Assume $f$ and $g$ are convex, lower semi-continuous, proper functions defined respectively on $X$ and $Y$ which are (real) Hilbert spaces and $A$ is a bounded linear operator from $X$ into $Y$. Fenchel-Rockafellar primal-dual problems are defined to be:
\begin{equation} \begin{split} &(P) \qquad \min_{x \in X} f(x)+g(Ax) \\&(D) \qquad \min_{y \in Y} f^*(A^*y) + g^*(-y) \end{split} \end{equation}
where $f^*$ and $g^*$ are convex conjugates of $f$ and $g$ and $A^*$ is the adjoint of $A$.
Now recall the so called pirmal-dual relations in linear programming:
\begin{equation} \begin{split} &(p) \qquad \min \{\langle c,x \rangle: x\geq 0, \, Ax=b \} \\&(d) \qquad \min \{\langle b,y \rangle: y\in \mathbb{R}^m, \, A^*y\leq c \} \end{split} \end{equation}
Our goal is to assume $PD$ and derive $pd$. If we take $$ f = \langle .,c \rangle + \delta_{\mathbb{R}_+^n} \quad \text{and} \quad g=\delta_{\{b\}} $$ and inject them into $P$ we readily obtain $p$. However for $D \Rightarrow d$ I can't seem to get anywhere. This is my attempt in this regard (taking the same $f$ and $g$):
$\begin{equation} \begin{split} \min_y f^*(A^*y) + g^*(-y) &= \min_y \, \Big( \max_x \, \langle x,A^*y \rangle - f(x)\Big) + g^*(-y) \\& = \min_y \, \Big( \max_x \, \langle x,A^*y \rangle - \langle x,c \rangle - \delta_{\mathbb{R}_+^n} \Big) + \delta^*_{\{b\}} (-y) \\& = \min_y \, \Big( \max_x \, \langle x,A^*y \rangle - \langle x,c \rangle \Big) + \langle b,-y \rangle; \quad \, x \in \mathbb{R}_+^n \end{split} \end{equation}$
I'm not sure where to go from here. Any help will be appreciated.
Edit: by $\delta _ S$ we mean the characteristic function of the set $S$.
\begin{eqnarray} f^*(y) &=& \sup_x \langle y,x\rangle - f(x) \\ &=& \sup_x \langle y-c,x\rangle - \delta_{\mathbb{R}^n_+}(x) \\ &=& \sup_{x \ge 0} \langle y-c,x\rangle \\ &=& \delta_{\mathbb{R}^n_+}(c-y) \end{eqnarray} It is straightforward to compute $g^*(y) = -\langle b,y \rangle$.
Now note that $f^* (A^*y) = \delta_{\mathbb{R}^n_+}(c-A^*y)$, so we obtain $\inf \{ \langle b,y \rangle | A^* y \le c \}$.