v2007.09.13 - Convex Optimization

4.3. RANK REDUCTION 249A rank-reduced optimal solution is theni∑X ⋆ ← X ⋆ + t j B j (603)j=14.3.2 Perturbation formThe perturbations are independent of constants C ∈ S n and b∈R m in primaland dual programs (546). Numerical accuracy of any rank-reduced result,found by perturbation of an initial optimal solution X ⋆ , is therefore quitedependent 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(604)The value −1 is taken for indefinite or nonzero negative semidefiniteargument.△Deza & Laurent [77,31.5.3] 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 (605)where∑i−1X ⋆ = ∆ R 1 R1 T , X ⋆ + t j B ∆ j = R i Ri T ∈ S n (606)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(607)and where matrix Z i ∈ S ρ is found at each iteration i by solving a very