Most of time Newton Method in optimization is used to find the local minimum of a function. I am wondering what would happen if we have an maximization problem. What happened to the update equation in the maximization case:
X_k+1= x_k-t*dx OR X_k+1=x_k+t*dx
This question was asked here
Basic Question about Newton's Method for Optimization
However there two different and opposing responses and they both some votes. I would like to know can anyone clarify this ambiguity.
That's because it depends a bit on which Newton method you refer to.
In the one case, it's Newton's root-finding algorithm applied to the gradient of the function: this method will find a local extremum which may be a minimum or a maximum (or a saddle point). To find which, you need further exporation (for instance, looking at second order information or at the values of the function at different extrema).
In the other case, it's the Newton gradient descent method. In this case we take steps in direction of the gradient $\nabla f$ to increase the function (to find maxima) and in the direction of the negative gradient $-\nabla f$ to decrease the function (to find minima).