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 $x^\*$