v2010.10.26 - Convex Optimization

734 APPENDIX E. PROJECTIONE.9.0.0.1 Theorem. (Bunt-Motzkin) Convex set if projections unique.[371,7.5] [196] 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 to C ,then C is convex.⋄Borwein & Lewis propose, for closed convex set C [55,3.3 exer.12d]for any point x whereas, for x /∈ C∇‖x − Px‖ 2 2 = 2(x − Px) (2007)∇‖x − Px‖ 2 = (x − Px) ‖x − Px‖ −12 (2008)E.9.0.0.2 Exercise. Norm gradient.Prove (2007) and (2008). (Not proved in [55].)A well-known equivalent characterization of projection on a convex set isa generalization of the perpendicularity condition (1907) for projection on asubspace:E.9.1Dual interpretation of projection on convex setE.9.1.0.1 Definition. Normal vector. [307, p.15]Vector z is normal to convex set C at point Px∈ C if〈z , y−Px〉 ≤ 0 ∀y ∈ C (2009)△A convex set has at least one nonzero normal at each of its boundarypoints. [307, p.100] (Figure 67) Hence, the normal or dual interpretation ofprojection:E.9.1.0.2 Theorem. Unique minimum-distance projection. [199,A.3.1][250,3.12] [109,4.1] [78] (Figure 169b p.752) 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 (2010)⋄