v2009.01.01 - Convex Optimization

C.4. TWO-SIDED ORTHOGONAL PROCRUSTES 605
C.4.0.2
Maximization
Any permutation matrix is an orthogonal matrix. Defining a row- and
column-swapping permutation matrix (a reflection matrix,B.5.2)
⎡ ⎤
0 1
·
Ξ = Ξ T =
⎢ ·
⎥
(1608)
⎣ 1 ⎦
1 0
then an optimal solution R ⋆ to the maximization problem [sic]
minimizes tr(A T R T BR) : [174] [206,2.1] [207]
maximize ‖A − R T BR‖ F
R
(1609)
subject to R T = R −1
R ⋆ = Q B ΞQ T A ∈ R N×N (1610)
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
(1611)
while the corresponding trace minimization has optimal value
C.4.1
inf tr(A T R T BR) = tr(A T R ⋆T BR ⋆ ) = tr(Λ A ΞΛ B Ξ) (1612)
R T =R −1
Procrustes’ relation to linear programming
Although these two-sided Procrustes problems are nonconvex, a connection
with linear programming [82] was discovered by Anstreicher & Wolkowicz
[11,3] [206,2.1] [207]: Given A,B∈ S N , this semidefinite program in S
and T
minimize tr(A T R T BR) = maximize tr(S + T ) (1613)
R
S , T ∈S N
subject to R T = R −1 subject to A T ⊗ B − I ⊗ S − T ⊗ I ≽ 0