v2009.01.01 - Convex Optimization

E.10. ALTERNATING PROJECTION 697
E.10.2.1.1 Example. Affine subset ∩ positive semidefinite cone.
Consider the problem of finding X ∈ S n that satisfies
X ≽ 0, 〈A j , X〉 = b j , j =1... m (1948)
given nonzero A j ∈ S n and real b j . Here we take C 1 to be the positive
semidefinite cone S n + while C 2 is the affine subset of S n
C 2 = A = ∆ {X | tr(A j X)=b j , j =1... m} ⊆ S n
⎡ ⎤
svec(A 1 ) T
= {X | ⎣ . ⎦svec X = b}
svec(A m ) T
∆
= {X | A svec X = b}
(1949)
where b = [b j ] ∈ R m , A ∈ R m×n(n+1)/2 , and symmetric vectorization svec is
defined by (49). Projection of iterate X i ∈ S n on A is: (E.5.0.0.6)
P 2 svec X i = svec X i − A † (A svec X i − b) (1950)
Euclidean distance from X i to A is therefore
dist(X i , A) = ‖X i − P 2 X i ‖ F = ‖A † (A svec X i − b)‖ 2 (1951)
Projection of P 2 X i ∆ = ∑ j
λ j q j q T j on the positive semidefinite cone (7.1.2) is
found from its eigen decomposition (A.5.2);
P 1 P 2 X i =
n∑
max{0 , λ j }q j qj T (1952)
j=1
Distance from P 2 X i to the positive semidefinite cone is therefore
n∑
dist(P 2 X i , S+) n = ‖P 2 X i − P 1 P 2 X i ‖ F = √ min{0,λ j } 2 (1953)
When the intersection is empty A ∩ S n + = ∅ , the iterates converge to that
positive semidefinite matrix closest to A in the Euclidean sense. Otherwise,
convergence is to some point in the nonempty intersection.
j=1