Euclidean projection on a set

An Euclidean projection of a point $x_0 in mathbf{R}^n$ on a set $mathbf{S} subseteq mathbf{R}^n$ is a point that achieves the smallest Euclidean distance from x_0 to the set. That is, it is any solution to the optimization problem

$min_x : |x-x_0|_2 ~:~ x in mathbf{S}.$

When the set mathbf{S} is convex, there is a unique solution to the above problem. In particular, the projection on an affine subspace is unique.

Example: assume that mathbf{S} is the hyperplane

$mathbf{S} = left{ x in mathbf{R}^3 ~:~ 2x_1 + x_2 -x_3 = 1 right}.$

The projection problem reads as a linearly constrained least-squares problem, of particularly simple form:

The projection of x_0 = 0 on mathbf{S} turns out to be aligned with the coefficient vector a = (2,1,-1) . Indeed, components of orthogonal to don't appear in the constraint, and only increase the objective value. Setting x = t a in the equation defining the hyperplane and solving for the scalar we obtain t = 1/(a^Ta) = 1/6 , so that the projection is x^ast = a/(a^Ta) = (1/3,1/6,-1/6) .