I am presented with the following minimization problem:
$$\text{minimize}\,\, g(x) \\ \text{s.t.}~~ Ax=b \\ $$
$\{x,b\}$ are vectors and $A$ is a matrix and $g(x)$ is twice continuously differential.
and I am tasked with finding the gradient $\nabla q(z)$ and the curvature $\nabla^2 q(z)$ where $q(z)=g(x_o+Hz)$. $H$ is an orthogonal projector onto the nullspace of $A$, and $x_o$ is a feasible point.
I do not understand how I can compute this without knowing what $g(x)$ is.
I tried a few things. If we know that $x_o$ is feasible, and we know that $Hz$ is in the nullspace of $A$, then we can say $A(x_o+Hz)=b$. But this only tells me that it is feasible at that point, and not much else.
I looked at this related question: Calculating the gradient without knowing the function but it did not help me much.
EDIT: some have suggested apply the chain rule, but I have not found anything useful in it. I looked at this resource: https://math.dartmouth.edu/archive/m9f07/public_html/m9lect1119.pdf, and this question: What is a Chain rule for gradient?. But both of them seem to still require knowing what $g(x)$ is.
EDIT 2: I tried to apply the extreme value theorem to this problem: http://www.ma.utexas.edu/users/rodin/408/ExtremeValues.pdf, figuring that maybe there was a way to abstract away from just computing the gradient. But I cannot see a way around knowing g(x)$.
How I can compute this without knowing what $g(x)$ is?