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

Andy Conn, Nick Gould, Philippe Toint

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    Abstract

    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. © 1991 The Mathematical Programming Society, Inc.
    Original languageEnglish
    Pages (from-to)177-195
    Number of pages19
    JournalMathematical Programming Series B
    Volume50
    Issue number1-3
    DOIs
    Publication statusPublished - 1 Apr 1991

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