Newton's method for optimization

Question

I have been reading about Newton's method and know that you can use it for optimization problems. However, does Newton's method only guarantee convergence to a local minimum or maximum, or can it be used to guarantee convergence to a global minimum or maximum?

Answer 1

It only guarantees convergence to a local minimum or maximum. But if the function is convex, then local minimum is the global minimum.

Answer 2

Newton's method only guarantee the convergence to a stationary point $\bar x$ with $\nabla f(\bar x) = 0$ if you start closely enough to $\bar x$ (and if the Hessian $\nabla^2 f(\bar x)$ is non-singular.