v2010.10.26 - Convex Optimization

E.9. PROJECTION ON CONVEX SET 743E.9.3nonexpansivityE.9.3.0.1 Theorem. Nonexpansivity. [175,2] [109,5.3]When C ⊂ R n is an arbitrary closed convex set, projector P projecting on Cis nonexpansive in the sense: for any vectors x,y ∈ R n‖Px − Py‖ ≤ ‖x − y‖ (2037)with equality when x −Px = y −Py . E.17⋄Proof. [54]‖x − y‖ 2 = ‖Px − Py‖ 2 + ‖(I − P )x − (I − P )y‖ 2+ 2〈x − Px, Px − Py〉 + 2〈y − Py , Py − Px〉(2038)Nonnegativity of the last two terms follows directly from the uniqueminimum-distance projection theorem (E.9.1.0.2).The 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 (2039)Deutsch shows yet another: [109,5.5]E.9.4‖Px − Py‖ 2 ≤ 〈x − y , Px − Py〉 (2040)Easy projectionsTo project any matrix H ∈ R n×n orthogonally in Euclidean/Frobeniussense on subspace of symmetric matrices S n in isomorphic R n2 , takesymmetric 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 symmetric hollow subspaceS n h in isomorphic Rn2 (2.2.3.0.1,7.0.1), take symmetric part then zeroall entries along main diagonal or vice versa (because this is projectionon intersection of two subspaces); id est, (H + H T )/2 − δ 2 (H).E.17 This condition for equality corrects an error in [78] (where the norm is applied to eachside of the condition given here) easily revealed by counterexample.