v2009.01.01 - Convex Optimization

696 APPENDIX E. PROJECTION
By the results in Example E.5.0.0.6
([ ]) ([ ])
xi PC (x
P 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 )
] 1
2
(1943)
This means the proposed variation of alternating projection is equivalent to
an 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 point
in the intersection of C and D .
We need not apply equal weighting to the projections, as supposed in
(1939). In that case, definition of R would change accordingly.
E.10.2.1
Relative measure of convergence
Inspired by Fejér monotonicity, the alternating projection algorithm from
the example of convergence illustrated by Figure 146 employs a redundant
∏
sequence: The first sequence (indexed by j) estimates point ( ∞ ∏
P k )b in
the presumably nonempty intersection, then the quantity
( ∞ ∥ x ∏∏
i − P k
)b
∥
j=1
k
j=1
k
(1944)
in second sequence x i is observed per iteration i for convergence. A priori
knowledge of a feasible point (1929) is both impractical and antithetical. We
need 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 ‖ (1945)
l
l
where
x ki ∆ = P k x k+1,i ∈ R n (1946)
represents unique minimum-distance projection of x k+1,i on convex set k at
iteration i . So a good convergence measure is the total monotonic sequence
ε i ∆ = ∑ k
‖x ki − x k,i+1 ‖ , i=0, 1, 2... (1947)
where lim
i→∞
ε i = 0 whether or not the intersection is nonempty.