Let X and Y be two Hilbert spaces. Let $f : X \rightarrow ]-\infty, +\infty]$ be convex and lower semicontinuous, let $A : X \rightarrow Y$ be linear, and let $g: Y \rightarrow ]-\infty,+\infty]$ be convex and lower semicontinuous. Recall that in the context of Fenchel–Rockafellar duality, the primal problem is defined by
$(P)\qquad p := \inf_{x\in X} f(x) + g(Ax)$
and the dual problem is
$(D)\qquad d := \inf_{y\in Y} f^*(-A^*y) + g^*(y),$
where $A^*$ denote the conjugate/transpose of A. Let $\bar{x}\in X$ and $\bar{y}\in Y$ . Show that the following are equivalent:
(i) $\;\;\bar{x} $ is a primal solution, $\bar{y}$ is a dual solution, and $p = -d$.
(ii) $-A^*\bar{y}\in\partial f(\bar{x})$ and $\bar{y}\in\partial g(A\bar{x}).$
(iii) $\;\;\bar{x}\in\partial f^*(-A^* \bar{y})\cap A^{-1}\partial g^*(\bar{y})$.
I believe I can show $(i)\Rightarrow (ii)$ as follows
\begin{align} (i)&\Rightarrow f(\bar{x}) + g(A\bar{x})=p=-d=-f^*(-A^*\bar{y}) + g^*(\bar{y})\\ &\Rightarrow f(\bar{x})+g(A\bar{x})=-f^*(-A^*\bar{y}) + g^*(\bar{y})\\ &\Rightarrow (f(\bar{x})+f^*(-A^*\bar{y})-\langle \bar{x},-A^*\bar{y}\rangle)+(g(A\bar{x})+g^*(\bar{y})-\langle A\bar{x},\bar{y}\rangle)=0\\ &\Rightarrow -A^*\bar{y}\in\partial f(\bar{x}) \text{ and } \bar{y}\in\partial g(A\bar{x}). \end{align}
but I am looking for pointers with the other two implication.
The proof (i) implies (ii) is correct and reversible, so (i) and (ii) are equivalent. To see that (ii) and (iii) are equivalent, observe that $(\partial f)^{-1} = \partial f^*$ and $(\partial g)^{-1}=\partial g^*$.