Superlinear convergence of the Sheng–Zou–Broyden method for nonlinear least squares problems

We are concerned with nonlinear least squares problems. It is known that structured quasi-Newton methods perform well for solving these problems. In this strategy, two kinds of factorized structured quasi-Newton methods have been independently proposed by Yabe and Takahashi, and Sheng and Zou. Sheng and Zou introduced a Broyden–Fletcher–Goldfarb–Shanno (BFGS)-like update by considering how the normal equation based on an affine model may consist with the Newton equation, and dealt with a hybrid method that combines the Gauss–Newton method and their BFGS-like method. In this paper, we deal with the Sheng–Zou–Broyden family proposed by Yabe, which is an extension of the update of Sheng and Zou to the Broyden-like family. Local and q-superlinear convergence of the method with this family is established for nonzero residual problems.