Journal of Global Optimization

, Volume 56, Issue 3, pp 1101–1121 | Cite as

On interval branch-and-bound for additively separable functions with common variables

  • J. L. Berenguel
  • L. G. Casado
  • I. García
  • E. M. T. Hendrix
  • F. Messine
Open Access


Interval branch-and-bound (B&B) algorithms are powerful methods which look for guaranteed solutions of global optimisation problems. The computational effort needed to reach this aim, increases exponentially with the problem dimension in the worst case. For separable functions this effort is less, as lower dimensional sub-problems can be solved individually. The question is how to design specific methods for cases where the objective function can be considered separable, but common variables occur in the sub-problems. This paper is devoted to establish the bases of B&B algorithms for separable problems. New B&B rules are presented based on derived properties to compute bounds. A numerical illustration is elaborated with a test-bed of problems mostly generated by combining traditional box constrained global optimisation problems, to show the potential of using the derived theoretical basis.


Branch-and-bound Interval arithmetic Separable functions 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.


  1. 1.
    Baumann E.: Optimal centered forms. BIT 28(1), 80–87 (1988)CrossRefGoogle Scholar
  2. 2.
    C-XSC: University Wuppertal.
  3. 3.
    Du K., Kearfott R.B.: The cluster problem in multivariate global optimization. J. Glob. Optim. 5(3), 253–265 (1994)CrossRefGoogle Scholar
  4. 4.
    Hansen E.R., Walster G.W.: Global Optimization Using Interval Analysis. Marcel Dekker, New York (2004)Google Scholar
  5. 5.
    Hansen P., Lagouanelle J.-L., Messine F.: Comparison between Baumann and admissible simplex forms in interval analysis. J. Glob. Optim. 37(2), 215–228 (2007)CrossRefGoogle Scholar
  6. 6.
    Kearfott R.B.: Rigorous Global Search: Continuous Problems. Kluwer, Dordrecht (1996)CrossRefGoogle Scholar
  7. 7.
    Markot M.C., Fernandez J., Casado L.G., Csendes T.: New interval methods for constrained global optimization. Math. Program. Ser. A 106(2), 287–318 (2006)CrossRefGoogle Scholar
  8. 8.
    Messine F., Lagouanelle J.-L.: Enclosure methods for multivariate differentiable functions and application to global optimization. J. Univers. Comput. Sci. 4(6), 589–603 (1998)Google Scholar
  9. 9.
    Messine F., Nogarede B.: Optimal design of multi-airgap electrical machines: an unknown size mixed-constrained global optimization formulation. IEEE Trans. Magn. 42(12), 3847–3853 (2006)CrossRefGoogle Scholar
  10. 10.
    Moore R.E.: Interval Analysis. Prentice-Hall, New Jersey (1966)Google Scholar
  11. 11.
    Moore R.E., Kearfott R.B., Cloud M.J.: Introduction to Interval analysis. SIAM, Philadelphia (2009)CrossRefGoogle Scholar
  12. 12.
    Neumaier A., Shcherbina O., Huyer W., Vinko T.: A comparison of complete global optimization solvers. Math. Program. Ser. B 103(2), 335–356 (2005)CrossRefGoogle Scholar
  13. 13.
    Nocedal J., Wright S.J.: Numerical Optimization. Springer, New York (1999)CrossRefGoogle Scholar
  14. 14.
    Ratschek H., Rokne J.: Computer Methods for the Range of Functions. Ellis Horwood, Chichester (1984)Google Scholar
  15. 15.
    Ratschek H., Rokne J.: New Computer Methods for Global Optimization. Ellis Horwood, Chichester (1988)Google Scholar
  16. 16.
    Tóth B., Casado L.G.: Multi-dimensional pruning from Baumann point in an interval global optimization algorithm. J. Glob. Optim. 38(2), 215–236 (2007)CrossRefGoogle Scholar
  17. 17.
    Tóth B., Csendes T.: Empirical investigation of the convergence speed of inclusion functions in a global optimization context. Reliab. Comput. 11(4), 253–273 (2005)CrossRefGoogle Scholar

Copyright information

© The Author(s) 2012

Authors and Affiliations

  • J. L. Berenguel
    • 1
  • L. G. Casado
    • 2
  • I. García
    • 3
  • E. M. T. Hendrix
    • 3
    • 4
  • F. Messine
    • 5
  1. 1.TIC 146: “Supercomputing-Algorithms” Research GroupUniversity of AlmeríaAlmeríaSpain
  2. 2.Department of Computer Architecture and ElectronicsUniversity of AlmeríaAlmeríaSpain
  3. 3.Department of Computer ArchitectureUniversity of MálagaMálagaSpain
  4. 4.Operations Research and LogisticsWageningen UniversityWageningenThe Netherlands
  5. 5.ENSEEIHT-IRIT UMR-CNRS-5505University of ToulouseToulouseFrance

Personalised recommendations