Mathematical Programming

, Volume 55, Issue 1–3, pp 319–337

The bisection method in higher dimensions

  • G. R. Wood
Article

Abstract

Is the familiar bisection method part of some larger scheme? The aim of this paper is to present a natural and useful generalisation of the bisection method to higher dimensions. The algorithm preserves the salient features of the bisection method: it is simple, convergence is assured and linear, and it proceeds via a sequence of brackets whose infinite intersection is the set of points desired. Brackets are unions of similar simplexes. An immediate application is to the global minimisation of a Lipschitz continuous function defined on a compact subset of Euclidean space.

AMS 1980 Subject Classification

49D37 

Key words

Bisection simplex global optimisation linear convergence zonotope tiling 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Patricio Basso, “Iterative methods for the localization of the global maximum,”SIAM Journal on Numerical Analysis 19 (1982) 781–792.Google Scholar
  2. [2]
    Gustave Choquet,Lectures on Analysis, Vol. 1 (Benjamin, New York, 1969).Google Scholar
  3. [3]
    A. Eiger, K. Sikorski and F. Stenger, “A bisection method for systems of nonlinear equations,”ACM Transactions on Mathematical Software 10 (1984) 367–377.Google Scholar
  4. [4]
    P. McMullen, “Space tiling zonotopes,”Mathematika 22 (1975) 202–211.Google Scholar
  5. [5]
    Regina Hunter Mladineo, “An algorithm for finding the global maximum of a multimodal, multivariate function,”Mathematical Programming 34 (1986) 188–200.Google Scholar
  6. [6]
    S.A. Piyavskii, “An algorithm for finding the absolute extremum of a function,”USSR Computational Mathematics and Mathematical Physics 12 (1972) 57–67.Google Scholar
  7. [7]
    J.E. Dennis Jr. and Robert B. Schnabel,Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, NJ, 1983).Google Scholar
  8. [8]
    Bruno O. Shubert, “A sequential method seeking the global maximum of a function,”SIAM Journal on Numerical Analysis 9 (1972) 379–388.Google Scholar
  9. [9]
    G.R. Wood, “On computing the dispersion function,”Journal of Optimization Theory and Applications 58 (1988) 331–350.Google Scholar
  10. [10]
    G.R. Wood, “Multidimensional bisection applied to global optimisation,”Computers and Mathematics with Applications 21 (1991) 161–172.Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1992

Authors and Affiliations

  • G. R. Wood
    • 1
  1. 1.Mathematics DepartmentUniversity of CanterburyChristchurchNew Zealand

Personalised recommendations