I am trying to derive the optimality conditions of a conic program which is a minimization problem in the form: $$ \mathrm{Minimize}_{x,s} c'x\\ s\in C\\ Ax + s = b. $$ Here $C$ is a cone with nonempty relative interior. But I would like to do so using the subdifferential-based Fermat's theorem which states that for a convex lower-semicontinuous function $F:\Re^n\to\Re\cup\{+\infty\}$, the optimization problem $$ \mathrm{Minimize}_x F(x), $$ has the optimality condition $0\in\partial F(x)$.
I am trying to apply this theorem to the conic problem above and derive the standard (primal-dual) optimality conditions. For that, I defined the function $F(x) = c'x + \delta_C(b-Ax)$, where $\delta_C$ is the indicator function of $C$, so the first-order optimality conditions become
$$ 0\in \partial (c'x + \delta_C(b-Ax)) = c - A' (\partial \delta_C)(b-Ax) $$
equivalently
$$ c \in A' (\partial \delta_C)(b-Ax) $$
or
$$ c = A'y\\ y \in \partial \delta_C(b-Ax) = N_C(b-Ax), $$
where $N_C$ is the normal cone of $C$.
At the same time, it is quite standard that the primal-dual optimality conditions of the above conic problem are:
$$ b - Ax \in C\\ y \in C^*\\ A'y + c = 0\\ c'x + b'y = 0, $$
where $C^*=\{z: z'x\geq 0, x \in C\}$ is the dual cone of $C$.
I can't see how to solve the above optimality conditions to yield these standard optimality conditions. I don't know how to go from $N_C$ to a term involving $C^*$.
You seem to be looking for the following identity which can be found in Hiriart-Urruty and Lemaréchal's book "Fundamentals of Convex Analysis," ex. 5.2.6:
$$ N_C(z) = C^\circ \cap \{z\}^\perp, $$
where $C^\circ$ is the polar of $C$. Can you take it from there?