I have an optimization problem operating on the implicit function $$F(x, y, z) = 0$$ and a vector $$(u, v, w)$$
where the solution to the problem is the point $(x_0, y_0, z_0)$ that satisfies the property that $$\nabla F \parallel (u, v, w)$$
F defines a smooth strictly convex body, so we are guaranteed that this point exists, and is unique.I'm interested the first and second order derivatives of the solution point $(x_0, y_0, z_0)$ with respect to the variables u, v, w.
The first order derivatives are relatively easy to obtained, however I'm having trouble establishing the identities of the second order derivatives.
In order to solve this, we treat z as an implicit function of x and y, and then solve the function$$G(x,y) = (\frac{dz}{dx}, \frac{dz}{dy})$$for the point $(x_0, y_0)$$$G(x_0, y_0) = (\frac{u}{w}, \frac{v}{w})$$ using newton's method.
Getting the gradients of the underlying variables x, and z with respect to u, and v is relatively simple (falls directly out of using newton's method), solving the system:
$$Jac(G) \begin{bmatrix} dx \\\ dy \\ \end{bmatrix} = \begin{bmatrix} \frac{d^2z}{dx^2} & \frac{d^2z}{dxdy} \\\ \frac{d^2z}{dxdy} & \frac{d^2z}{dy^2} \\ \end{bmatrix} = \frac{1}{w}\begin{bmatrix} du \\\ dv \\ \end{bmatrix}$$
which we can recognize as an application of the inverse function theorem, thus making the first derivatives the entries of the inverse jacobian:
$$\begin{bmatrix} \frac{dx}{du} & \frac{dy}{du} \\\ \frac{dx}{dv} & \frac{dy}{dv} \\ \end{bmatrix} = Jac(G)^{-1} = \frac{1}{det(Jac(G))} \begin{bmatrix} \frac{d^2z}{dy^2} & -\frac{d^2z}{dxdy} \\\ -\frac{d^2z}{dxdy} & \frac{d^2z}{dx^2} \\ \end{bmatrix}$$
We can further solve for the derivatives of x and y with respect to w:
$$Jac(G) \begin{bmatrix} dx \\\ dy \\ \end{bmatrix} = \begin{bmatrix} \frac{u}{w^2}dw \\\ \frac{v}{w^2}dw \\ \end{bmatrix}$$$$\begin{bmatrix} \frac{dx}{dw} \\\ \frac{dy}{dw} \\ \end{bmatrix} = Jac(G)^{-1} \begin{bmatrix} \frac{u}{w^2} \\\ \frac{v}{w^2} \\ \end{bmatrix}$$
and finally the inverse z derivatives:
$$\frac{dz}{du} = \frac{dx}{du}\frac{dz}{dx} + \frac{dy}{du}\frac{dz}{dy} = \frac{dx}{du}\frac{u}{w} + \frac{dy}{du}\frac{v}{w}$$$$\frac{dz}{dv} = \frac{dx}{dv}\frac{dz}{dx} + \frac{dy}{dv}\frac{dz}{dy} = \frac{dx}{dw}\frac{u}{w} + \frac{dy}{dw}\frac{v}{w}$$$$\frac{dz}{dw} = \frac{dx}{dw}\frac{dz}{dx} + \frac{dy}{dw}\frac{dz}{dy} = \frac{dx}{dw}\frac{u}{w} + \frac{dy}{dw}\frac{v}{w}$$
I have verified all of these derivatives against an various function with finite difference tests, and found them to be correct.
I've attempted a number of methods to get the second derivatives of the inverse, however have yet to find one that matches with my calculate finite difference test cases.
What is the correct derivation of these? This seems like a problem that should have been studied / have the solution published in a textbook or paper, however I've had trouble finding any.