In Convex Optimization by Boyd and Vandenberghe, I don't understand how the formulation for Example 2.9 works.
Example 2.9: Polyhedron. The polyhedron $ \{ x | Ax \preceq b, Cx=d \}$ can be expressed as the inverse image of the Cartesian product of the nonnegative orthant and the origin under the affine function $f(x) = (b-Ax, d-Cx)$ $$ \{ x | Ax \preceq b, Cx=d \} = \{ x | f(x) \in \mathbb{R}^m_+ \times \{0\} \} $$
If I understand correctly, $\mathbb{R}^m \times \{ 0 \}$ looks something like $(x,y,0)$ if $m=2$.
I'm assuming that $Cx=d$ is reformatted to $d-Cx=0$, which can represented matrix-wise as $[d-Cx | 0]$, which kind of makes sense to me.
How then, does $b-Ax \succeq 0$ work? Since $b-Ax$ produces a vector, would this be vector based (component-wise inequality) comparison and not positive semidefinite? How do you represent that in the form of $f(x)=(z_1,...,z_m,0)$?
And how does the original equation represent both inequalities at the same time? I suspect that I'm missing some background/definitions with notation.