v2010.10.26 - Convex Optimization

756 APPENDIX E. PROJECTIONthe nonnegative orthant with affine subset A an intersection of hyperplanes.Projection of an iterate x i ∈ R n on A is calculatedP 2 x i = x i − A T (AA T ) −1 (Ax i − β) (1953)while, thereafter, projection of the result on the orthant is simplyx i+1 = P 1 P 2 x i = max{0,P 2 x i } (2062)where the maximum is entrywise (E.9.2.2.3).One realization of this problem in R 2 is illustrated in Figure 170: ForA = [ 1 1 ] , β =1, and x 0 = b = [ −3 1/2 ] T , iterates converge to a feasiblesolution 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 171. 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. [58,3]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... (2063)Because the Cartesian product of convex sets remains convex, (2.1.8) wecan reformulate this problem.Consider the convex set [ ] CS (2064)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 )(2065)x i P D (x i )