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On the Convergence of Newton Iterations to Non-Stationary Points

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R. Byrd, M. Marazzi, J. Nocedal

We study conditions under which line search Newton methods for nonlinear
systems of equations and optimization fail due to the presence of singular
non-stationary points. These points are not solutions of the problem and
are characterized by the fact that Jacobian or Hessian matrices are singular.
It is shown that, for systems of nonlinear equations, the interaction between
the Newton direction and the merit function can prevent the iterates from
escaping such non-stationary points. The unconstrained minimization problem is
also studied, and conditions under which false convergence cannot occur are
presented. Several examples illustrating failure of Newton iterations for
constrained optimization are also presented. The paper concludes by showing
that a class of line search feasible interior methods cannot exhibit
convergence to non-stationary points.
Report OTC 2001/7 Optimization Technology Center
April, 2001

Contact: nocedal@ece.nwu.edu