I want to minimize $$\|Ax-b\|_2^2 + \|Cx-d\|_2^2 + \|x\|_1$$
I know we can introduce an auxiliary variable and use ADMM to separate the $\ell_2$ and $\ell_1$ norms.
But is there a more convenient way, such as direcly combine $\|Ax-b\|_2^2 + \|Cx-b\|_2^2$ into one $\ell_2$ norm? Just like we can transform numeral problem such as minimizing $(ax+b)^2+(cx+d)^2$ to minimizing $(ex+f)^2$.
I tried this way. But I found that it's hard to deal with $(Ax-b)^T(Ax-b)+(Cx-d)^T(Cx-d)$. The core is that I don't know how to transform $x^T(A^TA+C^TC)x$ to $x^T(E^TE)x$. Maybe this method is called completing the squares?