Here is my problem. I am trying to compute the following quantity: $A = \sup_{x \in X} H(x)$. In my problem, $x$ is not a scalar variable ($X$ is actually infinite-dimensional) so performing the $\sup$ seems untractable. However, note that if I could express $H(x) = \sup_{y \in \mathbb{R}} F(x,y)$, then I would have $$A = \sup_y \sup_x F(x,y)$$ and now if $F$ is 'simple' such that doing $\sup_x F(x,y)$ is analytically tractable, I would have won as I reduced the calculation of $A$ to optimization over a finite-dimensional (here, one-dimensional) variable.
However, in my case what I have is that I can epxress $H(x) = \mathrm{Re}[\mathrm{extr}_{z \in \mathbb{C}} F(x,z)]$. Here, $F(x,z)$ is an holomorphic function of $z$ and notation "extr" actually means that I can show that there is a unique solution $z^\star(x)$ in $\mathbb{C}$ to $\frac{\mathrm{d} F}{\mathrm{d}z}(x,z) = 0$, and I define $\mathrm{extr}_{z \in \mathbb{C}} F(x,z) \equiv F(x,z^\star(x))$. Again, $F(x,z)$ is 'simple' as a function of $x$, for instance I am able to do analyticaly $\sup_x \mathrm{Re}[F(x,z)]$ for any $z$.
My question is thus: can I use the form $H(x) = \mathrm{Re}[\mathrm{extr}_{z \in \mathbb{C}} F(x,z)]$ to simplify my original optimization problem ? I tried but could not find any significant simplification.
Thanks a lot for your help !