Superlinear convergence of factorized structured quasi-Newton methods for nonlinear optimization

This paper considers factorized structured quasi-Newton methods for solving nonlinear optimization problems. These methods were first suggested by Yabe and Takahashi to generate descent search directions for the objective function of the nonlinear least squares problem, and in particular, factorized BFGS-like and DFP-like updates were proposed. Subsequently, Yabe and Takahashi proved local and q-superlinear convergence of these methods. Yabe and Yamaki extended the convergence results to a factorized Broyden-like family. In this paper, we present a general framework of factorized structured quasi-Newton methods for nonlinear optimization, and develop a convergence theory for establishing local and q-superlinear convergence of the methods with updates derived from the bounded Broyden class. This includes as a special case Yabe and Yamaki's results for the nonlinear least squares problems.