I want to solve the optimization:
$$\arg\underset{x}{\min}f(x) = \arg\underset{x}{\min}\lambda\lVert Mx\rVert_1 + \frac{1}{2}\lVert x-y\rVert_2^2$$
Where $x,y\in\mathbb{R}^{n}$ and $M\in\mathbb{R}^{n\times n}$ (a positive semi-definite matrix representing a metric in my case).
Taking n=2 as example, after trying to develop the sub gradient of this target function I got (for component $x_1$) : $$\frac{\partial{f}}{\partial{x_1}}=\sum_{i=1}^2 m_{i1}\ \text{sign}(m_{i1}x_1 + m_{i2}x_2)+(x_1-y_1)$$
By following the derivation of $L_1$ regularized least squares (ref. https://angms.science/doc/CVX/ISTA0.pdf), we can draw polylines in $x_1x_2$ plane for a $y$ value, and finally find the intersection point where $\vec{0}\in\frac{\partial{f}}{\partial{x}}$.
But this method is hard to be generalized to 3D and higher dimension. I'm looking forward to some general solutions.
Thanks a lot!