A polyhedral cone can be defined as
$$ \mathcal{K} = \{x~|~Ax\preceq 0\}, $$
where $A \in \mathbb{R}^{m \times n}$, $x\in \mathbb{R}^n$ and $\preceq$ denotes component-wise less than and equal to.
The dual cone of $\mathcal{K}$ is denoted as $\mathcal{K}^*$, and according to the definitions of the dual cone, we have
$$ \mathcal{K}^* = \{ y~|~ y^Tx \geq 0,~ Ax\preceq 0 \}.$$
There are already some equivalent form, for example, in Expressions for the dual of a polyhedral cone:
$$\mathcal{K}^* = \{y ~|~ y = -A^Tz,~ z\succeq 0\}.$$
But I am wondering if there is another compact form that does not depend on any other variable (like $z$). That is, I want the following form:
$$\mathcal{K}^* = \{y ~|~ By \preceq 0\}.$$
But I need to find out what the relationship is between $B$ and $A$. I initially guessed $B^T = A^{\dagger}$, where $A^{\dagger}$ is the Moore–Penrose inverse of $A$. However, after some attempts, I found that this conclusion seems to be incorrect.
Could someone help me find a solution?
