On a New Updating Rule of the Levenberg-Marquardt Parameter

A new Levenberg-Marquardt (LM) algorithm is proposed for nonlinear equations, where the iterate is updated according to the ratio of the actual reduction to the predicted reduction as usual, but the update of the LM parameter is no longer just based on that ratio. When the iteration is unsuccessful, the LM parameter is increased; but when the iteration is successful, it is updated based on the value of the gradient norm of the merit function. The algorithm converges globally under certain conditions. It also converges quadratically under the local error bound condition, which does not require the nonsingularity of the Jacobian at the solution.