On Grid Aware Refinement of the Unit Hypercube and Simplex: Focus on the Complete Tree Size

  • L. G. Casado
  • E. M. T. Hendrix
  • J. M. G. Salmerón
  • B. G.-Tóth
  • I. García
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10406)


Branch and bound (BnB) Global Optimization algorithms can be used to find the global optimum (minimum) of a multiextremal function over the unit hypercube and unit simplex with a guaranteed accuracy. Subdivision strategies can take the information of the evaluated points into account leading to irregular shaped subsets. This study focuses on the passive generation of spatial subdivisions aiming at evaluating points on a predefined grid. The efficiency measure is in terms of the complete tree size, or worst case BnB scenario, with a termination criterion on the subset size. Longest edge bisection is used as a benchmark. It is shown that taking the grid for a given termination tolerance into account, other general partitions exist that improve the BnB upper bound on the number of evaluated points and subsets.


Branch and bound Division Covering Unit hypercube Unit simplex 



This work has been funded by grants from the Spanish Ministry (TIN2015-66680), in part financed by the European Regional Development Fund (ERDF). J.M.G. Salmerón is a fellow of the Spanish FPU program.


  1. 1.
    Aparicio, G., Casado, L.G., G-Tóth, B., Hendrix, E.M.T., García, I.: Heuristics to reduce the number of simplices in longest edge bisection refinement of a regular n-simplex. In: Murgante, B., Misra, S., Rocha, A.M.A.C., Torre, C., Rocha, J.G., Falcão, M.I., Taniar, D., Apduhan, B.O., Gervasi, O. (eds.) ICCSA 2014. LNCS, vol. 8580, pp. 115–125. Springer, Cham (2014). doi: 10.1007/978-3-319-09129-7_9 Google Scholar
  2. 2.
    Berenguel, J.L., Casado, L.G., García, I., Hendrix, E.M.T.: On estimating workload in interval branch-and-bound global optimization algorithms. J. Global Optim. 56(3), 821–844 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Casado, G.L., García, I., Csendes, T.: A new multisection technique in interval methods for global optimization. Computing 65(3), 263–269 (2000)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Casado, L.G., Hendrix, E.M.T., García, I.: Infeasibility spheres for finding robust solutions of blending problems with quadratic constraints. J. Global Optim. 39(2), 215–236 (2007)MathSciNetMATHGoogle Scholar
  5. 5.
    Csendes, T., Ratz, D.: Subdivision direction selection in interval methods for global optimization. SIAM J. Numer. Anal 34, 922–938 (1997)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    G.-Tóth, B., Hendrix, E.M.T., Casado, L.G., García, I.: On refinement of the unit simplex using regular simplices. J. Global Optim. 64(2), 305–323 (2016)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Hendrix, E.M.T., Pínter, J.: An application of Lipschitzian global optimization to product design. J. Global Optim. 1, 389–401 (1991)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hendrix, E.M.T., Tóth, B.G.: Introduction to Nonlinear and Global Optimization. Springer, New York (2010)CrossRefMATHGoogle Scholar
  9. 9.
    Hendrix, E.M.T., Casado, L.G., García, I.: The semi-continuous quadratic mixture design problem: Description and branch-and-bound approach. Eur. J. Oper. Res. 191(3), 803–815 (2008)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Horst, R.: On generalized bisection of \(n\)-simplices. Math. Comput. 66(218), 691–698 (1997)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Horst, R., Tuy, H.: Global Optimization (Deterministic Approaches). Springer, Berlin (1990)CrossRefMATHGoogle Scholar
  12. 12.
    Kuno, T., Ishihama, T.: A generalization of \(\omega \)-subdivision ensuring convergence of the simplicial algorithm. Comput. Optim. Appl. 64(2), 535–555 (2016)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Locatelli, M., Schoen, F.: Global Optimization: Theory, Algorithms, and Applications. SIAM, Philadelphia (2013)CrossRefMATHGoogle Scholar
  14. 14.
    Markót, M., Fernández, J., Casado, L., Csendes, T.: New interval methods for constrained global optimization. Math. Program. 106(2), 287–318 (2006)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Salmerón, J.M.G., Aparicio, G., Casado, L.G., García, I., Hendrix, E.M.T., G.-Tóth, B.: Generating a smallest binary tree by proper selection of the longest edges to bisect in a unit simplex refinement. J. Comb. Optim. 3, 389–402 (2017)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Todd, M.J.: The Computation of Fixed Points and Applications. Lecture Notes in Economics and Mathematical Systems, vol. 124. Springer, Heidelberg (1976)Google Scholar
  17. 17.
    Tuy, H.: Effect of the subdivision strategy on convergence and efficiency of some global optimization algorithms. J. Global Optim. 1(1), 23–36 (1991)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • L. G. Casado
    • 1
  • E. M. T. Hendrix
    • 2
  • J. M. G. Salmerón
    • 1
  • B. G.-Tóth
    • 3
  • I. García
    • 2
  1. 1.University of Almería (ceiA3)AlmeríaSpain
  2. 2.Universidad de MálagaMálagaSpain
  3. 3.Szeged UniversitySzegedHungary

Personalised recommendations