v2009.01.01 - Convex Optimization

674 APPENDIX E. PROJECTION
E.9.0.0.1 Theorem. (Bunt-Motzkin) Convex set if projections unique.
[324,7.5] [171] If C ⊆ R n is a nonempty closed set and if for each and every
x in R n there is a unique Euclidean projection Px of x on C belonging to
C , then C is convex.
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Borwein & Lewis propose, for closed convex set C [48,3.3, exer.12(d)]
for any point x whereas, for x /∈ C
∇‖x − Px‖ 2 2 = 2(x − Px) (1883)
∇‖x − Px‖ 2 = (x − Px) ‖x − Px‖ −1
2 (1884)
E.9.0.0.2 Exercise. Norm gradient.
Prove (1883) and (1884). (Not proved in [48].)
A well-known equivalent characterization of projection on a convex set is
a generalization of the perpendicularity condition (1783) for projection on a
subspace:
E.9.1
Dual interpretation of projection on convex set
E.9.1.0.1 Definition. Normal vector. [266, p.15]
Vector z is normal to convex set C at point Px∈ C if
〈z , y−Px〉 ≤ 0 ∀y ∈ C (1885)
A convex set has a nonzero normal at each of its boundary points.
[266, p.100] Hence, the normal or dual interpretation of projection:
E.9.1.0.2 Theorem. Unique minimum-distance projection. [173,A.3.1]
[215,3.12] [92,4.1] [69] (Figure 144(b), p.691) Given a closed convex set
C ⊆ 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 if
Px ∈ C , 〈x − Px , y − Px〉 ≤ 0 ∀y ∈ C (1886)
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