v2010.10.26 - Convex Optimization

5.13. RECONSTRUCTION EXAMPLES 485This quadratic program can be converted to a semidefinite program viaSchur-form (3.5.2); we get the equivalent problemminimizet∈R , σsubject tot[tI σ − Πd(σ − Πd) T 1]≽ 0(1153)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 (1149a) 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+subject to δ(Υ) ∈0(1154)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 an isometry because Frobenius’ norm is orthogonally invariant(48). This isometric isomorphism T thus maps a nonconvex problem to aconvex one that preserves distance.Secondly, the second half (1149b) of the alternation takes place in adifferent vector space; S N h (versus S N−1 ). From5.6 we know these twovector spaces are related by an isomorphism, S N−1 =V N (S N h ) (1018), butnot by an isometry.We have, therefore, no guarantee from theory of alternating projectionthat the alternation (1149) converges to a point, in the set of allEDMs corresponding to affine dimension not in excess of 3, belonging todvec EDM N ∩ Π T K M+ .