Set $g(u)=tf(u)+\|u\|^2/2$. Expanding the square gives
$$
\boxed{M_tf(x)=\frac1{2t}\|x\|^2-\frac1t g^*(x).}
$$
The function $g$ is one-<strongly convex function>[strongly convex], so its <Fenchel conjugate> is differentiable with one-Lipschitz gradient. The displayed identity therefore proves that the <Moreau envelope> $M_tf$ is differentiable even when $f$ is nonsmooth.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 339 / 1 / c / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-339/1/c/ii/solution