In this paper two simple examples of a twice continuously differentiable strictly convex function $f$ are presented for which Newton's method with line search converges to a point where the gradient of $f$ is not zero. The first example uses a line search based on the Wolfe conditions. For the second example, some strictly convex function $f$ is defined as well as a sequence of descent directions for which exact line searches do not converge to the minimizer of $f$. Then $f$ is perturbed such that these search directions coincide with the Newton directions for the perturbed function while leaving the exact line search invariant.
Report naXys-11-2014, Namur Centre for Complex Systems, University of Namur, Belgium