v2007.09.13 - Convex Optimization

4.3. RANK REDUCTION 251(t ⋆ i) −1 = max {ψ(Z i )λ(Z i ) j , j =1... ρ} (611)When Z i is indefinite, the direction of perturbation (determined by ψ(Z i )) isarbitrary. We may take an early exit from the Procedure were Z i to become0 or wererank [ svec R T iA 1 R i svec R T iA 2 R i · · · svec R T iA m R i]= ρ(ρ + 1)/2 (612)which characterizes the rank ρ of any [sic] extreme point in A ∩ S n + .[174,2.4]Proof. Assuming the form of every perturbation matrix is indeed (605),then by (608)svec Z i ⊥ [ svec(R T iA 1 R i ) svec(R T iA 2 R i ) · · · svec(R T iA m R i ) ] (613)By orthogonal complement we haverank [ svec(R T iA 1 R i ) · · · svec(R T iA m R i ) ] ⊥+ rank [ svec(R T iA 1 R i ) · · · svec(R T iA m R i ) ] = ρ(ρ + 1)/2(614)When Z i can only be 0, then the perturbation is null because an extremepoint has been found; thus[svec(RTi A 1 R i ) · · · svec(R T iA m R i ) ] ⊥= 0 (615)from which the stated result (612) directly follows.