I am trying to make a transformation on the set of parameters within the Ising model namely, $a_i$ and $b_{i,j}$. The Hamiltonian is: $H = \sum_{i} a_ix_i + \sum_i \sum_j b_{i,j}x_ix_j $
They need to be normalized between -1 and 1, but I cannot have values of order less than 1e-3, which I do after normalizing linearly/by a scaling factor. I figured one way of getting rid of this is by doing a transformation using $tanh$ or the logistic function on my parameters in order to normalize them.
However, the goal is to find the global minima (hopefully) of the Hamiltonian by solving for $x_i$. I'm sure if I applied this transformation to the entire Hamiltonian I could just apply the inverse transformation and I would get the same answer back since the transformations I am using are monotonic, but I am not certain if I apply it only to the given constants $a_i$ and $b_{i,j}$.
Would this be true that any minima I find in the transformed Hamiltonian would correspond to minima in the original Hamiltonian?