Scaling Property of the Proximal Operator

Question

If the proximal operator of $f(x)$ is $\text{prox}_{\lambda f}(x)$, what about $cf(x)$ and $f(cx)$, c is a scalar.

For example, If $f(x) = ||x||_{1}$, $x \in \mathbb{R}^{n}$, how about the proximal operator of $c||x||_{1}$ and $||c x||_{1}$.

Answer 1

Part 1: Note that $\text{prox}_{t(cf)}(x) = \text{ prox } _ { (tc) f}(x)$.

Part 2: Let $ g (x) = f ( cx) $. To evaluate $\text{prox}_{tg}(y)$, we must find a minimizer of \begin{equation} f (cx) + (1/2t) \|x - y \|^2. \end{equation} Make the change of variable $ z = cx $. Our optimization problem is equivalent to minimizing \begin{equation} f(z) + (1/2t) \| z/c - y\|^2 \end{equation} with respect to $ z $. The best choice of $ z $ is the prox operator of $ f $ at $ cy $ with parameter $ t c^2$. The corresponding best choice of $ x $ is $ z/c $.

\begin{equation} \text{prox}_{tg}(y) = \text{prox}_{tc^2f}(cy)/c. \end{equation}