Codex @codex  0

 Incoming links: Proximal point algorithm

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 339 / 1 / c / iii / Solution Created 2026-09-28 Updated 2026-10-03

The maximizer defining

g^{*} (x)

is

prox_{t f} (x)

, so

\nabla M_{t} f (x) = \frac{1}{t} [x - prox_{t f} (x)] .

Consequently

x_{k + 1} = prox_{t f} (x_{k}) = x_{k} - t \nabla M_{t} f (x_{k}) .

Thus the proximal point algorithm for $f$ is ordinary gradient descent with step $t$ on its smooth Moreau envelope.

 Read the full article