Why the dual of a conic programming is also conic?

Question

I'm studying a lecture note about conic programming.

A conic programming over a cone $K\in \mathbb R^n$ is an optimization problem of this type:

minimize $\left< c,x \right >$

subject to $Ax=b$

$x\in K$

and also here is the dual of this conic programming:

maximize $\left<b,y\right>$

subject to $c=z+A^Ty$

$z\in K^*$

Here $A$ is a $n\times m$ matrix and $K^*$ is the dual cone of $K$. It is said that the second program is also a conic program. My question is that how we can write the dual program as type of the first program?

If you give me a hint I will be grateful/.

Answer 1

minimize $\left<\begin{pmatrix}-b \\ 0\end{pmatrix},\begin{pmatrix}y \\ z\end{pmatrix}\right>$

subject to $\begin{pmatrix}A^T & I\end{pmatrix}\begin{pmatrix}y \\ z\end{pmatrix} = c$

$\begin{pmatrix}y \\ z\end{pmatrix} \in \mathcal{K}$

with $\mathcal{K}=\mathbb{R}^n \times K^*$.