v2006.03.09 - Convex Optimization

E.5. PROJECTION EXAMPLES 541E.5.0.0.5 Example. Projecting the origin on a hyperplane.(confer2.4.2.0.1) Given the hyperplane representation having b ∈ R andnonzero normal a∈R m ∂H = {y | a T y = b} ⊂ R m (90)the orthogonal projection of the origin P0 on that hyperplane is the solutionto a minimization problem: (1469)‖P0 − 0‖ 2 = infy∈∂H ‖y − 0‖ 2= infξ∈R m−1 ‖Zξ + x‖ 2(1507)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 argumentsoand from (1471)P0 = y ⋆ = a(a T a) −1 a T x =ξ ⋆ = −(Z T Z) −1 Z T x (1508)y ⋆ = ( I − Z(Z T Z) −1 Z T) x (1509)a Ta‖a‖ ‖a‖ x = ∆ AA † x = a b (1510)‖a‖ 2In words, any point x in the hyperplane ∂H projected on its normal a(confer (1535)) 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 ofhyperplanesA = {y | Ay = b} ⊂ R m (1511)where each row of A ∈ R m×n is nonzero and b ∈ R(A) , then the orthogonalprojection Px of any point x∈ R n on A is the solution to a minimizationproblem:‖Px − x‖ 2 = infy∈A‖y − x‖ 2= infξ∈R n−rank A ‖Zξ + y p − x‖ 2(1512)