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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Csendes, T., Ratz, D.: Subdivision direction selection in interval methods for global optimization. SIAM J. Numer. Anal 34, 922–938 (1997)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hendrix, E.M.T., Tóth, B.G.: Introduction to Nonlinear and Global Optimization. Springer, New York (2010)CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Horst, R.: On generalized bisection of \(n\)-simplices. Math. Comput. 66(218), 691–698 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Horst, R., Tuy, H.: Global Optimization (Deterministic Approaches). Springer, Berlin (1990)CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Locatelli, M., Schoen, F.: Global Optimization: Theory, Algorithms, and Applications. SIAM, Philadelphia (2013)CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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

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