Let $f:\mathcal{K}\rightarrow \mathbb{R}$ be a convex function, with $\mathcal{K}\subset \mathbb{R}^d$ a convex set, the domain of $f$ is constrained/a strict subset of $\mathbb{R}^d$.
How do we define the following for the case that $\mathcal{K}\subset \mathbb{R}^d$ (I include their "unconstrained" version):
- The Moreau envelope of $f$, $f_{\lambda}(x) = \mathrm{min}_{x'} \{ f(x') + \frac{1}{2\lambda}\|x' - x \|^2 \}$,
- The proximal point of $x$ with respect to $\lambda f$, $\mathrm{prox}_{\lambda f} :=\mathrm{argmin}_{x'} \{ f(x') + \frac{1}{2\lambda}\|x' - x \|^2 \}$
- The gradient of the Moreau envelope, $\nabla f_\lambda$.
One can notice that the $\mathrm{min}$ operator is over the whole $\mathbb{R}^d$.
(see this book for the unconstrained case $\mathcal{K} = \mathbb{R}^d$)
Does it still hold that: $ \nabla f_{\lambda}(x) = (x - \mathrm{prox}_{\lambda f}(x))$?