v2007.09.13 - Convex Optimization

E.5. PROJECTION EXAMPLES 599E.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 (1711)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(1712)where y p is any solution to Ay = b , and where the columns ofZ ∈ R n×n−rank A constitute a basis for N(A) so that y = Zξ + y p ∈ A forall ξ ∈ R n−rank A .The infimum is found by setting the gradient of the strictly convexnorm-square to 0. The minimizing argument isξ ⋆ = −(Z T Z) −1 Z T (y p − x) (1713)soand from (1673),y ⋆ = ( I − Z(Z T Z) −1 Z T) (y p − x) + x (1714)Px = y ⋆ = x − A † (Ax − b)= (I − A † A)x + A † Ay p(1715)which is a projection of x on N(A) then translated perpendicularly withrespect 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 givenminimal set of generators arranged columnar in X ∈ R n×N (65); id est,A ∆ = aff X = {Xa | a T 1=1} ⊆ R n (1716)Here minimal set means XV N = [x 2 −x 1 x 3 −x 1 · · · x N −x 1 ]/ √ 2 (772) isfull-rank (2.4.2.2) where V N ∈ R N×N−1 is the Schoenberg auxiliary matrix