v2007.09.13 - Convex Optimization

4.3. RANK REDUCTION 2474.3 Rank reduction...it is not clear generally how to predict rankX ⋆ or rankS ⋆before solving the SDP problem.−Farid Alizadeh (1995) [9, p.22]The premise of rank reduction in semidefinite programming is: an optimalsolution found does not satisfy Barvinok’s upper bound (232) on rank. Theparticular numerical algorithm solving a semidefinite program may haveinstead returned a high-rank optimal solution (4.1.1; e.g., (557)) when alower-rank optimal solution was expected.4.3.1 Posit a perturbation of X ⋆Recall from4.1.1.1, there is an extreme point of A ∩ S n + (549) satisfyingupper bound (232) on rank. [21,2.2] It is therefore sufficient to locatean extreme point of the intersection whose primal objective value (546P) isoptimal: 4.16 [77,31.5.3] [174,2.4] [5,3] [214]Consider again the affine subsetA = {X ∈ S n | A svec X = b} (549)where for A i ∈ S n ⎡A =∆ ⎣⎤svec(A 1 ) T. ⎦ ∈ R m×n(n+1)/2 (547)svec(A m ) TGiven any optimal solution X ⋆ tominimizeX∈ S n 〈C , X 〉subject to X ∈ A ∩ S n +(546)(P)4.16 There is no known construction for Barvinok’s tighter result (237). −Monique Laurent