v2007.09.13 - Convex Optimization

632 APPENDIX E. PROJECTION(a feasible point) whose existence is guaranteed by virtue of the fact that eachand every point in the convex intersection is in one-to-one correspondencewith fixed points of the nonexpansive projection product.Bauschke & Borwein [23,2] argue that any sequence monotonic in thesense of Fejér is convergent. E.19E.10.2.0.1 Definition. Fejér monotonicity. [197]Given closed convex set C ≠ ∅ , then a sequence x i ∈ R n , i=0, 1, 2..., ismonotonic in the sense of Fejér with respect to C iff‖x i+1 − c‖ ≤ ‖x i − c‖ for all i≥0 and each and every c ∈ C (1821)Given x 0 ∆ = b , if we express each iteration of alternating projection byx i+1 = P 1 P 2 x i , i=0, 1, 2... (1822)and define any fixed point a =P 1 P 2 a , then sequence x i is Fejér monotonewith respect to fixed point a because‖P 1 P 2 x i − a‖ ≤ ‖x i − a‖ ∀i ≥ 0 (1823)by nonexpansivity. The nonincreasing sequence ‖P 1 P 2 x i − a‖ is boundedbelow hence convergent because any bounded monotonic sequence in Ris convergent; [188,1.2] [30,1.1] P 1 P 2 x i+1 = P 1 P 2 x i = x i+1 . Sequencex i therefore converges to some fixed point. If the intersection C 1 ∩ C 2is nonempty, convergence is to some point there by the distance theorem.Otherwise, x i converges to a point in C 1 of minimum distance to C 2 .E.10.2.0.2 Example. Hyperplane/orthant intersection.Find a feasible point (1816) belonging to the nonempty intersection of twoconvex sets: given A∈ R m×n , β ∈ R(A)C 1 ∩ C 2 = R n + ∩ A = {y | y ≽ 0} ∩ {y | Ay = β} ⊂ R n (1824)the 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) (1715)E.19 Other authors prove convergence by different means; e.g., [124] [48].△