Optimization using Newton's method gradient hessian

[proglearn]

Hello,

This is my first post here. So I hope I'm posting in the right place, sorry if not.
http://homes.soic.indiana.edu/classes/spring2012/csci/b553-hauserk/Newtons_method.pdf

I am trying to solve the following numerical optimization function using Netwon's Method:

So, if I have the gradient of a function f(x) in xk=(1, -2) and also the hessian at the same point, how can I calculate which is the quadratic aproximation of this function using Newton's method multivariate.

 

[jvicens]

Hello,

Welcome to the forum! It looks like you are working on a numerical optimization problem using Newton's method. Newton's method is a powerful tool for finding the minimum or maximum of a function, and it relies on the first and second derivatives of the function.

To calculate the quadratic approximation of the function, you will need to use the Hessian matrix, which is the matrix of second derivatives of the function. In your case, since you have the gradient and Hessian at a specific point, xk=(1, -2), you can use the following formula to calculate the quadratic approximation:

f(x) ≈ f(xk) + ∇f(xk)^T(x-xk) + 1/2(x-xk)^TH(x-xk)

Where ∇f(xk) is the gradient at xk, H is the Hessian matrix at xk, and x is the variable you are optimizing for. This formula represents the quadratic approximation of the function around the point xk.

Now that you have the quadratic approximation, you can use Newton's method to find the minimum or maximum of the function. The general formula for Newton's method is:

xk+1 = xk - [H^-1∇f(xk)]

Where H^-1 is the inverse of the Hessian matrix. This formula tells us that to find the next point, xk+1, we need to take the current point, xk, and subtract the inverse of the Hessian matrix multiplied by the gradient at xk. This process is repeated until we reach a point where the gradient is close to zero, indicating that we have found a local minimum or maximum.

I hope this helps with your problem. Keep in mind that Newton's method can be sensitive to the starting point, so make sure to choose a good initial guess. Also, be careful when using the Hessian matrix, as it can be computationally expensive for large problems. Good luck with your optimization!