Journal of Global Optimization

, Volume 57, Issue 3, pp 771–782 | Cite as

Geometric branch-and-bound methods for constrained global optimization problems

Open Access


Geometric branch-and-bound methods are popular solution algorithms in deterministic global optimization to solve problems in small dimensions. The aim of this paper is to formulate a geometric branch-and-bound method for constrained global optimization problems which allows the use of arbitrary bounding operations. In particular, our main goal is to prove the convergence of the suggested method using the concept of the rate of convergence in geometric branch-and-bound methods as introduced in some recent publications. Furthermore, some efficient further discarding tests using necessary conditions for optimality are derived and illustrated numerically on an obnoxious facility location problem.


Global optimization Geometric branch-and-bound Approximation algorithms Continuous location 



The author would like to thank Anita Schöbel for fruitful suggestions for improving the paper. Furthermore, the author gratefully acknowledges the anonymous referees for their helpful comments.

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. Adjiman C.S., Dallwig S., Floudas C.A., Neumaier A.: A global optimization method, αBB, for general twice-differentiable constrained NLPs—I. Theoretical advances. Comput. Chem. Eng. 22, 1137–1158 (1998)CrossRefGoogle Scholar
  2. Androulakis I.P., Maranas C.D., Floudas C.A.: αBB: a global optimization method for general constrained nonconvex problems. J. Glob. Optim. 7, 337–363 (1995)CrossRefGoogle Scholar
  3. Bazaraa M.S., Sherali H.D., Shetty C.M.: Nonlinear Programming: Theory and Algorithms. 2nd edn. Wiley-Interscience, New York (1993)Google Scholar
  4. Blanquero R., Carrizosa E.: Continuous location problems and big triangle small triangle: Constructing better bounds. J. Glob. Optim. 45, 389–402 (2009)CrossRefGoogle Scholar
  5. Craven B.D., Mond B.: Sufficient Fritz John optimality conditions for nondifferentiable convex programming. J. Aust. Math. Soc. 19, 462–468 (1976)CrossRefGoogle Scholar
  6. Drezner Z., Suzuki A.: The big triangle small triangle method for the solution of nonconvex facility location problems. Oper. Res. 52, 128–135 (2004)CrossRefGoogle Scholar
  7. Floudas C.A.: Deterministic Global Optimization: Theory, Methods and Applications. 1st edn. Springer, New York (1999)Google Scholar
  8. Hansen E.: Global Optimization Using Interval Analysis. 1st edn. Marcel Dekker, New York (1992)Google Scholar
  9. Hansen P., Peeters D., Richard D., Thisse J.F.: The minisum and minimax location problems revisited. Oper. Res. 33, 1251–1265 (1985)CrossRefGoogle Scholar
  10. Horst R., Pardalos P.M., Thoai N.V.: Introduction to Global Optimization. 2nd edn. Springer, Berlin (2000)Google Scholar
  11. Horst R., Tuy H.: Global Optimization: Deterministic Approaches. 3rd edn. Springer, Berlin (1996)CrossRefGoogle Scholar
  12. Kearfott R.B.: An interval branch and bound algorithm for bound constrained optimization problems. J. Glob. Optim. 2, 259–280 (1992)CrossRefGoogle Scholar
  13. Neumaier A.: Interval Methods for Systems of Equations. 1st edn. Cambridge University Press, New York (1990)Google Scholar
  14. Plastria F.: GBSSS: The generalized big square small square method for planar single-facility location. Eur. J. Oper. Res. 62, 163–174 (1992)CrossRefGoogle Scholar
  15. Plastria F.: Continuous location problems. In: Drezner , Z. (ed.) Facility Location, pp. 225–262. Springer, Berlin (1995)CrossRefGoogle Scholar
  16. Ratschek H., Rokne J.: New Computer Methods for Global Optimization. 1st edn. Ellis Horwood, Chichester, England (1988)Google Scholar
  17. Ratschek H., Voller R.L: What can interval analysis do for global optimization?. J. Glob. Optim. 1, 111–130 (1991)CrossRefGoogle Scholar
  18. Schöbel A., Scholz D.: The theoretical and empirical rate of convergence for geometric branch-and-bound methods. J. Glob. Optim. 48, 473–495 (2010)CrossRefGoogle Scholar
  19. Scholz, D.: Theoretical rate of convergence for interval inclusion functions. J. Glob. Optim. (2011). doi: 10.1007/s10898-011-9735-9
  20. Scholz D.: Deterministic Global Optimization: Geometric Branch-and-bound Methods and Their Applications. 1st edn. Springer, New York (2012)CrossRefGoogle Scholar
  21. Sun M., Johnson A.W.: Interval branch and bound with local sampling for constrained global optimization. J. Glob. Optim. 33, 61–82 (2005)CrossRefGoogle Scholar

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© The Author(s) 2012

Authors and Affiliations

  1. 1.GöttingenGermany

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