I would like to develop a formula for the Gradient and Hessian of $f(x)= \phi(Ax)$ where $f $ is a nonlinear vector function $x$ is a vector $A$ is a matrix and $\phi$ is a nonlinear vector function.
$x$ is a vector in $R^n$.
$A$ is an $m*n$ real matrix.
$\phi$ is nonlinear vector function from $R^n$ to $R$
I tried the following: $Ax=u$
$du=Adx$
$df=<\nabla \phi,du>$
but how can I achieve from that the Gradient and Hessian of $f$?
You know the gradient $(g)$ and Hessian $(H)$ of a function $(f)$ with respect to the variable $u$.
And now you want to find the gradient $(p)$ and Hessian $(Q)$ with respect to a new independent variable $(x)$, which is related to the original variable by $$\eqalign{ u &= Ax \cr du &= A\,dx \cr }$$ For convenience, let me denote the inner/Frobenius product by a colon rather than angle brackets. To find the new gradient, start with the differential of the function and then switch to the new independent variable $$\eqalign{ df &= g:du = g:A\,dx = A^Tg:dx \cr \frac{\partial f}{\partial x} &= A^Tg = p \cr }$$ To find the new Hessian, start by taking the differential of the new gradient $$\eqalign{ dp &= A^T\,dg = A^TH\,du = A^THA\,dx \cr \frac{\partial p}{\partial x} &= A^THA = Q \cr }$$