On the convergence of Newton iterations to non-stationary points

Richard H. Byrd, Marcelo Marazzi, Jorge Nocedal

Research output: Contribution to journalArticlepeer-review

35 Scopus citations


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 also shows that a class of line search feasible interior methods cannot exhibit convergence to non-stationary points.

Original languageEnglish (US)
Pages (from-to)127-148
Number of pages22
JournalMathematical Programming
Issue number1
StatePublished - Jan 2004

ASJC Scopus subject areas

  • Software
  • Mathematics(all)


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