v2007.09.13 - Convex Optimization

452 CHAPTER 7. PROXIMITY PROBLEMS7.1.3.1 Orthant is best spectral cone for Problem 1This means unique minimum-distance projection of γ on the nearestspectral member of the rank ρ subset is tantamount to presorting γ intononincreasing order. Only then does unique spectral projection on a subsetK ρ M+of the monotone nonnegative cone become equivalent to unique spectralprojection on a subset R ρ + of the nonnegative orthant (which is simpler);in other words, unique minimum-distance projection of sorted γ on thenonnegative orthant in a ρ-dimensional subspace of R N is indistinguishablefrom its projection on the subset K ρ M+of the monotone nonnegative cone inthat same subspace.7.1.4 Closest-EDM Problem 1, “nonconvex” caseTrosset’s proof of solution (1118), for projection on a rank ρ subset of thepositive semidefinite cone S N−1+ , was algebraic in nature. [262,2] Here wederive that known result but instead using a more geometric argument viaspectral projection on a polyhedral cone (subsuming the proof in7.1.1).In so doing, we demonstrate how nonconvex Problem 1 is transformed to aconvex optimization:7.1.4.0.1 Proof. Solution (1118), nonconvex case.As explained in7.1.2, we may instead work with the more facile genericproblem (1125). With diagonalization of unknownB ∆ = UΥU T ∈ S N−1 (1136)given desired affine dimension 0 ≤ ρ ≤ N −1 and diagonalizableA ∆ = QΛQ T ∈ S N−1 (1137)having eigenvalues in Λ arranged in nonincreasing order, by (40) the genericproblem is equivalent tominimize ‖B − A‖ 2B∈S N−1Fsubject to rankB ≤ ρB ≽ 0≡minimize ‖Υ − R T ΛR‖ 2 FR , Υsubject to rank Υ ≤ ρ (1138)Υ ≽ 0R −1 = R T