v2010.10.26 - Convex Optimization

4.3. RANK REDUCTION 301A rank-reduced optimal solution is theni∑X ⋆ ← X ⋆ + t j B j (707)j=14.3.2 Perturbation formPerturbations of X ⋆ are independent of constants C ∈ S n and b∈R m inprimal and dual problems (649). Numerical accuracy of any rank-reducedresult, found by perturbation of an initial optimal solution X ⋆ , is thereforequite dependent upon initial accuracy of X ⋆ .4.3.2.0.1 Definition. Matrix step function. (conferA.6.5.0.1)Define the signum-like quasiconcave real function ψ : S n → Rψ(Z) { 1, Z ≽ 0−1, otherwise(708)The value −1 is taken for indefinite or nonzero negative semidefiniteargument.△Deza & Laurent [113,31.5.3] prove: every perturbation matrix B i ,i=1... n , is of the formB i = −ψ(Z i )R i Z i R T i ∈ S n (709)where∑i−1X ⋆ R 1 R1 T , X ⋆ + t j B j R i Ri T ∈ S n (710)j=1where the t j are scalars and R i ∈ R n×ρ is full-rank and skinny where( )∑i−1ρ rank X ⋆ + t j B jj=1(711)