eprintid: 13654
rev_number: 8
eprint_status: archive
userid: 353
dir: disk0/00/01/36/54
datestamp: 2016-08-09 14:55:37
lastmod: 2021-08-27 17:27:35
status_changed: 2016-08-09 14:55:37
type: article
metadata_visibility: show
item_issues_count: 2
creators_name: Nazareth, J.
creators_id: AL1035
title: Analogues of Dixon’s and Powell’s Theorems for Unconstrained Minimization with Inexact Line Searches
ispublished: pub
abstract: Dixon’s theorem (Math. Programming, 2 (1972), PP. 383–387) states that all variable metric methods in the Broyden class develop identical iterates when line searches are exact. Powell’s theorem (Rep. TP 495, AERE, Harwell, England, 1972) is a variant on this, which states that under similar conditions, the Hessian approximation developed by a BFGS update at any step is independent of the updates used at earlier steps.

By modifying the way in which search directions are defined, we show how to remove the restrictive assumption on line searches in these two theorems. We show also that the BFGS algorithm, modified in this way, is equivalent to the three-term-recurrence (TTR) method on quadratic functions.

Algorithmic implications are discussed and the results of some numerical experimentation are reported.
date: 1986-02
date_type: published
publisher: Society for Industrial and Applied Mathematics
id_number: 10.1137/0723012
creators_browse_id: 2226
full_text_status: none
publication: SIAM Journal on Numerical Analysis
volume: 23
number: 1
pagerange: 170-177
refereed: TRUE
issn: 0036-1429
coversheets_dirty: FALSE
fp7_project: no
fp7_type: info:eu-repo/semantics/article
citation:   Nazareth, J. <https://pure.iiasa.ac.at/view/iiasa/2226.html>  (1986).  Analogues of Dixon’s and Powell’s Theorems for Unconstrained Minimization with Inexact Line Searches.   SIAM Journal on Numerical Analysis 23 (1) 170-177. 10.1137/0723012 <https://doi.org/10.1137/0723012>.