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.