Critical solutions of nonlinear equations: local attraction for Newton-type methods (Q1702779)

This paper deals with convergence properties of Newton-type methods for solving a nonlinear equation \(\Phi(u)=0\) where the mapping \(\Phi\) is smooth enough. The convergence results assume a certain 2-regularity property of the solution of the nonlinear equation which implies that this solution is critical. In this paper, it is shown that, if \(\Phi\) is 2-regular at a (critical) solution \(\bar{u}\) in some direction \(v\), then \(v\) defines a domain star-like with respect to \(\bar{u}\) with nonempty interior, from which the iterates that satisfy the perturbed Newton method (pNM) framework necessarily converge to \(\bar{u}.\) The authors demonstrate how the general results for the pNM framework apply to some specific Newton-type methods. These include the classical Levenberg-Marquardt method and the LP-Newton method for nonlinear equations, and the stabilized Newton-Lagrange method for optimization (or stabilized sequential quadratic programming).