v2007.11.26 - Convex Optimization

490 CHAPTER 7. PROXIMITY PROBLEMSdimension not in excess of that ρ desired; id est, spectral projection on⎡ ⎤⎣ Rρ+1 +0 ⎦ ∩ ∂H ⊂ R N+1 (1234)R −where∂H = {λ ∈ R N+1 | 1 T λ = 0} (945)is a hyperplane through the origin. This pointed polyhedral cone (1234), towhich membership subsumes the rank constraint, has empty interior.Given desired affine dimension 0 ≤ ρ ≤ N −1 and diagonalization (A.5)of unknown EDM D[0 1T1 −D]∆= UΥU T ∈ S N+1h(1235)and given symmetric H in diagonalization[ ] 0 1T∆= QΛQ T ∈ S N+1 (1236)1 −Hhaving eigenvalues arranged in nonincreasing order, then by (958) problem(1232) is equivalent to∥minimize ∥δ(Υ) − π ( δ(R T ΛR) )∥ ∥ 2Υ , R⎡ ⎤subject to δ(Υ) ∈ ⎣ Rρ+1 +0 ⎦ ∩ ∂H(1237)R −δ(QRΥR T Q T ) = 0R −1 = R Twhere π is the permutation operator from7.1.3 arranging its vectorargument in nonincreasing order, 7.17 whereR ∆ = Q T U ∈ R N+1×N+1 (1238)in U on the set of orthogonal matrices is a bijection, and where ∂H insuresone negative eigenvalue. Hollowness constraint δ(QRΥR T Q T ) = 0 makesproblem (1237) difficult by making the two variables dependent.7.17 Recall, any permutation matrix is an orthogonal matrix.