v2007.09.13 - Convex Optimization

614 APPENDIX E. PROJECTIONAs for subspace projection, operator P is idempotent in the sense: foreach 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 whichprojection is made) is a subspace. (E.4)E.9.1.0.3 Theorem. Unique projection via normal cone. E.14 [73,4.3]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) ∗ (1774)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 [76, thm.2.3] [75,2] and Luenberger [181, p.134] carry forwardNirenberg’s dual interpretation of projection [204] 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 20(a)) has form∂H − = { y ∈ R n | a T y = σ C (a) } (108)where the support function is convex, definedσ C (a) ∆ = supz∈Ca T z (458)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) ) (1775)E.14 −(C − Px) ∗ is the normal cone to set C at point Px. (E.10.3.2)