Convergence of a Regularized Euclidean Residual Algorithm 
                    for Nonlinear Least-Squares

             S. Bellavia, C. Cartis, N. I. M. Gould,  
                    B. Morini and Ph. L. Toint

               Report TR08/11   August 2008

The convergence properties of the new Regularized Euclidean Residual method
for solving general nonlinear least-squares and nonlinear equations problems
are investigated. This method, derived from a proposal by Nesterov (2007),
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
exact subproblem 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.