v2007.09.13 - Convex Optimization

E.10. ALTERNATING PROJECTION 639bH 1y 21x 22x 21Hy 211x 12H 1 ∩ H 2x 11Figure 127: H 1 and H 2 are the same halfspaces as in Figure 122.Dykstra’s alternating projection algorithm generates the alternationsb, x 21 , x 11 , x 22 , x 12 , x 12 ...,. The path illustrated from b to x 12 in R 2terminates at the desired result, Pb . The alternations are not so robust inpresence of noise as for the example in Figure 121.E.10.3Optimization and projectionUnique projection on the nonempty intersection of arbitrary convex sets tofind the closest point therein is a convex optimization problem. The firstsuccessful application of alternating projection to this problem is attributedto Dykstra [83] [47] who in 1983 provided an elegant algorithm that prevailstoday. In 1988, Han [127] rediscovered the algorithm and provided aprimal−dual convergence proof. A synopsis of the history of alternatingprojection E.20 can be found in [49] where it becomes apparent that Dykstra’swork is seminal.E.20 For a synopsis of alternating projection applied to distance geometry, see [262,3.1].