v2010.10.26 - Convex Optimization

E.9. PROJECTION ON CONVEX SET 735As for subspace projection, convex operator P is idempotent in the sense:for each and every x∈ R n , P(Px)=Px . Yet operator P is nonlinear;Projector P is a linear operator if and only if convex set C (on whichprojection is made) is a subspace. (E.4)E.9.1.0.3 Theorem. Unique projection via normal cone. E.15 [109,4.3]Given closed convex set C ⊆ R n , point Px is the unique projection of agiven point x∈ R n on C if and only ifPx ∈ C , Px − x ∈ (C − Px) ∗ (2011)In other words, Px is that point in C nearest x if and only if Px − x belongsto that cone dual to translate C − Px .⋄E.9.1.1Dual interpretation as optimizationDeutsch [111, thm.2.3] [112,2] and Luenberger [250, p.134] carry forwardNirenberg’s dual interpretation of projection [279] as solution to amaximization problem: Minimum distance from a point x∈ R n to a convexset C ⊂ R n can be found by maximizing distance from x to hyperplane ∂Hover the set of all hyperplanes separating x from C . Existence of aseparating hyperplane (2.4.2.7) presumes point x lies on the boundary orexterior to set C .The optimal separating hyperplane is characterized by the fact it alsosupports C . Any hyperplane supporting C (Figure 29a) has form∂H − = { y ∈ R n | a T y = σ C (a) } (129)where the support function is convex, definedσ C (a) = supz∈Ca T z (553)When point x is finite and set C contains finite points, under this projectioninterpretation, if the supporting hyperplane is a separating hyperplane thenthe support function is finite. From Example E.5.0.0.8, projection P ∂H− x ofx on any given supporting hyperplane ∂H − isP ∂H− x = x − a(a T a) −1( a T x − σ C (a) ) (2012)E.15 −(C − Px) ∗ is the normal cone to set C at point Px. (E.10.3.2)