v2009.01.01 - Convex Optimization

E.9. PROJECTION ON CONVEX SET 679
Projection on K of any point x∈−K ∗ , belonging to the negative dual
cone, is the origin. By (1897): the set of all points reaching the origin, when
projecting on K , constitutes the negative dual cone; a.k.a, the polar cone
E.9.2.1
K ◦ = −K ∗ = {x∈ R n | Px = 0} (1898)
Relation to subspace projection
Conditions 1 and 2 of the theorem are common with orthogonal projection
on a subspace R(P ) : Condition 1 is the most basic requirement;
namely, Px∈ R(P ) , the projection belongs to the subspace. Invoking
perpendicularity condition (1783), we recall the second requirement for
projection on a subspace:
Px − x ⊥ R(P ) or Px − x ∈ R(P ) ⊥ (1899)
which corresponds to condition 2. Yet condition 3 is a generalization
of subspace projection; id est, for unique minimum-distance projection on
a closed convex cone K , polar cone −K ∗ plays the role R(P ) ⊥ plays
for subspace projection (P R x = x − P R ⊥ x). Indeed, −K ∗ is the algebraic
complement in the orthogonal vector sum (p.708) [232] [173,A.3.2.5]
K ⊞ −K ∗ = R n ⇔ cone K is closed and convex (1900)
Also, given unique minimum-distance projection Px on K satisfying
Theorem E.9.2.0.1, then by projection on the algebraic complement via I −P
inE.2 we have
−K ∗ = {x − Px | x∈ R n } = {x∈ R n | Px = 0} (1901)
consequent to Moreau (1904). Recalling any subspace is a closed convex
cone E.16 K = R(P ) ⇔ −K ∗ = R(P ) ⊥ (1902)
meaning, when a cone is a subspace R(P ) , then the dual cone becomes its
orthogonal complement R(P ) ⊥ . [53,2.6.1] In this circumstance, condition 3
becomes coincident with condition 2.
The properties of projection on cones following inE.9.2.2 further
generalize to subspaces by: (4)
K = R(P ) ⇔ −K = R(P ) (1903)
E.16 but a proper subspace is not a proper cone (2.7.2.2.1).