v2010.10.26 - Convex Optimization

584 CHAPTER 7. PROXIMITY PROBLEMSprojection on a positive semidefinite cone (1336); zeroing those eigenvalueshere in Problem 3 would place an elbow in the projection path (Figure 153)thereby producing a result that is necessarily suboptimal. Problem 3 isinstead a projection on the EDM cone whose associated spectral cone isconsiderably different. (5.11.2.3) Proper choice of spectral cone is demandedby diagonalization of that variable argument to the objective:7.3.3.1 Cayley-Menger formWe use Cayley-Menger composition of the Euclidean distance matrix to solvea problem that is the same as Problem 3 (1383): (5.7.3.0.1)[ ] [ ]∥ minimize0 1T 0 1T ∥∥∥2D∥ −1 −D 1 −H[ ]F0 1T(1401)subject to rank≤ ρ + 21 −DD ∈ EDM Na projection of H on a generally nonconvex subset (when ρ < N −1) of theEuclidean distance matrix cone boundary rel∂EDM N ; id est, projectionfrom the EDM cone interior or exterior on a subset of its relative boundary(6.5, (1179)).Rank of an optimal solution is intrinsically bounded above and below;[ ] 0 1T2 ≤ rank1 −D ⋆ ≤ ρ + 2 ≤ N + 1 (1402)Our proposed strategy ([ for low-rank ]) solution is projection on that subset0 1Tof a spectral cone λ1 −EDM N (5.11.2.3) corresponding to affinedimension not in excess of that ρ desired; id est, spectral projection on⎡ ⎤⎣ Rρ+1 +0 ⎦ ∩ ∂H ⊂ R N+1 (1403)R −where∂H = {λ ∈ R N+1 | 1 T λ = 0} (1112)is a hyperplane through the origin. This pointed polyhedral cone (1403), towhich membership subsumes the rank constraint, is not full-dimensional.