v2009.01.01 - Convex Optimization

686 APPENDIX E. PROJECTION
E.9.5
Projection on convex set in subspace
Suppose a convex set C is contained in some subspace R n . Then unique
minimum-distance projection of any point in R n ⊕ R n⊥ on C can be
accomplished by first projecting orthogonally on that subspace, and then
projecting the result on C ; [92,5.14] id est, the ordered product of two
individual projections that is not commutable.
Proof. (⇐) To show that, suppose unique minimum-distance projection
P C x on C ⊂ R n is y as illustrated in Figure 141;
‖x − y‖ ≤ ‖x − q‖ ∀q ∈ C (1920)
Further suppose P R
n x equals z . By the Pythagorean theorem
‖x − y‖ 2 = ‖x − z‖ 2 + ‖z − y‖ 2 (1921)
because x − z ⊥ z − y . (1783) [215,3.3] Then point y = P C x is the same
as P C z because
‖z − y‖ 2 = ‖x − y‖ 2 − ‖x − z‖ 2 ≤ ‖z − q‖ 2 = ‖x − q‖ 2 − ‖x − z‖ 2
which holds by assumption (1920).
(⇒) Now suppose z = P R
n x and
∀q ∈ C
(1922)
‖z − y‖ ≤ ‖z − q‖ ∀q ∈ C (1923)
meaning y = P C z . Then point y is identical to P C x because
‖x − y‖ 2 = ‖x − z‖ 2 + ‖z − y‖ 2 ≤ ‖x − q‖ 2 = ‖x − z‖ 2 + ‖z − q‖ 2
by assumption (1923).
∀q ∈ C
(1924)
This proof is extensible via translation argument. (E.4) Unique
minimum-distance projection on a convex set contained in an affine subset
is, therefore, similarly accomplished.
Projecting matrix H ∈ R n×n on convex cone K = S n ∩ R n×n
+ in isomorphic
R n2 can be accomplished, for example, by first projecting on S n and only then
projecting the result on R n×n
+ (confer7.0.1). This is because that projection
product is equivalent to projection on the subset of the nonnegative orthant
in the symmetric matrix subspace.