Euclidean projection onto a convex set

For a nonempty closed convex subset $C$ of a finite-dimensional inner-product space, ${\Pi }_C(x)$ is the unique point of $C$ minimizing $\Vert x-z\Vert$.

This closest-point map is the metric projection onto a closed convex set. Repeated projections onto several sets form the separate projections onto convex sets method.

For $p\in C$,

The metric projection onto a nonempty closed convex set in a Hilbert space is nonexpansive:

For $C=\{x:0\le x\le 1,a^{T}x=b\}$, the projection of $y$ has coordinates $x_i=\min (1,\max (0,y_i-\nu a_i))$, where the scalar multiplier $\nu$ is chosen so that $a^{T}x=b$. Thus the projection reduces to a continuous one-dimensional root-finding problem.