Linear optimization problem of $argmin$

Question

Consider a vector $a \in R^n$.

I want to know how I can find analytically the solution of the following optimization problem: $x^* = argmin_{x \in R^n} f(x)$, where

• $f(x) = ||x-a||_{2}^2 + \lambda ||x||_1$
• $\lambda > 0$ and
• $||.||_p$ is the p-norm in $R^n$.

Thanks in advance.

Answer 1

In response to the comments suggesting subgradient descent methods: the problem in the question does not contain any sum or similar terms, therefore the variables are independent of each other and there is a closed-form solution: $$ x^* = \begin{cases} a - \frac{\lambda}{2},& \text{if } a > \frac{\lambda}{2}\\ a + \frac{\lambda}{2},& \text{if } -a > \frac{\lambda}{2}\\ 0, & \text{otherwise} \end{cases} $$