v2006.03.09 - Convex Optimization

E.10. ALTERNATING PROJECTION 573E.10.1Distance and existenceExistence of a fixed point is established:E.10.1.0.1 Theorem. Distance. [45]Given any two closed convex sets C 1 and C 2 in R n , then P 1 b∈ C 1 is a fixedpoint of the projection product P 1 P 2 if and only if P 1 b is a point of C 1nearest C 2 .⋄Proof. (⇒) Given fixed point a = P 1 P 2 a ∈ C 1 with b ∆ = P 2 a ∈ C 2 intandem so that a = P 1 b , then by the unique minimum-distance projectiontheorem (E.9.1.0.1)(1611)(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 2by Schwarz inequality ‖〈x,y〉‖ ≤ ‖x‖ ‖y‖ [140] [194].(⇐) Suppose a∈ C 1 and ‖a − P 2 a‖ ≤ ‖u − P 2 u‖ ∀u∈ C 1 . Now suppose wechoose 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 (1612)Thus a = P 1 b (with b =P 2 a∈ C 2 ) is a fixed point in C 1 of the projectionproduct P 1 P 2 . E.18E.10.2Feasibility and convergenceThe set of all fixed points of any nonexpansive mapping is a closed convexset. [86, lem.3.4] [20,1] The projection product P 1 P 2 is nonexpansive byTheorem 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‖ (1613)If the intersection of two closed convex sets C 1 ∩ C 2 is empty, then the iteratesconverge to a point of minimum distance, a fixed point of the projectionproduct. Otherwise, convergence is to some fixed point in their intersectionE.18 Point b=P 2 a can be shown, similarly, to be a fixed point of the product P 2 P 1 .