Broyden's quasi-Newton methods for a nonlinear system of equations and unconstrained optimization: A review and open problems

Mehiddin Al-Baali*, Emilio Spedicato, Francesca Maggioni

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

22 Citations (Scopus)

Abstract

Quasi-Newton methods were introduced by Charles Broyden [A class of methods for solving nonlinear simultaneous equations, Math Comp. 19 (1965), pp. 577-593] as an alternative to Newton's method for solving nonlinear algebraic systems; in 1970 Broyden [The convergence of a class of double rank minimization algorithms, IMA J Appl Math. 6, part I and II (1970), pp. 76-90, 222-231] extended them to nonlinear unconstrained optimization as a generalization of the DFP method which is proposed by Davidon [Variable metric method for minimization (revised), Technical Report ANL-5990, Argonne National Laboratory, USA, 1959] and investigated by Fletcher and Powell [A rapidly convergent descent method for minimization, Comput J. 6 (1963), pp. 163-168]. Such methods (in particular, the BFGS (Broyden-Fletcher-Goldfarb-Shanno) method) are very useful in practice and have been subject to substantial theoretical analysis, albeit some problems are still open. In this paper we describe properties of these methods as derived by Broyden and then further developed by other researchers, especially with reference to improvement of their computational performance.

Original languageEnglish
Pages (from-to)937-954
Number of pages18
JournalOptimization Methods and Software
Volume29
Issue number5
DOIs
Publication statusPublished - Sep 3 2014

Keywords

  • ABS methods
  • finite termination
  • line search technique
  • modified methods
  • nonlinear algebraic equations
  • optimal conditioning
  • quasi-Newton methods
  • unconstrained optimization

ASJC Scopus subject areas

  • Control and Optimization
  • Software
  • Applied Mathematics

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