v2009.01.01 - Convex Optimization

274 CHAPTER 4. SEMIDEFINITE PROGRAMMING
4.3.3.0.1 Example. Aδ(X) = b .
This academic example demonstrates that a solution found by rank reduction
can certainly have rank less than Barvinok’s upper bound (245): Assume a
given vector b∈ R m belongs to the conic hull of columns of a given matrix
A∈ R m×n ;
⎡
⎤ ⎡ ⎤
A =
⎢
⎣
−1 1 8 1 1
−3 2 8
1
2
−9 4 8
1
4
1
3
1
9
Consider the convex optimization problem
⎥
⎦ , b =
⎢
⎣
1
1
2
1
4
⎥
⎦ (658)
minimize trX
X∈ S 5
subject to X ≽ 0
Aδ(X) = b
(659)
that minimizes the 1-norm of the main diagonal; id est, problem (659) is the
same as
minimize
X∈ S 5 ‖δ(X)‖ 1
subject to X ≽ 0
Aδ(X) = b
(660)
that finds a solution to Aδ(X)=b. Rank-3 solution X ⋆ = δ(x M
) is optimal,
where (confer (618))
⎡ 2 ⎤
128
0
x M
=
5
⎢ 128 ⎥
(661)
⎣ 0 ⎦
90
128
Yet upper bound (245) predicts existence of at most a
(⌊√ ⌋ )
8m + 1 − 1
rank-
= 2
2
(662)
feasible solution from m = 3 equality constraints. To find a lower
rank ρ optimal solution to (659) (barring combinatorics), we invoke
Procedure 4.3.1.0.1: