v2006.03.09 - Convex Optimization

C.5. TWO-SIDED ORTHOGONAL PROCRUSTES 491(where ⊗ signifies Kronecker product (D.1.2.1)) has optimal objective value(1317). These two problems are strong duals (2.13.1.0.1). Given ordereddiagonalizations (1308), make the observation:infR tr(AT R T BR) = inf tr(Λ ˆRT A Λ B ˆR) (1319)ˆRbecause ˆR =Q ∆ B TRQ A on the set of orthogonal matrices (which includes thepermutation matrices) is a bijection. This means, basically, diagonal matricesof eigenvalues Λ A and Λ B may be substituted for A and B , so only the maindiagonals of S and T come into play;maximizeS,T ∈S N 1 T δ(S + T )subject to δ(Λ A ⊗ (ΞΛ B Ξ) − I ⊗ S − T ⊗ I) ≽ 0(1320)a linear program in δ(S) and δ(T) having the same optimal objective valueas the semidefinite program.We relate their results to Procrustes problem (1310) by manipulatingsigns (1260) and permuting eigenvalues:maximize tr(A T R T BR) = minimize 1 T δ(S + T )RS , T ∈S Nsubject to R T = R −1 subject to δ(I ⊗ S + T ⊗ I − Λ A ⊗ Λ B ) ≽ 0= minimize tr(S + T )(1321)S , T ∈S Nsubject to I ⊗ S + T ⊗ I − A T ⊗ B ≽ 0This formulation has optimal objective value identical to that in (1312).C.5.2Two-sided orthogonal Procrustes via SVDBy making left- and right-side orthogonal matrices independent, we can pushthe upper bound on trace (1312) a little further: Given real matrices A,Beach 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 (1322)then a well-known optimal solution R ⋆ , S ⋆ to the problemminimize ‖A − SBR‖ FR , Ssubject to R H = R −1(1323)S H = S −1