v2009.01.01 - Convex Optimization

342 CHAPTER 4. SEMIDEFINITE PROGRAMMING
The fact
Λ ≽ 0 ⇔ X ≽ 0 (1353)
allows splitting semidefinite feasibility problem (776) into two parts:
( ρ
) minimize
Λ
∥ A svec ∑
λ i Q ii − b
∥
i=1
[ R
ρ
] (779)
+
subject to δ(Λ) ∈
0
( ρ
) minimize
Q
∥ A svec ∑
λ ⋆ i Q ii − b
∥
i=1
(780)
subject to Q T = Q −1
The linear equality constraint A svec X = b has been moved to the objective
within a norm because these two problems (779) (780) are iterated; equality
might only become feasible near convergence. This iteration always converges
to a local minimum because the sequence of objective values is monotonic
and nonincreasing; any monotonically nonincreasing real sequence converges.
[222,1.2] [37,1.1] A rank-ρ matrix X feasible to the original problem (776)
is found when the objective converges to 0. Positive semidefiniteness of
matrix X with an upper bound ρ on rank is assured by the constraint on
eigenvalue matrix Λ in convex problem (779); it means, the main diagonal
of Λ must belong to the nonnegative orthant in a ρ-dimensional subspace
of R n .
The second problem (780) in the iteration is not convex. We propose
solving it by convex iteration: Make the assignment
⎡
G = ⎣
⎡
= ⎣
⎤
q 1 [q1 T · · · qρ T ]
.
⎦
∈ S nρ
q ρ
⎤ ⎡
Q 11 · · · Q 1ρ q 1 q
.
... . ⎦ ∆ 1 T · · · q 1 qρ
T
= ⎣
.
... .
Q T 1ρ · · · Q ρρ q ρ q1 T · · · q ρ qρ
T
⎤
⎦
(781)