Local Convergence of the Gauss–Newton–Broyden Method for Solving Nonlinear Least Squares Problems

The Gauss–Newton–Broyden method is proposed and investigated for solving a nonlinear least squares problem with operator decomposition. This method is obtained from the Gauss–Newton method by replacing the Jacobian matrix of a nonlinear operator with the sum of the derivative of the differentiable part of the operator and the matrix computed by the Broyden update formula for the other part of the nonlinear vector function. A local convergence theorem for the proposed method, under the classical Lipschitz conditions and for problems with zero residual, is proved. The results of numerical experiments are also presented.