---
tag: E239APE78P
kind: concept
name: Proximal operator
msc: [90C25, 47H09, 49J53]
sources:
  - url: https://en.wikipedia.org/wiki/Proximal_operator
same_as:
  wikidata: Q28130527
  wikipedia: Proximal operator
---

## Summary

The proximal operator of a convex function $f$ on a Hilbert space sends a point $v$ to a nearby point where $f$ is small.
The new point minimizes $f(x) + \tfrac12 \norm{x - v}^2$, which balances the two aims.
It generalizes the projection onto a closed convex set, which is the proximal operator of the indicator function of the set.
Proximal operators are firmly nonexpansive, and they are the building blocks of proximal gradient methods and other splitting algorithms in convex optimization.

## Explanations

### L1 human

Imagine standing at a point $v$ and wanting to move to a place where $f$ is small, but each step away costs you.
The proximal point is the best compromise: it minimizes $f(x)$ plus half the squared distance from $x$ to $v$.
If $f$ is $0$ on a convex region and $+\infty$ outside, the compromise is simply the nearest point of the region.
For $f(x) = \abs{x}$ on $\R$, the proximal operator moves $v$ toward $0$ by $1$ and stops at $0$: it sends $3$ to $2$ and $0.4$ to $0$.