Mathematical Programming

, Volume 50, Issue 1, pp 177–195

Convergence of quasi-Newton matrices generated by the symmetric rank one update

  • A. R. Conn
  • N. I. M. Gould
  • Ph. L. Toint

DOI: 10.1007/BF01594934

Cite this article as:
Conn, A.R., Gould, N.I.M. & Toint, P.L. Mathematical Programming (1991) 50: 177. doi:10.1007/BF01594934


Quasi-Newton algorithms for unconstrained nonlinear minimization generate a sequence of matrices that can be considered as approximations of the objective function second derivatives. This paper gives conditions under which these approximations can be proved to converge globally to the true Hessian matrix, in the case where the Symmetric Rank One update formula is used. The rate of convergence is also examined and proven to be improving with the rate of convergence of the underlying iterates. The theory is confirmed by some numerical experiments that also show the convergence of the Hessian approximations to be substantially slower for other known quasi-Newton formulae.

Key words

Quasi-Newton updatesconvergence theory

Copyright information

© The Mathematical Programming Society, Inc. 1991

Authors and Affiliations

  • A. R. Conn
    • 1
  • N. I. M. Gould
    • 2
  • Ph. L. Toint
    • 3
  1. 1.IBM T. J. Watson Research CenterYorktown HeightsUSA
  2. 2.Rutherford Appleton LaboratoryChiltonUK
  3. 3.Department of MathematicsFacultés Universitaires ND de la PaixNamurBelgium