Newton Direction

Resources: page=43

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Assuming that ${\nabla }^2{f}_k$ is a Positive-Definite Matrix, we find the Newton Direction by setting the derivative of $m_k(p)$ to zero.

$$p_k^N=-({\nabla }^2{f}_k{)}^{-1}\nabla f_k$$

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A important search direction, derived from the second-order Taylor series approximation to $f(x_k+p)$.

$$f(x_k+p)\approx \underset{m_k(p)}{\underbrace{f_k+p^T\nabla f_k+\frac{1}{2}{p}^T\nabla f^2{f}_{k}p}}$$

A way to find optimums, with potential fewer iterations and greater accuracy than methods that rely solely on the gradient.

Positives and Drawbacks

Very fast convergence, typically quadratic. However, we need the Hessian ${\nabla }^{2}f(x)$ which is an error-prone and expensive process.