v2009.01.01 - Convex Optimization

E.5. PROJECTION EXAMPLES 659
E.5.0.0.5 Example. Projecting the origin on a hyperplane.
(confer2.4.2.0.2) Given the hyperplane representation having b∈R and
nonzero normal a∈ R m ∂H = {y | a T y = b} ⊂ R m (105)
orthogonal projection of the origin P0 on that hyperplane is the unique
optimal solution to a minimization problem: (1784)
‖P0 − 0‖ 2 = inf
y∈∂H ‖y − 0‖ 2
= inf
ξ∈R m−1 ‖Zξ + x‖ 2
(1820)
where x is any solution to a T y=b , and where the columns of Z ∈ R m×m−1
constitute a basis for N(a T ) so that y = Zξ + x ∈ ∂H for all ξ ∈ R m−1 .
The infimum can be found by setting the gradient (with respect to ξ) of
the strictly convex norm-square to 0. We find the minimizing argument
so
and from (1786)
P0 = y ⋆ = a(a T a) −1 a T x = a
‖a‖ ‖a‖ x =
ξ ⋆ = −(Z T Z) −1 Z T x (1821)
y ⋆ = ( I − Z(Z T Z) −1 Z T) x (1822)
a T
a
‖a‖ 2 aT x ∆ = A † Ax = a b
‖a‖ 2 (1823)
In words, any point x in the hyperplane ∂H projected on its normal a
(confer (1848)) yields that point y ⋆ in the hyperplane closest to the origin.
E.5.0.0.6 Example. Projection on affine subset.
The technique of Example E.5.0.0.5 is extensible. Given an intersection of
hyperplanes
A = {y | Ay = b} ⊂ R m (1824)
where each row of A ∈ R m×n is nonzero and b ∈ R(A) , then the orthogonal
projection Px of any point x∈ R n on A is the solution to a minimization
problem:
‖Px − x‖ 2 = inf
y∈A
‖y − x‖ 2
= inf
ξ∈R n−rank A ‖Zξ + y p − x‖ 2
(1825)