v2007.09.13 - Convex Optimization

E.10. ALTERNATING PROJECTION 635By the results in Example E.5.0.0.6([ ]) ([ ])xi PC (xP RS = P i )x R =i P D (x i )[PC (x i ) + P D (x i )P C (x i ) + P D (x i )] 12(1830)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(1826). 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 125 employs a redundant∏sequence: The first sequence (indexed by j) estimates point ( ∞ ∏P k )b inthe presumably nonempty intersection, then the quantity( ∞ ∥ x ∏∏i − P k)b∥j=1kj=1k(1831)in second sequence x i is observed per iteration i for convergence. A prioriknowledge of a feasible point (1816) is both impractical and antithetical. Weneed another measure:Nonexpansivity implies( ( ∥ ∏ ∏ ∥∥∥∥ P∥ l)x k,i−1 − P l)x ki = ‖x ki − x k,i+1 ‖ ≤ ‖x k,i−1 − x ki ‖ (1832)llwherex ki ∆ = P k x k+1,i ∈ R n (1833)represents unique minimum-distance projection of x k+1,i on convex set k atiteration i . So a good convergence measure is the total monotonic sequenceε i ∆ = ∑ k‖x ki − x k,i+1 ‖ , i=0, 1, 2... (1834)where limi→∞ε i = 0 whether or not the intersection is nonempty.