v2009.01.01 - Convex Optimization

692 APPENDIX E. PROJECTION
E.10.1
Distance and existence
Existence of a fixed point is established:
E.10.1.0.1 Theorem. Distance. [69]
Given any two closed convex sets C 1 and C 2 in R n , then P 1 b∈ C 1 is a fixed
point of the projection product P 1 P 2 if and only if P 1 b is a point of C 1
nearest C 2 .
⋄
Proof. (⇒) Given fixed point a =P 1 P 2 a∈ C 1 with b =P ∆ 2 a∈ C 2 in
tandem so that a =P 1 b , then by the unique minimum-distance projection
theorem (E.9.1.0.2)
(1931)
(b − a) T (u − a) ≤ 0 ∀u∈ C 1
(a − b) T (v − b) ≤ 0
⇔
∀v ∈ C 2
‖a − b‖ ≤ ‖u − v‖ ∀u∈ C 1 and ∀v ∈ C 2
by Schwarz inequality ‖〈x,y〉‖ ≤ ‖x‖ ‖y‖ [197] [266].
(⇐) Suppose a∈ C 1 and ‖a − P 2 a‖ ≤ ‖u − P 2 u‖ ∀u∈ C 1 . Now suppose we
choose u =P 1 P 2 a . Then
‖u − P 2 u‖ = ‖P 1 P 2 a − P 2 P 1 P 2 a‖ ≤ ‖a − P 2 a‖ ⇔ a = P 1 P 2 a (1932)
Thus a = P 1 b (with b =P 2 a∈ C 2 ) is a fixed point in C 1 of the projection
product P 1 P 2 . E.19
E.10.2
Feasibility and convergence
The set of all fixed points of any nonexpansive mapping is a closed convex
set. [131, lem.3.4] [27,1] The projection product P 1 P 2 is nonexpansive by
Theorem E.9.3.0.1 because, for any vectors x,a∈ R n
‖P 1 P 2 x − P 1 P 2 a‖ ≤ ‖P 2 x − P 2 a‖ ≤ ‖x − a‖ (1933)
If the intersection of two closed convex sets C 1 ∩ C 2 is empty, then the iterates
converge to a point of minimum distance, a fixed point of the projection
product. Otherwise, convergence is to some fixed point in their intersection
E.19 Point b=P 2 a can be shown, similarly, to be a fixed point of the product P 2 P 1 .