v2010.10.26 - Convex Optimization

302 CHAPTER 4. SEMIDEFINITE PROGRAMMINGand where matrix Z i ∈ S ρ is found at each iteration i by solving a verysimple feasibility problem: 4.20find Z i ∈ S ρsubject to 〈Z i , RiA T j R i 〉 = 0 ,j =1... m(712)Were there a sparsity pattern common to each member of the set{R T iA j R i ∈ S ρ , j =1... m} , then a good choice for Z i has 1 in each entrycorresponding to a 0 in the pattern; id est, a sparsity pattern complement.At iteration i∑i−1X ⋆ + t j B j + t i B i = R i (I − t i ψ(Z i )Z i )Ri T (713)j=1By fact (1450), therefore∑i−1X ⋆ + t j B j + t i B i ≽ 0 ⇔ 1 − t i ψ(Z i )λ(Z i ) ≽ 0 (714)j=1where λ(Z i )∈ R ρ denotes the eigenvalues of Z i .Maximization of each t i in step 2 of the Procedure reduces rank of (713)and locates a new point on the boundary ∂(A ∩ S n +) . 4.21 Maximization oft i thereby has closed form;4.20 A simple method of solution is closed-form projection of a random nonzero point onthat proper subspace of isometrically isomorphic R ρ(ρ+1)/2 specified by the constraints.(E.5.0.0.6) Such a solution is nontrivial assuming the specified intersection of hyperplanesis not the origin; guaranteed by ρ(ρ + 1)/2 > m. Indeed, this geometric intuition aboutforming the perturbation is what bounds any solution’s rank from below; m is fixed bythe number of equality constraints in (649P) while rank ρ decreases with each iteration i.Otherwise, we might iterate indefinitely.4.21 This holds because rank of a positive semidefinite matrix in S n is diminished belown by the number of its 0 eigenvalues (1460), and because a positive semidefinite matrixhaving one or more 0 eigenvalues corresponds to a point on the PSD cone boundary (193).Necessity and sufficiency are due to the facts: R i can be completed to a nonsingular matrix(A.3.1.0.5), and I − t i ψ(Z i )Z i can be padded with zeros while maintaining equivalencein (713).