I'm trying to do the derivation for the second problem on the page 13(slide 12) of this pdf : http://www.seas.ucla.edu/~vandenbe/ee236a/lectures/duality.pdf
I'm using Fenchel-Rockafellar duality. If we let $A$ from $X$ to $Y$, ( $(X,W)$ and $(Y,Z)$ are pairs hence $((X,Y),(W,Z))$ is a dual pair) and $$f(x)=c^{T}x+\delta_{R_{+}^{n}}(x)$$ $$g(y)=\delta_{R_{+}^{n}}(y)$$ then $f^\star(w)=\delta_{R_{+}^{n}}(c-w)$ and $g^\star(z)=b^{T}z$ then the duality will be $\sup-\{\delta_{R_{+}^{n}}(c+A^\star z)+b^{T}z\}$ according to the Fenchel-Rockafellar duality. But how I read this is maximimize $-b^{T}z$ subject to $c+A^\star z\geq 0$ which is not similar to the given duality. What am I doing wrong?
I'm using this pdf as a source https://people.math.ethz.ch/~patrickc/CA2013.pdf . You can find the definition I'm using from this pdf or Ican write them if needed.
Let $y = -z$. Then the dual problem you derived can be written as maximizing $b^T y$ subject to $A^T y \leq c$. This agrees with the dual problem shown on slide 12.
By the way, the Cheridito lecture notes you linked to look quite good, but Vandenberghe's lecture notes are self-contained. In this case you can derive the dual problem by first converting the primal problem to inequality form and then applying the result on slide 2.
You might also consider reading chapter 5 of the Boyd and Vandenberghe Convex Optimization textbook (which is free online). It gives a simple explanation of Lagrange duality, which is an easy way to derive the dual problem shown on slide 2. (Not that there is anything wrong with the Fenchel duality approach; it's also very useful.)