# Different norms in stacked form optimisation

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?

• I doubt this is possible for $p\ne q$. – daw Nov 22 '18 at 20:34
• If you think the minimum will be 0 then you can always write the system of equations $Ax=y$ and $Dx=0$. But in practice one uses regression for very overdetermined systems. – Michal Adamaszek Nov 23 '18 at 7:40