v2007.11.26 - Convex Optimization

E.9. PROJECTION ON CONVEX SET 635E.9.2.2 Salient properties: Projection Px on closed convex cone K[148,A.3.2] [73,5.6] For x, x 1 , x 2 ∈ R n1. P K (αx) = α P K x ∀α≥0 (nonnegative homogeneity)2. ‖P K x‖ ≤ ‖x‖3. P K x = 0 ⇔ x ∈ −K ∗4. P K (−x) = −P −K x5. (Jean-Jacques Moreau (1962)) [199]x = x 1 + x 2 , x 1 ∈ K , x 2 ∈−K ∗ , x 1 ⊥ x 2⇔x 1 = P K x , x 2 = P −K∗x(1822)6. K = {x − P −K∗x | x∈ R n } = {x∈ R n | P −K∗x = 0}7. −K ∗ = {x − P K x | x∈ R n } = {x∈ R n | P K x = 0} (1819)E.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 ∗ (1823)Now assume x − x 1 = P −K∗x . Then we havex − x 1 ∈ −K ∗ , −x 1 ⊥ x − x 1 , −x 1 ∈ −K (1824)But these two assumptions are apparently identical. We must therefore havex −P −K∗x = x 1 = P K x (1825)