v2009.01.01 - Convex Optimization

4.7. CONVEX ITERATION RANK-1 341
find X ∈ S n
subject to A svec X = b
X ≽ 0
rankX ≤ ρ
(776)
given an upper bound 0 < ρ < n on rank, vector b ∈ R m , and typically fat
full-rank
⎡
A =
∆ ⎣
⎤
svec(A 1 ) T
. ⎦∈ R m×n(n+1)/2 (585)
svec(A m ) T
where A i ∈ S n , i=1... m are also given. Symmetric matrix vectorization
svec is defined in (49). Thus
⎡
A svec X = ⎣
tr(A 1 X)
.
tr(A m X)
⎤
⎦ (586)
This program (776) states the classical problem of finding a matrix of
specified rank in the intersection of the positive semidefinite cone with
a number m of hyperplanes in the subspace of symmetric matrices S n .
[25,II.13] [23,2.2]
Express the nonincreasingly ordered diagonalization of variable matrix
X ∆ = QΛQ T =
n∑
λ i Q ii ∈ S n (777)
i=1
which is a sum of rank-1 orthogonal projection matrices Q ii weighted by
eigenvalues λ i where Q ij ∆ = q i q T j ∈ R n×n , Q = [q 1 · · · q n ]∈ R n×n , Q T = Q −1 ,
Λ ii = λ i ∈ R , and
⎡
Λ = ⎢
⎣
⎤
λ 1 0
λ 2 ⎥ ... ⎦ ∈ Sn (778)
0 T λ n