v2010.10.26 - Convex Optimization

7.3. THIRD PREVALENT PROBLEM: 585Given desired affine dimension 0 ≤ ρ ≤ N −1 and diagonalization (A.5)of unknown EDM D [ ] 0 1T UΥU T ∈ S N+11 −Dh(1404)and given symmetric H in diagonalization[ ] 0 1T QΛQ T ∈ S N+1 (1405)1 −Hhaving eigenvalues arranged in nonincreasing order, then by (1125) problem(1401) is equivalent to∥minimize ∥δ(Υ) − π ( δ(R T ΛR) )∥ ∥ 2Υ , R⎡ ⎤subject to δ(Υ) ∈ ⎣ Rρ+1 +0 ⎦ ∩ ∂H(1406)R −δ(QRΥR T Q T ) = 0R −1 = R Twhere π is the permutation operator from7.1.3 arranging its vectorargument in nonincreasing order, 7.20 whereR Q T U ∈ R N+1×N+1 (1407)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 (1406) difficult by making the two variables dependent.Our plan is to instead divide problem (1406) into two and then iteratetheir solution:minimizeΥsubject to δ(Υ) ∈(∥ δ(Υ) − π δ(R T ΛR) )∥ ∥ 2⎡ ⎤⎣ Rρ+1 +0 ⎦ ∩ ∂HR −(a)(1408)minimize ‖R Υ ⋆ R T − Λ‖ 2 FRsubject to δ(QR Υ ⋆ R T Q T ) = 0R −1 = R T(b)7.20 Recall, any permutation matrix is an orthogonal matrix.