Numerical comparison among structured quasi-Newton methods for nonlinear least squares problems

The purpose of this paper is to construct effective algorithms for solving nonlinear least squares problems. These methods are based on the idea of structured quasi-Newton methods, which use the structure of the Hessian matrix of the objective function. In order to obtain a descent search direction of the objective function, it has been proposed to approximate the Hessian matrix by the factorized form and the BFGS-like update and DEP-like update have been obtained. Independently, Sheng Songbai and Zou Zhihong (SZ) have been studying factorized versions of structured quasi-Newton methods. This paper constructs an update by a slightly different way from their formulation, in which the SZ update is contained. Further, it applies sizing techniques to the SZ method and proposes new sizing factors. Finally, computational experiments are shown in order to compare our factorized versions with the SZ method and investigate the effect of sizing techniques.