Scaling a vector-valued non-linear function for numerical optimization/minimization [closed]

I am trying to minimize a non-linear vector-valued function in MATLAB. As a test case for my code, I try to minimize a function whose solution I know apriori.

The problem is that one of the solutions is widely distributed in scale. They are [175,164,854,3.7e5,6000];

As you can see, the 3.7e5 has a wide difference in scale with the rest of the solution vector. As a result, despite changing tolerances, algorithms etc., the best that MATLAB can find for my x(4) solution is 5.2e5

Now, I know that using a diagonal (or close to diagonal) scaling can help in finding the minimizing solution. However, from my understanding, the Hessian matrix is needed to obtain the scaling factor, i.e. Ideally I would like to scale the solutions so that the Hessian of the objective function becomes the identity matrix at/near the solution.

But the problem is, the Hessian of a vector-valued function is a Tensor and not a matrix.

Is there a solution, or any other method for handling this problem ?

closed as unclear what you're asking by Anton Menshov♦Jun 6 at 20:26

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• You haven't told us how you are minimizing your vector valued function. Are you using some sort of scalarization? – Brian Borchers Mar 31 '16 at 21:00