v2007.09.13 - Convex Optimization

618 APPENDIX E. PROJECTIONProjection on K of any point x∈−K ∗ , belonging to the negative dualcone, is the origin. By (1784): the set of all points reaching the origin, whenprojecting on K , constitutes the negative dual cone; a.k.a, the polar coneE.9.2.1K ◦ = −K ∗ = {x∈ R n | Px = 0} (1785)Relation to subspace projectionConditions 1 and 2 of the theorem are common with orthogonal projectionon a subspace R(P ) : Condition 1 is the most basic requirement;namely, Px∈ R(P ) , the projection belongs to the subspace. Invokingperpendicularity condition (1670), we recall the second requirement forprojection on a subspace:Px − x ⊥ R(P ) or Px − x ∈ R(P ) ⊥ (1786)which corresponds to condition 2. Yet condition 3 is a generalizationof subspace projection; id est, for unique minimum-distance projection ona closed convex cone K , polar cone −K ∗ plays the role R(P ) ⊥ playsfor subspace projection (P R x = x − P R ⊥ x). Indeed, −K ∗ is the algebraiccomplement in the orthogonal vector sum (p.674) [196] [147,A.3.2.5]K ⊞ −K ∗ = R n ⇔ cone K is closed and convex (1787)Also, given unique minimum-distance projection Px on K satisfyingTheorem E.9.2.0.1, then by projection on the algebraic complement via I −PinE.2 we have−K ∗ = {x − Px | x∈ R n } = {x∈ R n | Px = 0} (1788)consequent to Moreau (1791). Recalling any subspace is a closed convexcone E.15 K = R(P ) ⇔ −K ∗ = R(P ) ⊥ (1789)meaning, when a cone is a subspace R(P ) then the dual cone becomes itsorthogonal complement R(P ) ⊥ . [46,2.6.1] In this circumstance, condition 3becomes coincident with condition 2.The properties of projection on cones following inE.9.2.2 furthergeneralize to subspaces by: (4)K = R(P ) ⇔ −K = R(P ) (1790)E.15 but a proper subspace is not a proper cone (2.7.2.2.1).