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!

How is this any different from the case you know of by simply identifying $A=I$ and $b = -a$. — Johan Löfberg, Oct 12 '18 at 19:51
This looks to be a convex optimization problem, since the L1 and L2 norms are both convex functions over the domain supplied, and $\lambda > 0$. — Nurmister, Oct 12 '18 at 20:31
Looks like Lagrangian form of Lasso. https://en.wikipedia.org/wiki/Lasso_(statistics) — mathreadler, Oct 12 '18 at 20:39

score 1 · Accepted Answer · answered Oct 14 '18 at 00:44

1

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

answered Oct 14 '18 at 00:44

Ryan Cory-Wright

1 Answers1