Good evening,
I'm looking for references for the existence of minimizers to the problem: \begin{equation} \arg\min_{x\in\mathbb{R}^d}\Vert Ax-y\Vert^2+\lambda\Vert Dx\Vert_1, \end{equation} where $A$ is a $k\times d$ real matrix, $\lambda\geq0$, $y\in\mathbb{R}^k$ and $D$ is a $N\times d$ matrix, $\Vert z\Vert^2=z_1^2+...+z_d^2$ and $\Vert{z}\Vert_1=|z_1|+...+|z_N|$.
I know it shall follow easily by sub gradient arguments, but I can't find any reference to a theorem from which it follows directly.