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

Research output: Contribution to journalArticle

17 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

Fingerprint

Broyden's Method
Nonlinear Systems of Equations
Quasi-Newton Method
Unconstrained Optimization
Newton-Raphson method
Nonlinear systems
Open Problems
Variable Metric Methods
Descent Method
Simultaneous equations
Nonlinear Optimization
Newton Methods
Theoretical Analysis
Nonlinear Equations
Review
Alternatives

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

Cite this

Broyden's quasi-Newton methods for a nonlinear system of equations and unconstrained optimization : A review and open problems. / Al-Baali, Mehiddin; Spedicato, Emilio; Maggioni, Francesca.

In: Optimization Methods and Software, Vol. 29, No. 5, 03.09.2014, p. 937-954.

Research output: Contribution to journalArticle

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