v2010.10.26 - Convex Optimization

664 APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTSto the two-sided orthogonal Procrustes problemminimize ‖A − R T BR‖ FR= minimize tr ( A T A − 2A T R T BR + B T B )subject to R T = R −1 Rsubject to R T = R −1 (1725)maximizes tr(A T R T BR) over the nonconvex manifold of orthogonal matrices.Optimal product R ⋆T BR ⋆ has the eigenvectors of A but the eigenvalues of B .[164,7.5.1] The optimal value for the objective of minimization is, by (48)‖Q A Λ A Q T A −R ⋆T Q B Λ B Q T BR ⋆ ‖ F = ‖Q A (Λ A −Λ B )Q T A ‖ F = ‖Λ A −Λ B ‖ F (1726)while the corresponding trace maximization has optimal valuesup tr(A T R T BR) = tr(A T R ⋆T BR ⋆ ) = tr(Λ A Λ B ) ≥ tr(A T B) (1727)R T =R −1The lower bound on inner product of eigenvalues is due to Fan (p.603).C.4.0.2MaximizationAny permutation matrix is an orthogonal matrix. Defining a row- andcolumn-swapping permutation matrix (a reflection matrix,B.5.2)⎡ ⎤0 1·Ξ = Ξ T =⎢ ·⎥(1728)⎣ 1 ⎦1 0then an optimal solution R ⋆ to the maximization problem [sic]minimizes tr(A T R T BR) : [200] [239,2.1] [240]maximize ‖A − R T BR‖ FR(1729)subject to R T = R −1R ⋆ = Q B ΞQ T A ∈ R N×N (1730)The optimal value for the objective of maximization is‖Q A Λ A Q T A − R⋆T Q B Λ B Q T B R⋆ ‖ F = ‖Q A Λ A Q T A − Q A ΞT Λ B ΞQ T A ‖ F= ‖Λ A − ΞΛ B Ξ‖ F(1731)while the corresponding trace minimization has optimal valueinf tr(A T R T BR) = tr(A T R ⋆T BR ⋆ ) = tr(Λ A ΞΛ B Ξ) (1732)R T =R −1