The maximizer defining $g^*(x)$ is $\operatorname{prox}_{tf}(x)$, so
$$
\boxed{\nabla M_tf(x)=\frac1t[x-\operatorname{prox}_{tf}(x)].}
$$
Consequently
$$
x_{k+1}=\operatorname{prox}_{tf}(x_k)
=x_k-t\nabla M_tf(x_k).
$$
Thus the <proximal point algorithm> for $f$ is ordinary <gradient descent> with step $t$ on its smooth Moreau envelope.

Solution

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

{wiki}

The proximal point algorithm seeks a zero of a <monotone operator> $F$ by repeatedly applying its resolvent:
$$
w_{k+1}=(I+\lambda F)^{-1}w_k.
$$
For $F=\partial f$, this is iteration of a <proximal operator>.

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