Skip to main content
SpringerLink
Log in

New results on verified global optimization

Neue Ergebnisse bei der verifizierten globalen Optimierung

  • Published:
Computing Aims and scope Submit manuscript

Abstract

We study the influence of different strategies for selecting the next box for subdivision in an interval branch and bound method for global optimization. A theoretical result establishes the optimality of a ‘best-first’ strategy. Moreover, we define a multisection algorithm for subdivision of boxes. Different possibilities for the choice of bisection directions and the number of bisections are compared by numerical results showing that multisection is usually superior to bisection. In addition, we identify crucial properties of a division strategy which yield convergence of an algorithm with multisection.

Zusammenfassung

Wir untersuchen den Einfluß verschiedener Strategien zur Auswahl der als nächstes zu teilenden Box in einem Branch-und-Bound-Verfahren zur globalen Optimierung. Ein theoretisches Resultat zeigt die Optimalität einer ‘best-first’-Strategie. Außerdem definieren wir einen Multisektions-Algorithmus zum Teilen von Boxen. Verschiedene Möglichkeiten für die Wahl der Halbierungsrichtungen und die Anzahl der Halbierungen werden anhand numerischer Ergebnisse verglichen, wobei deutlich wird, daß Multisektion der Bisektion immer überlegen ist. Zusätzlich identifizieren wir zwei wesentliche Eigenschaften für Teilungsstrategien, die zur Konvergenz von Algorithmen mit Multisektion führen.

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. Alefeld, G., Herzberger, J.: Introduction to interval computations. New York: Academic Press, 1983.

    Google Scholar 

  2. Baumann, E.: Optimal centered forms. BIT28, 80–87 (1988).

    Google Scholar 

  3. Berner, S.: Parallel validated global optimization. ZAMM “Numerical Analysis, Scientific Computing, Computer Science” (special volume) (1996).

  4. Berner, S.: Ein paralleles Verfahren zur verifizierten globalen Optimierung. Dissertation, Bergische Universität GH Wuppertal, 1995.

  5. Csendes, T., Ratz, D.: Subdivision direction selection in interval methods for global optimization. SIAM J. Numer. Anal. (1997) to appear.

  6. Hansen, E.: Global optimization using interval analysis: The one-dimensional case. J. Optim. Theory Appl.29, 331–344 (1979).

    Google Scholar 

  7. Hansen, E.: Global optimization using interval analysis — the multidimensional case. Numer. Math.34, 247–270 (1980).

    Google Scholar 

  8. Hansen, E.: Global optimization using interval analysis. New York: Marcel Dekker, 1992.

    Google Scholar 

  9. Henriksen, T., Madsen, K.: Use of a depth-first strategy in parallel global optimization. Tech. Report 92-10, Institute for Numerical Analysis, Technical University of Denmark, Lyngby, 1992.

    Google Scholar 

  10. Jansson, C.: On self-validating methods for optimization problems. In: Topics in validated computations (Herzberger, J., ed.), pp. 381–438. New York: Elsevier, 1994.

    Google Scholar 

  11. Kearfott, R. B.: Preconditioners for the interval Gauss-Seidel method. SIAM J. Numer. Anal.27, 804–822 (1990).

    Google Scholar 

  12. Leclerc, A.: Parallel interval global optimization and its implementation in C++. Interval Comput.3, 148–163 (1993).

    Google Scholar 

  13. Moore, R. E., Hansen, R., Leclerc, A.: Rigorous methods for global optimization. In: Recent Advances in global optimization (Floudas, C. A., Pardalos, P. M., eds.), pp. 321–342. Princeton: Princeton University Press, 1992.

    Google Scholar 

  14. Neumaier, A.: Interval methods for systems of equations. Cambridge: Cambridge University Press, 1990.

    Google Scholar 

  15. Rall, L.B.: Automatic differentiation: techniques and applications. Lecture Notes in Computer Science, Vol. 120. New York: Springer, 1981.

    Google Scholar 

  16. Ratschek, H., Rokne, J.: New computer methods for global optimization. Chichester: Ellis Horwood, 1988.

    Google Scholar 

  17. Ratz, D.: On branching rules in second-order branch-and-bound methods for global optimization. In: Scientific computing and validation (Alefeld, G., Frommer, A., Lang, B., eds.), pp. 221–227. Berlin: Akademie-Verlag, 1996.

    Google Scholar 

  18. Ratz, D., Csendes, T.: On the selection of subdivision directions in interval branch-and-bound methods for global optimization. J. Global Optim.7, 183–207 (1995).

    Google Scholar 

  19. Törn, A., Žilinskas, A.: Global optimization. Berlin, Heidelberg, New York, Tokyo: Springer, 1989.

    Google Scholar 

  20. Walster, G., Hansen, E., Sengupta, S.: Test results for a global optimization algorithm. In: Numerical optimization 1984 (Boggs, P., Byrd, R., Schnabel, R., eds.), pp. 272–287. Philadelphia: SIAM, 1985.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Bergische Universität Gesamthochschule Wuppertal, Gaußstraße 20, D-42097, Wuppertal

    S. Berner

Authors
  1. S. Berner
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Berner, S. New results on verified global optimization. Computing 57, 323–343 (1996). https://doi.org/10.1007/BF02252252

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02252252

AMS Subject Classification

Key words

Search

Navigation