v2009.01.01 - Convex Optimization

606 APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTS
(where ⊗ signifies Kronecker product (D.1.2.1)) has optimal objective
value (1612). These two problems are strong duals (2.13.1.0.3). Given
ordered diagonalizations (1603), make the observation:
inf
R tr(AT R T BR) = inf tr(Λ ˆRT A Λ B ˆR) (1614)
ˆR
because ˆR =Q ∆ B TRQ A on the set of orthogonal matrices (which includes the
permutation matrices) is a bijection. This means, basically, diagonal matrices
of eigenvalues Λ A and Λ B may be substituted for A and B , so only the main
diagonals of S and T come into play;
maximize
S,T ∈S N 1 T δ(S + T )
subject to δ(Λ A ⊗ (ΞΛ B Ξ) − I ⊗ S − T ⊗ I) ≽ 0
(1615)
a linear program in δ(S) and δ(T) having the same optimal objective value
as the semidefinite program (1613).
We relate their results to Procrustes problem (1605) by manipulating
signs (1560) and permuting eigenvalues:
maximize tr(A T R T BR) = minimize 1 T δ(S + T )
R
S , T ∈S N
subject to R T = R −1 subject to δ(I ⊗ S + T ⊗ I − Λ A ⊗ Λ B ) ≽ 0
= minimize tr(S + T )
(1616)
S , T ∈S N
subject to I ⊗ S + T ⊗ I − A T ⊗ B ≽ 0
This formulation has optimal objective value identical to that in (1607).
C.4.2
Two-sided orthogonal Procrustes via SVD
By making left- and right-side orthogonal matrices independent, we can push
the upper bound on trace (1607) a little further: Given real matrices A,B
each having full singular value decomposition (A.6.3)
A ∆ = U A Σ A Q T A ∈ R m×n , B ∆ = U B Σ B Q T B ∈ R m×n (1617)
then a well-known optimal solution R ⋆ , S ⋆ to the problem
minimize ‖A − SBR‖ F
R , S
subject to R H = R −1
(1618)
S H = S −1