Compute proximal mapping of this function

Question

I want to know how to compute the proximal mapping of this function:

$f(x) = \sup_y(yx - \frac{1}{2}\sigma y^2 ), \|y\|_{\infty} < \beta$

I know how to compute the proximal mapping when $\beta$ is 1, but I don't know how to do it when $\beta \neq 1$.

Do you guys have any ideas?

Answer 1

This is equivalent to computing the prox of $$ f_\beta(x) = \beta \sup_{y:\|y\|_\infty\le 1} (yx - \frac{\sigma\beta}2 y^2). $$