TY - JOUR
T1 - Broyden's quasi-Newton methods for a nonlinear system of equations and unconstrained optimization
T2 - A review and open problems
AU - Al-Baali, Mehiddin
AU - Spedicato, Emilio
AU - Maggioni, Francesca
PY - 2014/9/3
Y1 - 2014/9/3
N2 - 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.
AB - 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.
KW - ABS methods
KW - finite termination
KW - line search technique
KW - modified methods
KW - nonlinear algebraic equations
KW - optimal conditioning
KW - quasi-Newton methods
KW - unconstrained optimization
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U2 - 10.1080/10556788.2013.856909
DO - 10.1080/10556788.2013.856909
M3 - Article
AN - SCOPUS:84903521674
SN - 1055-6788
VL - 29
SP - 937
EP - 954
JO - Optimization Methods and Software
JF - Optimization Methods and Software
IS - 5
ER -