v2009.01.01 - Convex Optimization

E.10. ALTERNATING PROJECTION 695
while, thereafter, projection of the result on the orthant is simply
x i+1 = P 1 P 2 x i = max{0,P 2 x i } (1938)
where the maximum is entrywise (E.9.2.2.3).
One realization of this problem in R 2 is illustrated in Figure 145: For
A = [ 1 1 ] , β =1, and x 0 = b = [ −3 1/2 ] T , the iterates converge to the
feasible point Pb = [ 0 1 ] T .
To give a more palpable sense of convergence in higher dimension, we
do this example again but now we compute an alternating projection for
the case A∈ R 400×1000 , β ∈ R 400 , and b∈R 1000 , all of whose entries are
independently and randomly set to a uniformly distributed real number in
the interval [−1, 1] . Convergence is illustrated in Figure 146.
This application of alternating projection to feasibility is extensible to
any finite number of closed convex sets.
E.10.2.0.3 Example. Under- and over-projection. [50,3]
Consider the following variation of alternating projection: We begin with
some point x 0 ∈ R n then project that point on convex set C and then
project that same point x 0 on convex set D . To the first iterate we assign
x 1 = (P C (x 0 ) + P D (x 0 )) 1 . More generally,
2
x i+1 = (P C (x i ) + P D (x i )) 1 2
, i=0, 1, 2... (1939)
Because the Cartesian product of convex sets remains convex, (2.1.8) we
can reformulate this problem.
Consider the convex set [ ]
S =
∆ C
(1940)
D
representing Cartesian product C × D . Now, those two projections P C and
P D are equivalent to one projection on the Cartesian product; id est,
([ ]) [ ]
xi PC (x
P S = i )
(1941)
x i P D (x i )
Define the subspace
R ∆ =
{ [ ] ∣ }
R
n ∣∣∣
v ∈
R n [I −I ]v = 0
(1942)