v2010.10.26 - Convex Optimization

740 APPENDIX E. PROJECTIONP R x = x − P R ⊥ x). Indeed, −K ∗ is the algebraic complement in theorthogonal vector sum (p.773) [269] [199,A.3.2.5]K ⊞ −K ∗ = R n ⇔ cone K is closed and convex (2023)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} (2024)consequent to Moreau (2027). Recalling that any subspace is a closed convexcone E.16 K = R(P ) ⇔ −K ∗ = R(P ) ⊥ (2025)meaning, when a cone is a subspace R(P ) , then the dual cone becomes itsorthogonal complement R(P ) ⊥ . [61,2.6.1] 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 ) (2026)E.9.2.2 Salient properties: Projection Px on closed convex cone K[199,A.3.2] [109,5.6] 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)) [269]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(2027)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} (2024)E.16 but a proper subspace is not a proper cone (2.7.2.2.1).