Suppose we have the primal problem:
$\mathop{\min}\limits_{x\in X} F(Kx)+G(x)$,
where $F,K,G$ are linear operators. And I want to ask how is the primal-dual formulation work to get the saddle-point problem form:
$\mathop{\min}\limits_{x\in X}\mathop{\max}\limits_{y\in Y} \langle Kx,y\rangle +G(x)-F^{*}(y)$,
where $X,Y$ are two finite-dimensional spaces, and $F^{*}$ is the conjugate function of $F$.
$\newcommand{\<}{\left\langle}\newcommand{\>}{\right\rangle}\newcommand{\R}{\mathbb{R}}\newcommand{\Rbar}{\overline{\R}}$Recall that for a convex function $F:\R^n\to\Rbar$, its convex conjugate, $F^*:\R^n\to\Rbar$, is defined as
$$ F^*(y) = \sup_{x} \<x, y\> - F(x). $$
The conjugate of the conjugate, $F^{**}:\R^n\to\Rbar$ is
$$ F^*(x) = \sup_{y} \<x, y\> - F^*(y). $$
By the Fenchel-Moreau Theorem, if $F$ is convex then $F$ is lsc and convex if and only in $F=F^{**}$.
You can find this in several books such as (i) Theorem 12.2 in R.T. Rockafellar, "Convex Analysis," Princeton University Press, 1970, (ii) Theorem 13.32 in H.H. Bauschke and P.L. Combettes, "Convex Analysis and Monotone Operator Theory in Hilbert Spaces," Springer, 2011.
As a result, assuming that $F$ is proper and lsc,
$$ \operatorname*{Minimize}_{x} F(Kx) + G(x), $$
is equivalent to
$$ \operatorname*{Minimize}_{x} F^{**}(Kx) + G(x), $$
which is equivalent to
$$ \operatorname*{Minimize}_{x} \sup_{y} \<Kx, y\> - F^*(y) + G(x). $$
You need some additional assumptions to turn $\sup$ into a $\max$. In particular, you need to assume that $F^*$ is level-bounded.