A standard variational problem (arising, e.g., in imaging) reads
$$ \operatorname{argmin}_x \frac{1}{2}\Vert Ax - y\Vert ^2 + \Vert x\Vert_1 $$
where the $1$-norm serves as a sparsifying regularizer. In this case, I can clearly apply a forward-backward algorithm, since the fidelity is smooth and the regularizer is proximable.
My question: what if instead I have
$$ \operatorname{argmin}_x \frac{1}{2}\Vert Ax - y\Vert ^2 + \Vert x\Vert_1 + \iota_{\ge0}(x)$$
i.e. a positivity constraint on the variable entries (say, $x$ is an image and pixels have positive values)?
If I want to apply forward backward again, I should find explicitly the prox operator of $\Vert x\Vert_1 + \iota_{\ge0}(x)$, which amounts to solve the constrained problem
$$ \operatorname{argmin}_{x \ge 0} \Vert x \Vert_1 + \frac{1}{2\tau}\Vert x - \bar{x} \Vert^2 $$
Does anyone have hints on how to do that?