v2007.11.26 - Convex Optimization

7.1. FIRST PREVALENT PROBLEM: 469whereR ∆ = Q T U ∈ R N−1×N−1 (1169)in U on the set of orthogonal matrices is a linear bijection. We proposesolving (1168) by instead solving the problem sequence:minimize ‖Υ − R T ΛR‖ 2 FΥsubject to rank Υ ≤ ρΥ ≽ 0minimize ‖Υ ⋆ − R T ΛR‖ 2 FRsubject to R −1 = R T(a)(b)(1170)Problem (1170a) is equivalent to: (1) orthogonal projection of R T ΛRon an N − 1-dimensional subspace of isometrically isomorphic R N(N−1)/2containing δ(Υ)∈ R N−1+ , (2) nonincreasingly ordering the[result, (3) uniqueRρ]+minimum-distance projection of the ordered result on . (E.9.5)0Projection on that N−1-dimensional subspace amounts to zeroing R T ΛR atall entries off the main diagonal; thus, the equivalent sequence leading witha spectral projection:minimize ‖δ(Υ) − π ( δ(R T ΛR) ) ‖ 2Υ [ Rρ]+subject to δ(Υ) ∈0minimize ‖Υ ⋆ − R T ΛR‖ 2 FRsubject to R −1 = R T(a)(b)(1171)Because any permutation matrix is an orthogonal matrix, it is alwaysfeasible that δ(R T ΛR)∈ R N−1 be arranged in nonincreasing order; hence, thepermutation operator π . Unique minimum-distance projection of vectorπ ( δ(R T ΛR) ) [ Rρ]+on the ρ-dimensional subset of nonnegative orthant0