Newton’s method




     There  are  two  major shortcomings of the classical conver‐
gence analysis of Newton’s method.

Newton’s method: unconstrained minimization.



where f: Rˆn ‐> R is convex and twice continuously differentiable
(which implies that dom f is open).

A necessary and sufficient condition for a point x* to be optimal
is



Solving the unconstrained minimization problem  is  the  same  as
finding  a  solution of the gradient=0, which is a set of n equa‐
tions in the n variables x1,...,xn. In a few  special  cases,  we
can  find  a  solution  to  the problem by solving the optimality
equation, but usually the problem must be solved by an  iterative
algorithm.

By  this  we mean an algorithm that computes a sequence of points
x(0), x(1), ... such a sequence of points is called a  minimizing
sequence for the problem. The algorithm is terminated when



(some specified nonnegative tolerance)