Superlinear convergence of Broyden’s method and BFGS algorithm using Kantorovich-type assumptions

Abstract

Broyden’s method is a quasi-Newton method which is used to solve a system of nonlinear equations. Almost all convergence theories in the literature assume existence of a root and bounds on the nonlinear function and its derivative in some neighbourhood of the root. All these conditions cannot be checked in practice. The motivation of this work is to derive a local convergence theory where all assumptions can be verified, and the existence of a root and its superlinear rate of convergence are consequences of the theory. The BFGS algorithm is a quasi-Newton method for unconstrained minimization. Also, all known convergence theories assume existence of a solution and bounds of the function in a neighbourhood of the minimizer. The second main result of this paper is a local convergence theory where all assumptions are verifiable and existence of a minimizer and superlinear convergence of the iteration are conclusions. In addition, both theories are simple in the sense that they contain as few constants as possible.