Convergence of a regularized Euclidean residual algorithm for nonlinear least-squares

The convergence properties of the new regularized Euclidean residual method for solving general nonlinear least-squares and nonlinear equation problems are investigated. This method, derived from a proposal by Nesterov [Optim. Methods Softw., 22 (2007), pp. 469-483], uses a model of the objective function consisting of the unsquared Euclidean linearized residual regularized by a quadratic term. At variance with previous analysis, its convergence properties are here considered without assuming uniformly nonsingular globally Lipschitz continuous Jacobians nor an exact sub-problem solution. It is proved that the method is globally convergent to first-order critical points and, under stronger assumptions, to roots of the underlying system of nonlinear equations. The rate of convergence is also shown to be quadratic under stronger assumptions. © 2010 Society for Industrial and Applied Mathematics.