Quasi-newton Methods for Nonlinear Least Squares Focusing on Curvatures

Most existing quasi-Newton methods for nonlinear least squares problems incorporate both linear and nonlinear information in the secant update. These methods exhibit good theoretical properties, but are not especially accurate in practice. The objective of this paper is to propose quasi-Newton methods that only update the nonlinearities. We show two advantages of such updates. First, fast convergence is established for well-conditioned problems by interpolating also the previous step, thereby allowing diierent Broyden parameters in each iteration. Secondly, high accuracy is also possible to achieve for certain diicult problems. For example, ill-conditioned problems and problems with large curvature in the tangent space. Numerical results for artiicial problems and standard test problems are presented and discussed.