An example of slow convergence for Newton’s method on a function with globally Lipschitz continuous Hessian

An example is presented where Newton’s method for unconstrained minimization is applied to find an ǫ-approximate first-order critical point of a smooth function and takes a multiple of ǫ−2 iterations and function evaluations to terminate, which is as many as the steepest-descent method in its worst-case. The novel feature of the proposed example is that the objective function has a globally Lipschitz-continuous Hessian, while a previous example published by the same authors only ensured this critical property along the path of iterates, which is impossible to verify a priori.