v2009.01.01 - Convex Optimization

680 APPENDIX E. PROJECTION
E.9.2.2 Salient properties: Projection Px on closed convex cone K
[173,A.3.2] [92,5.6] For x, x 1 , x 2 ∈ R n
1. 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 x
5. (Jean-Jacques Moreau (1962)) [232]
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
(1904)
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} (1901)
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. Then
x −P −K
∗x is the unique minimum-distance projection of x∈ R n on K if and
only if P −K
∗x is the unique minimum-distance projection of x on −K ∗ the
polar cone.
⋄
Proof. Assume x 1 = P K x . Then by Theorem E.9.2.0.1 we have
x 1 ∈ K , x 1 − x ⊥ x 1 , x 1 − x ∈ K ∗ (1905)
Now assume x − x 1 = P −K
∗x . Then we have
x − x 1 ∈ −K ∗ , −x 1 ⊥ x − x 1 , −x 1 ∈ −K (1906)
But these two assumptions are apparently identical. We must therefore have
x −P −K
∗x = x 1 = P K x (1907)