v2009.01.01 - Convex Optimization

700 APPENDIX E. PROJECTION
b
H 1
x 22
x 21
H 2
x 12
H 1 ∩ H 2
x 11
Figure 148: H 1 and H 2 are the same halfspaces as in Figure 143.
Dykstra’s alternating projection algorithm generates the alternations
b, x 21 , x 11 , x 22 , x 12 , x 12 ... x 12 . The path illustrated from b to x 12 in R 2
terminates at the desired result, Pb in Figure 143. The {y ki } correspond
to the first two difference vectors drawn (in the first iteration i=1), then
oscillate between zero and a negative vector thereafter. These alternations
are not so robust in presence of noise as for the example in Figure 142.
E.10.3
Optimization and projection
Unique projection on the nonempty intersection of arbitrary convex sets to
find the closest point therein is a convex optimization problem. The first
successful application of alternating projection to this problem is attributed
to Dykstra [105] [54] who in 1983 provided an elegant algorithm that prevails
today. In 1988, Han [152] rediscovered the algorithm and provided a
primal−dual convergence proof. A synopsis of the history of alternating
projection E.21 can be found in [56] where it becomes apparent that Dykstra’s
work is seminal.
E.21 For a synopsis of alternating projection applied to distance geometry, see [306,3.1].