Newton optimization algorithm with non-positive definite Hessian In the newton optimization algorithm to find the local minimum $x^*$ of a non-linear function $f(x)$ with iteration sequence of $x_0 \rightarrow x_1 \rightarrow x_2 ... \rightarrow x^*$ all $\nabla ^2 f(x_k)$ should be pos. definite otherwise search direction might not be a descent direction. The solution to this problem is to use Hessian modification techniques.
I am looking for a simple function that has local minimum at $x^*$, but $\nabla ^2 f(x_k)$ is not positive definite, and hence the Newton algorithm fails. Then after using one of the Hessian modification techniques, Newton algorithm converges to $x^*$

If the Hessian is negative-definite, then the direction is automatically ascent.
Proof: if the Hessian is negative-definite, then its inverse exists and is also negative-definite. Newton's direction is $-[\nabla^2 f(x)]^{-1}(\nabla f)(x)$. Descent direction is $-(\nabla f)(x)$. Now the scalar multiplication of both is negative: $[(\nabla f)(x)]^{\text{T}}[\nabla^2 f(x)]^{-1}(\nabla f)(x)<0$. $\diamondsuit$
