Moreau envelope ($M_{t}f$)

The Moreau envelope of a proper closed convex function isIt is differentiable even when $f$ is not, with $\nabla M_{t}f(x)=[x-{\mathrm{prox}}_{tf}(x)]/t$ and $1/t$-Lipschitz gradient.

For a nonempty closed convex set $C$, the squared-distance function is the Moreau envelope of its indicator functional of a constraint set. It is convex and differentiable withand this gradient is one-Lipschitz.