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

The Fenchel conjugate of $h$ is

Set $g(u)=tf(u)+\Vert u{\Vert }^2/2$. Expanding the square givesThe function $g$ is one-strongly convex, so its Fenchel conjugate is differentiable with one-Lipschitz gradient. The displayed identity therefore proves that the Moreau envelope $M_{t}f$ is differentiable even when $f$ is nonsmooth.

The maximizer defining $g^{\ast }(x)$ is ${\mathrm{prox}}_{tf}(x)$, soConsequentlyThus the proximal point algorithm for $f$ is ordinary gradient descent with step $t$ on its smooth Moreau envelope.