v2009.01.01 - Convex Optimization

678 APPENDIX E. PROJECTION
Projection on cone K is
P K x = (I − 1
τ 2a⋆ a ⋆T )x (1894)
whereas projection on the polar cone −K ∗ is (E.9.2.2.1)
P K ◦x = x − P K x = 1
τ 2a⋆ a ⋆T x (1895)
Negating vector a , this maximization problem (1892) becomes a
minimization (the same problem) and the polar cone becomes the dual cone:
E.9.2
‖x − P K x‖ = − 1 τ minimize a T x
a
subject to ‖a‖ ≤ τ (1896)
a ∈ K ∗
Projection on cone
When convex set C
conditions:
is a cone, there is a finer statement of optimality
E.9.2.0.1 Theorem. Unique projection on cone. [173,A.3.2]
Let K ⊆ R n be a closed convex cone, and K ∗ its dual (2.13.1). Then Px is
the unique minimum-distance projection of x∈ R n on K if and only if
Px ∈ K , 〈Px − x, Px〉 = 0, Px − x ∈ K ∗ (1897)
In words, Px is the unique minimum-distance projection of x on K if
and only if
1) projection Px lies in K
2) direction Px−x is orthogonal to the projection Px
3) 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 .
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