v2010.10.26 - Convex Optimization

762 APPENDIX E. PROJECTIONbH 1x 22x 21H 2x 12H 1 ∩ H 2x 11Figure 173: H 1 and H 2 are the same halfspaces as in Figure 168.Dykstra’s alternating projection algorithm generates the alternationsb, x 21 , x 11 , x 22 , x 12 , x 12 ... x 12 . The path illustrated from b to x 12 in R 2terminates at the desired result, Pb in Figure 168. The {y ki } correspondto the first two difference vectors drawn (in the first iteration i=1), thenoscillate between zero and a negative vector thereafter. These alternationsare not so robust in presence of noise as for the example in Figure 167.Denoting by P k t the unique minimum-distance projection of t on C k , andfor convenience x L+1,i = x 1,i−1 (2070), iterate x 1i calculation proceeds: E.22for i=1, 2,...until convergence {for k=L... 1 {t = x k+1,i − y k,i−1x ki = P k ty ki = P k t − t}}(2086)E.22 We reverse order of projection (k=L...1) in the algorithm for continuity of exposition.