Copyright © 2009 Mohammedi R. Abdel-Aziz and Mahmoud M. El-Alem. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The minimization of a quadratic function within an ellipsoidal trust region is an important
subproblem for many nonlinear programming algorithms. When the number of variables is large,
one of the most widely used strategies is to project the original problem into a small dimensional
subspace. In this paper, we introduce an algorithm for solving nonlinear least squares problems.
This algorithm is based on constructing a basis for the Krylov subspace in conjunction with a
model trust region technique to choose the step. The computational step on the small dimensional
subspace lies inside the trust region. The Krylov subspace is terminated such that the termination
condition allows the gradient to be decreased on it. A convergence theory of this algorithm is
presented. It is shown that this algorithm is globally convergent.