I'm working on solving a pretty tricky optimization problem, where the constrain functions $g_i$ are not well behaved, so has been challenging for optimizers to work on directly. Instead of solving this problem directly, I've transformed it into a much better behaved problem where instead of solving for the values directly, we solve some related ones that forces all the constrains to have a certain derivative, and then recovers the original variables from there.
We end up solving a problem we know that at the current point $(x,y)$
$u g_x = v g_y$
where $(u, v)$ are the decision variables.
I want to understand how the gradient and hessian look with respect to u, v rather than x and y.
This hasn't been difficult for the first derivatives and diagonal entries of the hessian, however I've been having trouble discovering the rule that can give me the off-diagonal hessian entries / cross second derivatives:
$\frac{d^2x}{dudv}, \frac{d^2y}{dudv} $ ....
What is the rule that I can use to define these inverse second cross derivatives in term of derivatives of the original function g?
The $g_i$ functions are of up to 4 variables, although I just posted the 2 variable example for simplicity.