Encyclopedia of Optimization

2009 Edition
| Editors: Christodoulos A. Floudas, Panos M. Pardalos

Global Optimization: Interval Analysis and Balanced Interval Arithmetic

  • Julius Žilinskas
  • Ian David Lockhart Bogle
Reference work entry
DOI: https://doi.org/10.1007/978-0-387-74759-0_237

Article Outline

Keywords and Phrases

Introduction

Methods / Applications

  Interval Analysis in Global Optimization

  Underestimating Interval Arithmetic

  Random Interval Arithmetic

  Balanced Interval Arithmetic

See also

References

Keywords and Phrases

Global optimization Interval arithmetic Interval computations 
This is a preview of subscription content, log in to check access.

References

  1. 1.
    Alt R, Lamotte JL (2001) Experiments on the evaluation of functional ranges using random interval arithmetic. Math Comput Simul 56:17–34MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Csallner AE, Csendes T, Markót MC (2000) Multisection in interval branch-and-bound methods for global optimization I. Theoretical results. J Glob Optim 16:371–392CrossRefzbMATHGoogle Scholar
  3. 3.
    Hansen E (1978) A globally convergent interval method for computing and bounding real roots. BIT 18:415–424MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Hansen E (1978) Global optimization using interval analysis – the multidimensional case. Numer Math 34:247–270CrossRefGoogle Scholar
  5. 5.
    Hansen E, Walster G (2003) Global Optimization Using Interval Analysis, 2nd edn. Marcel Dekker, New YorkGoogle Scholar
  6. 6.
    Kaucher E (1977) Über Eigenschaften und Anwendungsmöglichkeiten der erweiterten Intervallrechnung und des hyperbolischen Fastkörpers über R′. Comput Suppl 1:81–94Google Scholar
  7. 7.
    Kreinovich V, Nesterov VM, Zheludeva NA (1996) Interval methods that are guaranteed to underestimate (and the resulting new justification of Kaucher arithmetic). Reliab Comput 2:119–124MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Markót MC, Csendes T, Csallner AE (2000) Multisection in interval branch-and-bound methods for global optimization II. Numerical tests. J Global Optim 16:219–228MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Markov S (1995) On directed interval arithmetic and its applications. J Univers Comput Sci 1:514–526zbMATHGoogle Scholar
  10. 10.
    Moore RE (1966) Interval Analysis. Prentice-Hall, Englewood CliffszbMATHGoogle Scholar
  11. 11.
    Moore RE (1977) A test for existence of solutions to non-linear systems. SIAM J Numer Anal 14:611–615MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Skelboe S (1974) Computation of rational interval functions. BIT 14:87–95MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Žilinskas A, Žilinskas J (2005) On underestimating in interval computations. BIT Numer Math 45:415–427CrossRefzbMATHGoogle Scholar
  14. 14.
    Žilinskas A, Žilinskas J (2006) On efficiency of tightening bounds in interval global optimization. Lect Note Comput Sci 3732:197–205CrossRefGoogle Scholar
  15. 15.
    Žilinskas J (2006) Estimation of functional ranges using standard and inner interval arithmetic. Inform 17:125–136zbMATHGoogle Scholar
  16. 16.
    Žilinskas J, Bogle IDL (2004) Balanced random interval arithmetic. Comput Chem Eng 28:839–851CrossRefGoogle Scholar
  17. 17.
    Žilinskas J, Bogle IDL (2006) Balanced random interval arithmetic in market model estimation. Eur J Oper Res 175:1367–1378CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Julius Žilinskas
    • 1
  • Ian David Lockhart Bogle
    • 2
  1. 1.Institute of Mathematics and InformaticsVilniusLithuania
  2. 2.Centre for Process Systems Engineering, Department of Chemical EngineeringUniversity College LondonLondonUK