v2010.10.26 - Convex Optimization

4.3. RANK REDUCTION 2994.3 Rank reduction...it is not clear generally how to predict rankX ⋆ or rankS ⋆before solving the SDP problem.−Farid Alizadeh (1995) [11, p.22]The premise of rank reduction in semidefinite programming is: an optimalsolution found does not satisfy Barvinok’s upper bound (272) on rank. Theparticular numerical algorithm solving a semidefinite program may haveinstead returned a high-rank optimal solution (4.1.2; e.g., (660)) when alower-rank optimal solution was expected. Rank reduction is a means toadjust rank of an optimal solution, returned by a solver, until it satisfiesBarvinok’s upper bound.4.3.1 Posit a perturbation of X ⋆Recall from4.1.2.1, there is an extreme point of A ∩ S n + (652) satisfyingupper bound (272) on rank. [24,2.2] It is therefore sufficient to locate anextreme point of the intersection whose primal objective value (649P) isoptimal: 4.19 [113,31.5.3] [239,2.4] [240] [7,3] [291]Consider again affine subsetA = {X ∈ S n | A svec X = b} (652)where for A i ∈ S n ⎡A ⎣⎤svec(A 1 ) T. ⎦ ∈ R m×n(n+1)/2 (650)svec(A m ) TGiven any optimal solution X ⋆ tominimizeX∈ S n 〈C , X 〉subject to X ∈ A ∩ S n +(649P)4.19 There is no known construction for Barvinok’s tighter result (277).−Monique Laurent (2004)