v2009.01.01 - Convex Optimization

E.10. ALTERNATING PROJECTION 689
b
H 1
H 2
P 2 b
H 1 ∩ H 2
Pb
P 1 P 2 b
Figure 143: The sets {C k } in this example comprise two halfspaces H 1 and
H 2 . The von Neumann-style alternating projection in R 2 quickly converges
to P 1 P 2 b (feasibility). The unique minimum-distance projection on the
intersection is, of course, Pb .
always yield the closest point, although we shall show it always yields some
point in the intersection or a point that attains the distance between two
convex sets.
Alternating projection is also known as successive projection [152] [149]
[55], cyclic projection [123] [224,3.2], successive approximation [69], or
projection on convex sets [282] [283,6.4]. It is traced back to von Neumann
(1933) [322] and later Wiener [328] who showed that higher iterates of a
product of two orthogonal projections on subspaces converge at each point
in the ambient space to the unique minimum-distance projection on the
intersection of the two subspaces. More precisely, if R 1 and R 2 are closed
subspaces of a Euclidean space and P 1 and P 2 respectively denote orthogonal
projection on R 1 and R 2 , then for each vector b in that space,
lim (P 1P 2 ) i b = P R1 ∩ R 2
b (1926)
i→∞
Deutsch [92, thm.9.8, thm.9.35] shows rate of convergence for subspaces to
be geometric [342,1.4.4]; bounded above by κ 2i+1 ‖b‖ , i=0, 1, 2..., where