Global convergence of Gauss-Newton-MBFGS method for solving the nonlinear least squares problem

In this paper, by using a modified BFGS (MBFGS) update, we propose a structured MBFGS update for the nonlinear least squares problem. We then propose a hybrid method that combines the Gauss-Newton method with the structured MBFGS method for solv-ing the nonlinear least squares problem. We show that the hybrid method is globally and quadratically convergent for zero residual problems, and globally and superlinearly con-vergent for the nonzero residual problems. We also show that the unit step is essentially accepted. We also present some preliminary numerical results which show that the hybrid method is comparable with existing structured BFGS methods.