v2009.01.01 - Convex Optimization

4.3. RANK REDUCTION 271
and where matrix Z i ∈ S ρ is found at each iteration i by solving a very
simple feasibility problem: 4.18
find Z i ∈ S ρ
subject to 〈Z i , RiA T j R i 〉 = 0 ,
j =1... m
(646)
Were there a sparsity pattern common to each member of the set
{R T iA j R i ∈ S ρ , j =1... m} , then a good choice for Z i has 1 in each entry
corresponding to a 0 in the pattern; id est, a sparsity pattern complement.
At iteration i
∑i−1
X ⋆ + t j B j + t i B i = R i (I − t i ψ(Z i )Z i )Ri T (647)
j=1
By fact (1353), therefore
∑i−1
X ⋆ + t j B j + t i B i ≽ 0 ⇔ 1 − t i ψ(Z i )λ(Z i ) ≽ 0 (648)
j=1
where λ(Z i )∈ R ρ denotes the eigenvalues of Z i .
Maximization of each t i in step 2 of the Procedure reduces rank of (647)
and locates a new point on the boundary ∂(A ∩ S n +) . 4.19 Maximization of
t i thereby has closed form;
4.18 A simple method of solution is closed-form projection of a random nonzero point on
that proper subspace of isometrically isomorphic R ρ(ρ+1)/2 specified by the constraints.
(E.5.0.0.6) Such a solution is nontrivial assuming the specified intersection of hyperplanes
is not the origin; guaranteed by ρ(ρ + 1)/2 > m. Indeed, this geometric intuition about
forming the perturbation is what bounds any solution’s rank from below; m is fixed by
the number of equality constraints in (584P) while rank ρ decreases with each iteration i.
Otherwise, we might iterate indefinitely.
4.19 This holds because rank of a positive semidefinite matrix in S n is diminished below
n by the number of its 0 eigenvalues (1363), and because a positive semidefinite matrix
having one or more 0 eigenvalues corresponds to a point on the PSD cone boundary (175).
Necessity and sufficiency are due to the facts: R i can be completed to a nonsingular matrix
(A.3.1.0.5), and I − t i ψ(Z i )Z i can be padded with zeros while maintaining equivalence
in (647).