v2010.10.26 - Convex Optimization

E.9. PROJECTION ON CONVEX SET 741E.9.2.2.1 Corollary. I −P for cones. (conferE.2)Denote by K ⊆ R n a closed convex cone, and call K ∗ its dual. Thenx −P −K∗x is the unique minimum-distance projection of x∈ R n on K if andonly if P −K∗x is the unique minimum-distance projection of x on −K ∗ thepolar cone.⋄Proof. Assume x 1 = P K x . Then by Theorem E.9.2.0.1 we havex 1 ∈ K , x 1 − x ⊥ x 1 , x 1 − x ∈ K ∗ (2028)Now assume x − x 1 = P −K∗x . Then we havex − x 1 ∈ −K ∗ , −x 1 ⊥ x − x 1 , −x 1 ∈ −K (2029)But these two assumptions are apparently identical. We must therefore havex −P −K∗x = x 1 = P K x (2030)E.9.2.2.2 Corollary. Unique projection via dual or normal cone.[109,4.7] (E.10.3.2, confer Theorem E.9.1.0.3) Given point x∈ R n andclosed convex cone K ⊆ R n , the following are equivalent statements:1. point Px is the unique minimum-distance projection of x on K2. Px ∈ K , x − Px ∈ −(K − Px) ∗ = −K ∗ ∩ (Px) ⊥3. Px ∈ K , 〈x − Px, Px〉 = 0, 〈x − Px, y〉 ≤ 0 ∀y ∈ KE.9.2.2.3 Example. Unique projection on nonnegative orthant.(confer (1320)) From Theorem E.9.2.0.1, to project matrix H ∈ R m×n onthe selfdual orthant (2.13.5.1) of nonnegative matrices R m×n+ in isomorphicR mn , the necessary and sufficient conditions are:⋄H ⋆ ≥ 0tr ( (H ⋆ − H) T H ⋆) = 0H ⋆ − H ≥ 0(2031)