Newton-Type Methods for Nonlinear Least Squares Using Restricted Second Order Information

In the paper, a special approximated Newton method for minimizing a sum of squares f(x) = 1 2 ‖F (x)‖ = 1 2 Pm i=1[Fi(x)] 2 is introduced. In this Restricted Newton method, the Hessian H = G + S of f where G = (F ′)T F ′, S = F ◦ F ′′, is approximated by ARN = G + B where B = Z2Z T 2 SZ2Z T 2 is the restriction of the second order term S on the subspace imZ2 spanned by the eigenvectors of the GaussNewton matrix G which belong to the q smallest eigenvalues of G. Some properties of this approximation are derived, and a related trust region method is tested on hand of some test functions from the literature.