v2007.09.17 - Convex Optimization

E.9. PROJECTION ON CONVEX SET 619Projection on cone K isP K x = (I − 1τ 2a⋆ a ⋆T )x (1786)whereas projection on the polar cone −K ∗ is (E.9.2.2.1)P K ◦x = x − P K x = 1τ 2a⋆ a ⋆T x (1787)Negating vector a , this maximization problem (1784) becomes aminimization (the same problem) and the polar cone becomes the dual cone:E.9.2‖x − P K x‖ = − 1 τ minimize a T xasubject to ‖a‖ ≤ τ (1788)a ∈ K ∗Projection on coneWhen convex set Cconditions:is a cone, there is a finer statement of optimalityE.9.2.0.1 Theorem. Unique projection on cone. [148,A.3.2]Let K ⊆ R n be a closed convex cone, and K ∗ its dual (2.13.1). Then Px isthe unique minimum-distance projection of x∈ R n on K if and only ifPx ∈ K , 〈Px − x, Px〉 = 0, Px − x ∈ K ∗ (1789)In words, Px is the unique minimum-distance projection of x on K ifand only if1) projection Px lies in K2) direction Px−x is orthogonal to the projection Px3) direction Px−x lies in the dual cone K ∗ .As the theorem is stated, it admits projection on K having empty interior;id est, on convex cones in a proper subspace of R n .⋄