v2007.09.13 - Convex Optimization

600 APPENDIX E. PROJECTION(B.4.2). Then the orthogonal projection Px of any point x∈ R n on A isthe solution to a minimization problem:‖Px − x‖ 2= inf ‖Xa − x‖ 2a T 1=1(1717)= inf ‖X(V N ξ + a p ) − x‖ 2ξ∈R N−1where a p is any solution to a T 1=1. We find the minimizing argumentξ ⋆ = −(V T NX T XV N ) −1 V T NX T (Xa p − x) (1718)and so the orthogonal projection is [152,3]Px = Xa ⋆ = (I − XV N (XV N ) † )Xa p + XV N (XV N ) † x (1719)a projection of point x on R(XV N ) then translated perpendicularly withrespect to that range until it meets the affine subset A . E.5.0.0.8 Example. Projecting on hyperplane, halfspace, slab.Given the hyperplane representation having b ∈ R and nonzero normala∈ R m ∂H = {y | a T y = b} ⊂ R m (94)the orthogonal projection of any point x∈ R m on that hyperplane isPx = x − a(a T a) −1 (a T x − b) (1720)Orthogonal projection of x on the halfspace parametrized by b ∈ R andnonzero normal a∈ R m H − = {y | a T y ≤ b} ⊂ R m (86)is the pointPx = x − a(a T a) −1 max{0, a T x − b} (1721)Orthogonal projection of x on the convex slab (Figure 9), for c < bB = ∆ {y | c ≤ a T y ≤ b} ⊂ R m (1722)is the point [98,5.1]Px = x − a(a T a) ( −1 max{0, a T x − b} − max{0, c − a T x} ) (1723)