v2009.01.01 - Convex Optimization

694 APPENDIX E. PROJECTION
y 2
θ
C 1 = R 2 +
b
Pb
y 1
C 2 = A = {y | [ 1 1 ]y = 1}
Figure 145: From Example E.10.2.0.2 in R 2 , showing von Neumann-style
alternating projection to find feasible point belonging to intersection of
nonnegative orthant with hyperplane. Point Pb lies at intersection of
hyperplane with ordinate axis. In this particular example, the feasible point
found is coincidentally optimal. Rate of convergence depends upon angle θ ;
as it becomes more acute, convergence slows. [149,3]
∥ x ∏
i − ( ∞ ∏
P k )b
∥
j=1
k
20
18
16
14
12
10
8
6
4
2
0
0 5 10 15 20 25 30 35 40 45
i
Figure 146: Geometric convergence of iterates in norm, for
Example E.10.2.0.2 in R 1000 .