v2007.11.26 - Convex Optimization

5.13. RECONSTRUCTION EXAMPLES 391where m=N(N ∆ −1)/2, we may rewrite (982b) as an equivalent quadraticprogram; a convex optimization problem [46,4] in terms of thehalfspace-description of K M+ :minimize (σ − Πd) T (σ − Πd)σsubject to Y † σ ≽ 0(985)This quadratic program can be converted to a semidefinite program viaSchur-form (3.1.7.2); we get the equivalent problemminimizet∈R , σsubject tot[tI σ − Πd(σ − Πd) T 1]≽ 0(986)Y † σ ≽ 05.13.2.3 ConvergenceInE.10 we discuss convergence of alternating projection on intersectingconvex sets in a Euclidean vector space; convergence to a point in theirintersection. Here the situation is different for two reasons:Firstly, sets of positive semidefinite matrices having an upper bound onrank are generally not convex. Yet in7.1.4.0.1 we prove (982a) is equivalentto a projection of nonincreasingly ordered eigenvalues on a subset of thenonnegative orthant:minimize ‖−VN T(D − O)V N ‖ FDsubject to rankVN TDV N ≤ 3D ∈ EDM N≡minimize ‖Υ − Λ‖ FΥ [ ]R3+ (987)subject to δ(Υ) ∈0∆where −VN TDV N =UΥU T ∈ S N−1 and −VN TOV N =QΛQ T ∈ S N−1 areordered diagonalizations (A.5). It so happens: optimal orthogonal U ⋆always equals Q given. Linear operator T(A) = U ⋆T AU ⋆ , acting on squarematrix A , is a bijective isometry because the Frobenius norm is orthogonallyinvariant (40). This isometric isomorphism T thus maps a nonconvexproblem to a convex one that preserves distance.∆