A modified inexact Levenberg–Marquardt method with the descent property for solving nonlinear equations

In this work, we propose a modified inexact Levenberg–Marquardt method with the descent property for solving nonlinear equations. A novel feature of the proposed method is that one can directly use the search direction generated by the approach to perform Armijo-type line search once the unit step size is not acceptable. We achieve this via properly controlling the level of inexactness such that the resulting search direction is automatically a descent direction for the merit function. Under the local Lipschitz continuity of the Jacobian, the global convergence of the proposed method is established, and an iteration complexity bound of $$O(1/\epsilon ^2)$$ O ( 1 / ϵ 2 ) to reach an $$\epsilon $$ ϵ -stationary solution is proved under some appropriate conditions. Moreover, with the aid of the designed inexactness condition, we establish the local superlinear rate of convergence for the proposed method under the Hölderian continuity of the Jacobian and the Hölderian local error bound condition. For some special parameters, the convergence rate is even quadratic. The numerical experiments on the underdetermined nonlinear equations illustrate the effectiveness and efficiency of the algorithm compared with a previously proposed inexact Levenberg–Marquardt method. Finally, applying it to solve the Tikhonov-regularized logistic regression shows that our proposed method is quite promising.