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

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.