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A quasi-Newton method with optimal R-order without independence assumption

Please always quote using this URN:urn:nbn:de:0296-matheon-3514
  • Quasi-Newton methods based on least change secant updating formulas that solve linear equations $Ax=b$ in $n=\dim(x)=\dim(b)$ steps can be expected to solve corresponding smooth nonlinear systems $n$-step quadratically, i.e. with an $r$-order of $\rho = 2^{1/n} = 1 + 1/n +O(1/n^2)$. The best rate one can possibly expect on general problems is given by the positive root $\rho_n$ of $\rho^n(\rho -1)=1$, for which $\rho_n-1 = \ln(n)/n + O(1/n^2)$. To show that this upper bound is actually achieved one usually has to impose a priori some kind of linear independence condition on the sequence of steps taken by the quasi-Newton iteration in question. Without any such assumptions we establish in this paper the convergence order $\rho_n$ for the two-sided rank one formula proposed by Schlenkrich et al in \cite{SGW06}. It requires the evaluation of adjoint vectors, is invariant with respect to linear transformations on the variable domain and combines the properties of bounded deterioration and heredity.

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Metadaten
Author:Andreas Griewank, Sebastian Schlenkrich, Andrea Walther
URN:urn:nbn:de:0296-matheon-3514
Referee:Peter Deuflhard
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2006/06/09
Release Date:2006/06/09
Tag:
Institute:Humboldt-Universität zu Berlin
MSC-Classification:46-XX FUNCTIONAL ANALYSIS (For manifolds modeled on topological linear spaces, see 57Nxx, 58Bxx) / 46Nxx Miscellaneous applications of functional analysis [See also 47Nxx] / 46N10 Applications in optimization, convex analysis, mathematical programming, economics
Preprint Number:340
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