Global Convergence of a Cass of Quasi-Newton Methods on Convex Problems

Abstract

We study the global convergence properties of the restricted Broyden class of quasi-Newton methods, when applied to a convex objective function. We assume that the line search satisfies a standard sufficient decrease condition and that the initial Hessian approximation is any positive definite matrix. We show global and superlinear convergence for this class of methods, except for DFP. This generalizes Powell’s well-known result for the BFGS method. The analysis gives us insight into the properties of these algorithms; in particular it shows that DFP lacks a very desirable self-correcting property possessed by BFGS.