v2010.10.26 - Convex Optimization

356 CHAPTER 4. SEMIDEFINITE PROGRAMMINGandminimizeW ∈ S 2n+1 〈G ⋆ , W 〉subject to 0 ≼ W ≼ ItrW = 2n(791)which has an optimal solution W that is known in closed form (p.658).These two problems are iterated until convergence and a rank-1 G matrix isfound. A good initial value for direction matrix W is 0. Optimal Q ⋆ equals[x ⋆ y ⋆ ].Numerically, this Procrustes problem is easy to solve; a solution seemsalways to be found in one or few iterations. This problem formulation isextensible, of course, to orthogonal (square) matrices Q . 4.6.0.0.3 Example. Combinatorial Procrustes problem.In case A,B∈ R n , when vector A = ΞB is known to be a permutation ofvector B , solution to orthogonal Procrustes problemminimize ‖A − XB‖ FX∈R n×n(1712)subject to X T = X −1is not necessarily a permutation matrix Ξ even though an optimal objectivevalue of 0 is found by the known analytical solution (C.3). The simplestmethod of solution finds permutation matrix X ⋆ = Ξ simply by sortingvector B with respect to A .Instead of sorting, we design two different convex problems each of whoseoptimal solution is a permutation matrix: one design is based on rankconstraint, the other on cardinality. Because permutation matrices are sparseby definition, we depart from a traditional Procrustes problem by insteaddemanding a vector 1-norm which is known to produce solutions more sparsethan Frobenius’ norm.There are two principal facts exploited by the first convex iteration design(4.4.1) we propose. Permutation matrices Ξ constitute:1) the set of all nonnegative orthogonal matrices,2) all points extreme to the polyhedron (100) of doubly stochasticmatrices.That means: