v2009.01.01 - Convex Optimization

E.9. PROJECTION ON CONVEX SET 675
As for subspace projection, operator P is idempotent in the sense: for
each and every x∈ R n , P(Px)=Px . Yet operator P is not linear;
projector P is a linear operator if and only if convex set C (on which
projection is made) is a subspace. (E.4)
E.9.1.0.3 Theorem. Unique projection via normal cone. E.15 [92,4.3]
Given closed convex set C ⊆ R n , point Px is the unique projection of a
given point x∈ R n on C if and only if
Px ∈ C , Px − x ∈ (C − Px) ∗ (1887)
In other words, Px is that point in C nearest x if and only if Px − x belongs
to that cone dual to translate C − Px .
⋄
E.9.1.1
Dual interpretation as optimization
Deutsch [94, thm.2.3] [95,2] and Luenberger [215, p.134] carry forward
Nirenberg’s dual interpretation of projection [240] as solution to a
maximization problem: Minimum distance from a point x∈ R n to a convex
set C ⊂ R n can be found by maximizing distance from x to hyperplane ∂H
over the set of all hyperplanes separating x from C . Existence of a
separating hyperplane (2.4.2.7) presumes point x lies on the boundary or
exterior to set C .
The optimal separating hyperplane is characterized by the fact it also
supports C . Any hyperplane supporting C (Figure 25(a)) has form
∂H − = { y ∈ R n | a T y = σ C (a) } (120)
where the support function is convex, defined
σ C (a) ∆ = sup
z∈C
a T z (491)
When point x is finite and set C contains finite points, under this projection
interpretation, if the supporting hyperplane is a separating hyperplane then
the support function is finite. From Example E.5.0.0.8, projection P ∂H− x of
x on any given supporting hyperplane ∂H − is
P ∂H− x = x − a(a T a) −1( a T x − σ C (a) ) (1888)
E.15 −(C − Px) ∗ is the normal cone to set C at point Px. (E.10.3.2)