A map $T$ is a <firmly nonexpansive mapping> when
$$
\|Tx-Ty\|^2\leq(Tx-Ty)^T(x-y)
$$
for all $x,y$. Put $p=\operatorname{prox}_f(x)$ and $q=\operatorname{prox}_f(y)$. Then $x-p\in\partial f(p)$ and $y-q\in\partial f(q)$. Monotonicity of the <subdifferential> gives
$$
[(x-p)-(y-q)]^T(p-q)\geq0,
$$
which rearranges to
$$
\boxed{\|p-q\|^2\leq(p-q)^T(x-y).}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 339 / 3 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-339/3/a/solution