v2009.01.01 - Convex Optimization

E.6. VECTORIZATION INTERPRETATION, 661
E.5.0.0.8 Example. Projecting on hyperplane, halfspace, slab.
Given the hyperplane representation having b ∈ R and nonzero normal
a∈ R m ∂H = {y | a T y = b} ⊂ R m (105)
the orthogonal projection of any point x∈ R m on that hyperplane is
Px = x − a(a T a) −1 (a T x − b) (1833)
Orthogonal projection of x on the halfspace parametrized by b ∈ R and
nonzero normal a∈ R m H − = {y | a T y ≤ b} ⊂ R m (97)
is the point
Px = x − a(a T a) −1 max{0, a T x − b} (1834)
Orthogonal projection of x on the convex slab (Figure 11), for c < b
is the point [123,5.1]
B ∆ = {y | c ≤ a T y ≤ b} ⊂ R m (1835)
Px = x − a(a T a) −1( max{0, a T x − b} − max{0, c − a T x} ) (1836)
E.6 Vectorization interpretation,
projection on a matrix
E.6.1
Nonorthogonal projection on a vector
Nonorthogonal projection of vector x on the range of vector y is
accomplished using a normalized dyad P 0 (B.1); videlicet,
〈z,x〉
〈z,y〉 y = zT x
z T y y = yzT
z T y x = ∆ P 0 x (1837)
where 〈z,x〉/〈z,y〉 is the coefficient of projection on y . Because P0 2 =P 0
and R(P 0 )= R(y) , rank-one matrix P 0 is a nonorthogonal projector dyad
projecting on R(y). Direction of nonorthogonal projection is orthogonal
to z ; id est,
P 0 x − x ⊥ R(P0 T ) (1838)