v2010.10.26 - Convex Optimization

E.5. PROJECTION EXAMPLES 719where Y = N(1 T ). We have M = R N , A = [ √ 2V N e 1 ] , and u = e 1 .Thus P u = e 1 1 T is a nonorthogonal projector projecting on R(u) in adirection parallel to a vector in Y (E.3.5), and P Y x = x − e 1 1 T x is anonorthogonal projection of x on Y in a direction parallel to u . E.5.0.0.5 Example. Projecting origin on a hyperplane.(confer2.4.2.0.2) Given the hyperplane representation having b∈R andnonzero normal a∈ R m ∂H = {y | a T y = b} ⊂ R m (114)orthogonal projection of the origin P0 on that hyperplane is the uniqueoptimal solution to a minimization problem: (1908)‖P0 − 0‖ 2 = infy∈∂H ‖y − 0‖ 2= infξ∈R m−1 ‖Zξ + x‖ 2(1945)where x is any solution to a T y=b , and where the columns of Z ∈ R m×m−1constitute 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) ofthe strictly convex norm-square to 0. We find the minimizing argumentξ ⋆ = −(Z T Z) −1 Z T x (1946)soy ⋆ = ( I − Z(Z T Z) −1 Z T) x (1947)and from (1910)a TP0 = y ⋆ = a(a T a) −1 a T x = a‖a‖ ‖a‖ x =a‖a‖ 2 aT x A † Ax = a b‖a‖ 2 (1948)In words, any point x in the hyperplane ∂H projected on its normal a(confer (1972)) yields that point y ⋆ in the hyperplane closest to the origin.