v2010.10.26 - Convex Optimization

4.8. CONVEX ITERATION RANK-1 391which is a sum of rank-1 orthogonal projection matrices Q ii weighted byeigenvalues λ i where Q ij q i q T j ∈ R n×n , Q = [q 1 · · · q n ]∈ R n×n , Q T = Q −1 ,Λ ii = λ i ∈ R , and⎡Λ = ⎢⎣⎤λ 1 0λ 2 ⎥ ... ⎦ ∈ Sn (871)0 T λ nThe factΛ ≽ 0 ⇔ X ≽ 0 (1450)allows splitting semidefinite feasibility problem (869) into two parts:( ρ) minimizeΛ∥ A svec ∑λ i Q ii − b∥i=1[ Rρ] (872)+subject to δ(Λ) ∈0( ρ) minimizeQ∥ A svec ∑λ ⋆ i Q ii − b∥i=1(873)subject to Q T = Q −1The linear equality constraint A svec X = b has been moved to the objectivewithin a norm because these two problems (872) (873) are iterated; equalitymight only become attainable near convergence. This iteration alwaysconverges to a local minimum because the sequence of objective values ismonotonic and nonincreasing; any monotonically nonincreasing real sequenceconverges. [258,1.2] [42,1.1] A rank ρ matrix X solving the originalproblem (869) is found when the objective converges to 0 ; a certificate ofglobal optimality for the iteration. Positive semidefiniteness of matrix Xwith an upper bound ρ on rank is assured by the constraint on eigenvaluematrix Λ in convex problem (872); it means, the main diagonal of Λ mustbelong to the nonnegative orthant in a ρ-dimensional subspace of R n .The second problem (873) in the iteration is not convex. We propose