I'm trying to generalise the stacked form of a minimisation problem:
$$\text{argmin}_x||Ax-y||_p^p+\alpha||Dx||_q^q$$
where the L2 norm is often used, so $p=q=2$. This can be brought to
$$\text{argmin}_x\left|\left|\begin{bmatrix}A\\\sqrt{\alpha}D\end{bmatrix}x-\begin{bmatrix}y\\0\end{bmatrix}\right|\right|_2^2$$
And the left side being equal to the right side would optimise the situation, so taking both the square and the norm out, we are left with a basic matrix equation.
However, if one was to use different norms, that is $p\not=q$, I imagine the solution should change accordingly.
So given the minimisation, for example
$$\text{argmin}_x||Ax-y||_2^2+\alpha||Dx||_1$$
how would one start deriving the stacked form?
I tried to search for "optimisation stacked form" and some similar things, but didn't find much. My terminology may be a bit off. Maybe there's a name for this sort of approach?