v2009.01.01 - Convex Optimization

4.3. RANK REDUCTION 275
Initialize:
C =I , ρ=3, A j ∆ = δ(A(j, :)) , j =1, 2, 3, X ⋆ = δ(x M
) , m=3, n=5.
{
Iteration i=1:
⎡
Step 1: R 1 =
⎢
⎣
√
2
128
0 0
0 0 0
0
√
5
0
128
0 0
√
0
0 0
90
128
⎤
.
⎥
⎦
find Z 1 ∈ S 3
subject to 〈Z 1 , R T 1A j R 1 〉 = 0, j =1, 2, 3
(663)
A nonzero randomly selected matrix Z 1 having 0 main diagonal
is feasible and yields a nonzero perturbation matrix. Choose,
arbitrarily,
Z 1 = 11 T − I ∈ S 3 (664)
then (rounding)
⎡
B 1 =
⎢
⎣
Step 2: t ⋆ 1= 1 because λ(Z 1 )=[−1 −1
⎡
X ⋆ ← δ(x M
) + B 1 =
⎢
⎣
0 0 0.0247 0 0.1048
0 0 0 0 0
0.0247 0 0 0 0.1657
0 0 0 0 0
0.1048 0 0.1657 0 0
2 ] T . So,
⎤
⎥
⎦
2
0 0.0247 0 0.1048
128
0 0 0 0 0
0.0247 0
5
128
0 0.1657
0 0 0 0 0
0.1048 0 0.1657 0
90
128
⎤
(665)
⎥
⎦ (666)
has rank ρ ←1 and produces the same optimal objective value.
}