Skip to main content
SpringerLink
Log in

Interval Branch and Bound with Local Sampling for Constrained Global Optimization

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

In this article, we introduce a global optimization algorithm that integrates the basic idea of interval branch and bound, and new local sampling strategies along with an efficient data structure. Also included in the algorithm are procedures that handle constraints. The algorithm is shown to be able to find all the global optimal solutions under mild conditions. It can be used to solve various optimization problems. The local sampling (even if done stochastically) is used only to speed up the convergence and does not affect the fact that a complete search is done. Results on several examples of various dimensions ranging from 1 to 100 are also presented to illustrate numerical performance of the algorithm along with comparison with another interval method without the new local sampling and several noninterval methods. The new algorithm is seen as the best performer among those tested for solving multi-dimensional problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. Alefeld J. Herzberger (1983) Introduction to Interval Computations Academic Press New York, NY

    Google Scholar 

  2. J. Clausen A. Zilinskas (2002) ArticleTitleSubdivision, sampling,and initialization strategies for simplical branch and bound in global optimization Computers & Mathematics with Applications 44 943–955

    Google Scholar 

  3. A.E. Csallner (2001) ArticleTitleLipschitz continuity and the termination of interval methods for global optimization Computers & Mathematics with Applications 42 1035–1042

    Google Scholar 

  4. J.E. Falk R.M. Soland (1969) ArticleTitleAn algorithm for separable nonconvex programming problems Management Science 15 550–569

    Google Scholar 

  5. C.A. Floudas P.M. Pardalos (1990) A Collection of Test Problems for Constrained Global Optimization Algorithms Springer-Verlag Berlin Heidelberg

    Google Scholar 

  6. D.E. Goldberg (1989) Genetic Algorithms in Search, Optimization and Machine Learning Addison-Wesley Reading MA

    Google Scholar 

  7. E.R. Hansen (1992) Global Optimization using Interval Analysis Marcel Dekker NY

    Google Scholar 

  8. R. Horst (1976) ArticleTitleAn algorithm for nonconvex programming problems Mathematical Programming 10 312–321 Occurrence Handle10.1007/BF01580678

    Article  Google Scholar 

  9. R. Horst H. Tuy (1990) Global Optimization, Deterministic Approaches Springer-Verlag Berlin

    Google Scholar 

  10. K. Ichida Y. Fujii (1979) ArticleTitleAn interval arithmetic method for global optimization Computing 23 85–97

    Google Scholar 

  11. Kearfott, R.B. (1996), A review of techniques in the verified solution of constrained global optimization problems. In: Applications of Interval Computations (El Paso, TX, 1995), Appl. Optim., vol. 3, Kluwer Acad. Publ., Dordrecht, 23–59.

  12. Z. Michalewicz (1996) Genetic Algorithms + Data Structures = Evolution Programs, 3rd ed. Springer-Verlag Berlin

    Google Scholar 

  13. S. Kirkpatrick C.D. Gelatt SuffixJr. M.P. Vecchi (1983) ArticleTitleOptimization by simulated annealing Science 220 671–680

    Google Scholar 

  14. R.E. Moore (1966) Interval Analysis Prentice-Hall Englewood Cliffs, NJ.

    Google Scholar 

  15. R.E. Moore (1979) Methods and Applications of Interval Analysis SIAM Publication Philadelphia, Pennsylvania

    Google Scholar 

  16. A. Neumaier (1990) Interval Methods for Systems of Equations Cambridge University Press Cambridge, U.K

    Google Scholar 

  17. H. Ratscheck J. Rokne (1984) Computer Methods for the Range of Functions Ellis Horwood Limited Chichester

    Google Scholar 

  18. H. Ratscheck J. Rokne (1988) New Computer Methods for Global Optimization Wiley New York, NY

    Google Scholar 

  19. S. Skelboe (1974) ArticleTitleComputation of rational interval functions BIT 14 87–95 Occurrence Handle10.1007/BF01933121

    Article  Google Scholar 

  20. Sun, M. (2002), Tree annealing for constrained optimization, In: Proceedings of 34th IEEE Southeastern Symposium on System Theory, Huntsville, AL, pp. 412–416.

  21. T. Van Voorhis (2002) ArticleTitleA global optimization algorithm using Lagrangian underestimates and the interval Newton method Journal of Global Optimization 24 349–370 Occurrence Handle10.1023/A:1020383700229

    Article  Google Scholar 

  22. Z.B. Xu J.S. Zhang Y.W. Leung (1997) ArticleTitleA general CDC formulation for specializing the cell exclusion algorithms of finding all zeros of vector functions Applied Mathematics and Computation 86 235–259 Occurrence Handle10.1016/S0096-3003(96)00191-9

    Article  Google Scholar 

  23. X. Yao Y. Liu G. Lin (1999) ArticleTitleEvolutionary programming made faster IEEE Transactions on Evolutionay Computation 3 82–102 Occurrence Handle10.1109/4235.771163

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The University of Alabama, Tuscaloosa, AL, 35487-0350, USA

    M. Sun

  2. Computer-Based Honors Program, The University of Alabama, Tuscaloosa, AL, 35487-0352, USA

    A.W. Johnson

Authors
  1. M. Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A.W. Johnson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Sun.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sun, M., Johnson, A. Interval Branch and Bound with Local Sampling for Constrained Global Optimization. J Glob Optim 33, 61–82 (2005). https://doi.org/10.1007/s10898-004-6097-6

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-004-6097-6

Keywords

Search

Navigation