Global Convergence of a New Hybrid Gauss--Newton Structured BFGS Method for Nonlinear Least Squares Problems

In this paper, we propose a hybrid Gauss-Newton structured BFGS method with a new update formula and a new switch criterion for the iterative matrix to solve nonlinear least squares problems. We approximate the second term in the Hessian by a positive definite BFGS matrix. Under suitable conditions, global convergence of the proposed method with a backtracking line search is established. Moreover, the proposed method automatically reduces to the Gauss-Newton method for zero residual problems and the structured BFGS method for nonzero residual problems in a neighborhood of an accumulation point. Locally quadratic convergence rate for zero residual problems and locally superlinear convergence rate for nonzero residual problems are obtained for the proposed method. Some numerical results are given to compare the proposed method with some existing methods.