v2007.09.13 - Convex Optimization

634 APPENDIX E. PROJECTIONwhile, thereafter, projection of the result on the orthant is simplyx i+1 = P 1 P 2 x i = max{0,P 2 x i } (1825)where the maximum is entrywise (E.9.2.2.3).One realization of this problem in R 2 is illustrated in Figure 124: ForA = [ 1 1 ] , β =1, and x 0 = b = [ −3 1/2 ] T , the iterates converge to thefeasible point Pb = [ 0 1 ] T .To give a more palpable sense of convergence in higher dimension, wedo this example again but now we compute an alternating projection forthe case A∈ R 400×1000 , β ∈ R 400 , and b∈R 1000 , all of whose entries areindependently and randomly set to a uniformly distributed real number inthe interval [−1, 1] . Convergence is illustrated in Figure 125. This application of alternating projection to feasibility is extensible toany finite number of closed convex sets.E.10.2.0.3 Example. Under- and over-projection. [43,3]Consider the following variation of alternating projection: We begin withsome point x 0 ∈ R n then project that point on convex set C and thenproject that same point x 0 on convex set D . To the first iterate we assignx 1 = (P C (x 0 ) + P D (x 0 )) 1 . More generally,2x i+1 = (P C (x i ) + P D (x i )) 1 2, i=0, 1, 2... (1826)Because the Cartesian product of convex sets remains convex, (2.1.8) wecan reformulate this problem.Consider the convex set [ ]S =∆ C(1827)Drepresenting Cartesian product C × D . Now, those two projections P C andP D are equivalent to one projection on the Cartesian product; id est,([ ]) [ ]xi PC (xP S = i )(1828)x i P D (x i )Define the subspaceR ∆ ={ [ ] ∣ }Rn ∣∣∣v ∈R n [I −I ]v = 0(1829)