v2010.10.26 - Convex Optimization

E.9. PROJECTION ON CONVEX SET 739R(P )x − (I −P )xx(I −P )xR(P ) ⊥Figure 165: (confer Figure 83) Given orthogonal projection (I −P )x of xon orthogonal complement R(P ) ⊥ , projection on R(P ) is immediate:x − (I −P )x .Projection on K of any point x∈−K ∗ , belonging to the negative dualcone, is the origin. By (2021): the set of all points reaching the origin, whenprojecting on K , constitutes the negative dual cone; a.k.a, the polar coneK ◦ = −K ∗ = {x∈ R n | Px = 0} (2022)E.9.2.1Relationship to subspace projectionConditions 1 and 2 of the theorem are common with orthogonal projection ona subspace R(P ) : Condition 1 corresponds to the most basic requirement;namely, the projection Px∈ R(P ) belongs to the subspace. Recall theperpendicularity requirement for projection on a subspace;Px − x ⊥ R(P ) or Px − x ∈ R(P ) ⊥ (1907)which corresponds to condition 2.Yet condition 3 is a generalization of subspace projection; id est, forunique minimum-distance projection on a closed convex cone K , polarcone −K ∗ plays the role R(P ) ⊥ plays for subspace projection (Figure 165: