What is the proximal operator of the $ \left\| x \right\|_{\infty} $ norm:
$$ \operatorname{Prox}_{\lambda \left\| \cdot \right\|_{\infty}} \left( v \right) = \arg \min_{x} \frac{1}{2} \left\| x - v \right\|_{2}^{2} + \lambda \left\| x \right\|_{\infty} $$
I know we have to take the subgradient and compute it but I am a bit stuck.
Can anyone show me steps?
Let $f(x) = \|x\|_{\infty}$, so $f^*$ is the indicator function of $B$, where $B$ is the 1-norm unit ball.
The Moreau decomposition expresses the prox operator of $f$ in terms of the prox operator of $f^*$, which simply projects onto $B$. So we need to know how to project onto $B$. This is explained in ch. 8 ("the proximal mapping") of Vandenberghe's 236c notes. See slide 8-15 here.