On the Local and Superlinear Convergence of Quasi-Newton Methods

This paper presents a local convergence analysis for several well-known quasi-Newton methods when used, without line searches, in an iteration of the form to solve for x* such that Fx* = 0. The basic idea behind the proofs is that under certain reasonable conditions on xo, F and xo, the errors in the sequence of approximations {Hk} to F′(x*)−1 can be shown to be of bounded deterioration in that these errors, while not ensured to decrease, can increase only in a controlled way. Despite the fact that Hk is not shown to approach F′(x*)−1, the methods considered, including those based on the single-rank Broyden and double-rank Davidon-Fletcher-Powell formulae, generate locally Q-superlinearly convergent sequences {xk}.