On the convergence of Newton iterations to non-stationary points

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 min imization problem is also studied and conditions under which false convergence cannot occur are presented Several examples illustrating failure of Newton iterations for con strained 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 Department of Computer Science University of Colorado Boulder CO richard cs colorado edu This author was supported by Air Force O ce of Scienti c Research grant F Army Research O ce Grant DAAG and National Science Foundation grant INT Department of Industrial Engineering and Management Sciences Northwestern University Sheri dan Road Evanston IL marazzi iems northwestern edu This author was supported by Department of Energy grant DE FG ER A Department of Electrical and Computer Engineering Northwestern University Evanston IL nocedal ece northwestern edu www ece northwestern edu nocedal This author was supported by National Science Foundation grant CCR and Department of Energy grant DE FG ER A