v2009.01.01 - Convex Optimization

E.10. ALTERNATING PROJECTION 691
a
(a)
{y | (b −P 1 b) T (y −P 1 b)=0}
b
(b)
C 1
y
C 1
C 1
P 1 b
C 2
C 2
b
C 2
Pb
P 1 P 2 b
(c)
Figure 144:
(a) (distance) Intersection of two convex sets in R 2 is empty. Method of
alternating projection would be applied to find that point in C 1 nearest C 2 .
(b) (distance) Given b ∈ C 2 , then P 1 b ∈ C 1 is nearest b iff
(y −P 1 b) T (b −P 1 b)≤0 ∀y ∈ C 1 by the unique minimum-distance projection
theorem (E.9.1.0.2). When P 1 b attains the distance between the two sets,
hyperplane {y | (b −P 1 b) T (y −P 1 b)=0} separates C 1 from C 2 . [53,2.5.1]
(c) (0 distance) Intersection is nonempty.
(optimization) We may want the point Pb in ⋂ C k nearest point b .
(feasibility) We may instead be satisfied with a fixed point of the projection
product P 1 P 2 b in ⋂ C k .