v2009.01.01 - Convex Optimization

268 CHAPTER 4. SEMIDEFINITE PROGRAMMING
4.3 Rank reduction
...it is not clear generally how to predict rankX ⋆ or rankS ⋆
before solving the SDP problem.
−Farid Alizadeh (1995) [10, p.22]
The premise of rank reduction in semidefinite programming is: an optimal
solution found does not satisfy Barvinok’s upper bound (245) on rank. The
particular numerical algorithm solving a semidefinite program may have
instead returned a high-rank optimal solution (4.1.1.1; e.g., (595)) when
a lower-rank optimal solution was expected.
4.3.1 Posit a perturbation of X ⋆
Recall from4.1.1.2, there is an extreme point of A ∩ S n + (587) satisfying
upper bound (245) on rank. [23,2.2] It is therefore sufficient to locate
an extreme point of the intersection whose primal objective value (584P) is
optimal: 4.17 [96,31.5.3] [206,2.4] [207] [6,3] [252]
Consider again the affine subset
A = {X ∈ S n | A svec X = b} (587)
where for A i ∈ S n ⎡
A =
∆ ⎣
⎤
svec(A 1 ) T
. ⎦ ∈ R m×n(n+1)/2 (585)
svec(A m ) T
Given any optimal solution X ⋆ to
minimize
X∈ S n 〈C , X 〉
subject to X ∈ A ∩ S n +
(584)(P)
4.17 There is no known construction for Barvinok’s tighter result (250). −Monique Laurent