regression optimisation problem with $l_1$ and $l_2$ norms for $x \in \Bbb R^n$

Question

I'm trying to solve an optimization problem: $$\text{argmin}_{x \in \Bbb R^n}~ f(x),~ f(x) = ||x - a||_2 ^2 + \lambda ||x||_1,~ \lambda>0.$$

Any thoughts on how to solve it?

Thanks in advance!

Answer 1

This is a soft-thresholding problem for linear regression, for which a closed-form solution is provided here: Derivation of Soft Thresholding Operator