Global convergence of a new hybrid Gauss-Newton structured BFGS method for nonlinear least squares problems

Abstract

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. A locally quadratic convergence rate for zero residual problems and a 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.