v2007.09.13 - Convex Optimization

C.4. TWO-SIDED ORTHOGONAL PROCRUSTES 545C.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 =⎢ ·⎥(1502)⎣ 1 ⎦1 0then an optimal solution R ⋆ to the maximization problem [sic]minimizes tr(A T R T BR) : [148] [174,2.1]maximize ‖A − R T BR‖ FR(1503)subject to R T = R −1R ⋆ = Q B ΞQ T A ∈ R N×N (1504)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(1505)while the corresponding trace minimization has optimal valueC.4.1inf tr(A T R T BR) = tr(A T R ⋆T BR ⋆ ) = tr(Λ A ΞΛ B Ξ) (1506)R T =R −1Procrustes’ relation to linear programmingAlthough these two-sided Procrustes problems are nonconvex, a connectionwith linear programming [64] was discovered by Anstreicher & Wolkowicz[10,3] [174,2.1]: Given A,B∈ S N , this semidefinite program in S and Tminimize tr(A T R T BR) = maximize tr(S + T ) (1507)RS , T ∈S Nsubject to R T = R −1 subject to A T ⊗ B − I ⊗ S − T ⊗ I ≽ 0