v2010.10.26 - Convex Optimization

E.10. ALTERNATING PROJECTION 757Define the subspaceR By the results in Example E.5.0.0.6P RS([xix i])= P R([PC (x i )P D (x i ){ [ ] ∣ }Rn ∣∣∣v ∈R n [I −I ]v = 0])=[PC (x i ) + P D (x i )P C (x i ) + P D (x i )] 12(2066)(2067)This means the proposed variation of alternating projection is equivalent toan alternation of projection on convex sets S and R . If S and R intersect,these iterations will converge to a point in their intersection; hence, to a pointin the intersection of C and D .We need not apply equal weighting to the projections, as supposed in(2063). In that case, definition of R would change accordingly. E.10.2.1Relative measure of convergenceInspired by Fejér monotonicity, the alternating projection algorithm fromthe example of convergence illustrated by Figure 171 employs a redundant∏sequence: The first sequence (indexed by j) estimates point ( ∞ L∏P k )b inj=1 k=1the presumably nonempty intersection of L convex sets, then the quantity( ∥ x ∏ ∞ L∏i − P k)b(2068)∥j=1 k=1in second sequence x i is observed per iteration i for convergence. A prioriknowledge of a feasible solution (2052) is both impractical and antithetical.We need another measure:Nonexpansivity implies( L( ∏L∥∏ ∥∥∥∥ P l)x k,i−1 − P l)x ki = ‖x ki − x k,i+1 ‖ ≤ ‖x k,i−1 − x ki ‖ (2069)∥l=1l=1wherex ki P k x k+1,i ∈ R n , x L+1,i x 1,i−1 (2070)