v2007.09.13 - Convex Optimization

E.9. PROJECTION ON CONVEX SET 613E.9.0.0.1 Theorem. (Bunt-Motzkin) Convex set if projections unique.[277,7.5] [145] If C ⊆ R n is a nonempty closed set and if for each and everyx in R n there is a unique Euclidean projection Px of x on C belonging toC , then C is convex.⋄Borwein & Lewis propose, for closed convex set C [41,3.3, exer.12(d)]for any point x whereas, for x /∈ C∇‖x − Px‖ 2 2 = 2(x − Px) (1770)∇‖x − Px‖ 2 = (x − Px) ‖x − Px‖ −12 (1771)E.9.0.0.2 Exercise. Norm gradient.Prove (1770) and (1771). (Not proved in [41].)A well-known equivalent characterization of projection on a convex set isa generalization of the perpendicularity condition (1670) for projection on asubspace:E.9.1Dual interpretation of projection on convex setE.9.1.0.1 Definition. Normal vector. [228, p.15]Vector z is normal to convex set C at point Px∈ C if〈z , y−Px〉 ≤ 0 ∀y ∈ C (1772)A convex set has a nonzero normal at each of its boundary points.[228, p.100] Hence, the normal or dual interpretation of projection:E.9.1.0.2 Theorem. Unique minimum-distance projection. [147,A.3.1][181,3.12] [73,4.1] [56] (Figure 123(b), p.630) Given a closed convex setC ⊆ R n , point Px is the unique projection of a given point x∈ R n on C(Px is that point in C nearest x) if and only ifPx ∈ C , 〈x − Px , y − Px〉 ≤ 0 ∀y ∈ C (1773)△⋄