I am working on a problem in which I need to find the supremum of an expression. Namely, the expression below:
$$ \sup_{w > 0}\Big\{\operatorname{tr}[ \rho \log w] - \log\operatorname{tr}[\sigma w ]\Big\} $$
where $\rho$ is a density matrix (positive semi-definite Hermitian operator of trace one) and $\sigma$ is a positive semi-definite operator. You can see this expression here.
The first thing that comes to my mind, is taking the derivative of the expression and setting it to zero to find the answer. However, I do not know how to calculate the derivative of the above expression. I know how to take the derivative of trace in basic cases (i.e. the cases in The Matrix Cookbook) but I need help in this case.
I have also looked at similar expressions where sup was calculated, to solve the above problem, but I did not understand those either. The examples are the following from this paper:
$$ \sup_{w {\geq} 0}\Big\{\operatorname{tr} \rho \log w - \log\operatorname{tr}\exp(\log w + \log \sigma)\Big\} $$ and $$ \sup_{w \in S(A)}\Big\{\operatorname{tr} H w - D(w\,\|\,\sigma)\Big\} $$
$S(A)$ is the set of density matrices on $A$, $H$ is a Hermitian operator and $D(w\,\|\,\sigma)$ is the relative entropy.
I would appreciate to know how to calculate the derivative or take the supremum in general or even a resource where I can learn from.
