v2007.11.26 - Convex Optimization

638 APPENDIX E. PROJECTIONThe foregoing proof reveals another flavor of nonexpansivity; for each andevery x,y ∈ R n‖Px − Py‖ 2 + ‖(I − P )x − (I − P )y‖ 2 ≤ ‖x − y‖ 2 (1834)Deutsch shows yet another: [73,5.5]E.9.4‖Px − Py‖ 2 ≤ 〈x − y , Px − Py〉 (1835)Easy projectionsProjecting any matrix H ∈ R n×n in the Euclidean/Frobenius senseorthogonally on the subspace of symmetric matrices S n in isomorphicR n2 amounts to taking the symmetric part of H ; (2.2.2.0.1) id est,(H+H T )/2 is the projection.To project any H ∈ R n×n orthogonally on the symmetric hollowsubspace S n h in isomorphic Rn2 (2.2.3.0.1), we may take the symmetricpart and then zero all entries along the main diagonal, or vice versa(because this is projection on the intersection of two subspaces); id est,(H + H T )/2 − δ 2 (H) .To project a matrix on the nonnegative orthant R m×n+ , simply clip allnegative entries to 0. Likewise, projection on the nonpositive orthantR m×n−sees all positive entries clipped to 0. Projection on other orthantsis equally simple with appropriate clipping.Clipping in excess of |1| each entry of a point x∈ R n is equivalentto unique minimum-distance projection of x on the unit hypercubecentered at the origin. (conferE.10.3.2)Projecting on hyperplane, halfspace, slab:E.5.0.0.8.Projection of x∈ R n on a (rectangular) hyperbox: [46,8.1.1]C = {y ∈ R n | l ≼ y ≼ u, l ≺ u} (1836)⎧⎨ l k , x k ≤ l kP(x) k = x k , l k ≤ x k ≤ u k (1837)⎩u k , x k ≥ u k