v2009.01.01 - Convex Optimization

270 CHAPTER 4. SEMIDEFINITE PROGRAMMING
A rank-reduced optimal solution is then
i∑
X ⋆ ← X ⋆ + t j B j (641)
j=1
4.3.2 Perturbation form
The perturbations are independent of constants C ∈ S n and b∈R m in primal
and dual programs (584). Numerical accuracy of any rank-reduced result,
found by perturbation of an initial optimal solution X ⋆ , is therefore quite
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 → R
ψ(Z) ∆ =
{ 1, Z ≽ 0
−1, otherwise
(642)
The value −1 is taken for indefinite or nonzero negative semidefinite
argument.
△
Deza & Laurent [96,31.5.3] prove: every perturbation matrix B i ,
i=1... n , is of the form
B i = −ψ(Z i )R i Z i R T i ∈ S n (643)
where
∑i−1
X ⋆ = ∆ R 1 R1 T , X ⋆ + t j B ∆ j = R i Ri T ∈ S n (644)
j=1
where the t j are scalars and R i ∈ R n×ρ is full-rank and skinny where
( )
∑i−1
ρ = ∆ rank X ⋆ + t j B j
j=1
(645)