, R., Tuy, H.: On the convergence of global methods in multiextremal optimization. Journal of Optimization Theory and Applications 54, 253–271 (1987)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Horst, R., Tuy, H.: Global Optimization (Deterministic Approaches). Springer, Berlin (1990)MATHGoogle Scholar
  12. 12.
    Locatelli, M., Raber, U.: On convergence of the simplicial branch-and-bound algorithm based on ω-subdivisions. J. Optim. Theory Appl. 107, 69–79 (2000)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Mladineo, R.H.: An algorithm for finding the global maximum of a multimodal multivariate function. Mathematical Programming 34, 188–200 (1986)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Paulavičius, R., Žilinskas, J., Grothey, A.: Investigation of selection strategies in branch and bound algorithm with simplicial partitions and combination of lipschitz bounds. Optimization Letters 4, 173–183 (2010)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Piyavski, S.A.: An algorithm for finding the absolute extremum of a function. USSR Computational Mathematics and Mathematical Physics 12, 57–67 (1972) (in Russian)CrossRefGoogle Scholar
  16. 16.
    Raber, U.: A simplicial branch-and-bound method for solving nonconvex all-quadratic programs. Journal of Global Optimization 13, 417–432 (1998)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Raber, U.: Nonconvex All-Quadratic Global Optimization Problems: Solution Methods, Application and Related Topics. Ph.D. thesis, Trier University (1999)Google Scholar
  18. 18.
    Shubert, B.O.: A sequential method seeking the global maximum of a function. SIAM Journal of Numerical Analysis 9, 379–388 (1972)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Zilinskas, A., Clausen, J.: Subdivision, sampling, and initialization strategies for simplicial branch and bound in global optimization. International Journal of Computers and Mathematics with Applications 44, 943–955 (2002)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Zilinskas, A., Zilinskas, J.: Global optimization based on a statistical model and simplicial partitioning. International Journal of Computers and Mathematics with Applications 44, 957–967 (2002)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Eligius M. T. Hendrix
    • 1
    • 2
  • Leocadio G. Casado
    • 3
  • Paula Amaral
    • 4
    • 5
  1. 1.Arquitectura de ComputadoresUniversidad de MálagaSpain
  2. 2.Logistics and Operations ResearchWageningen UniversityThe Netherlands
  3. 3.Dpt. de Arquitectura de Computadores y ElectronicaUniversidad de AlmeríaSpain
  4. 4.Department of MathematicsUniversidade Nova de LisboaPortugal
  5. 5.CMAUniversidade Nova de LisboaPortugal

Personalised recommendations