v2009.01.01 - Convex Optimization

660 APPENDIX E. PROJECTION
where y p is any solution to Ay = b , and where the columns of
Z ∈ R n×n−rank A constitute a basis for N(A) so that y = Zξ + y p ∈ A for
all ξ ∈ R n−rank A .
The infimum is found by setting the gradient of the strictly convex
norm-square to 0. The minimizing argument is
so
and from (1786),
ξ ⋆ = −(Z T Z) −1 Z T (y p − x) (1826)
y ⋆ = ( I − Z(Z T Z) −1 Z T) (y p − x) + x (1827)
Px = y ⋆ = x − A † (Ax − b)
= (I − A † A)x + A † Ay p
(1828)
which is a projection of x on N(A) then translated perpendicularly with
respect to the nullspace until it meets the affine subset A .
E.5.0.0.7 Example. Projection on affine subset, vertex-description.
Suppose now we instead describe the affine subset A in terms of some given
minimal set of generators arranged columnar in X ∈ R n×N (68); id est,
A ∆ = aff X = {Xa | a T 1=1} ⊆ R n (1829)
Here minimal set means XV N = [x 2 −x 1 x 3 −x 1 · · · x N −x 1 ]/ √ 2 (865) is
full-rank (2.4.2.2) where V N ∈ R N×N−1 is the Schoenberg auxiliary matrix
(B.4.2). Then the orthogonal projection Px of any point x∈ R n on A is
the solution to a minimization problem:
‖Px − x‖ 2
= inf ‖Xa − x‖ 2
a T 1=1
(1830)
= inf ‖X(V N ξ + a p ) − x‖ 2
ξ∈R N−1
where a p is any solution to a T 1=1. We find the minimizing argument
ξ ⋆ = −(V T NX T XV N ) −1 V T NX T (Xa p − x) (1831)
and so the orthogonal projection is [179,3]
Px = Xa ⋆ = (I − XV N (XV N ) † )Xa p + XV N (XV N ) † x (1832)
a projection of point x on R(XV N ) then translated perpendicularly with
respect to that range until it meets the affine subset A .