v2010.10.26 - Convex Optimization

358 CHAPTER 4. SEMIDEFINITE PROGRAMMINGcolumn-sum of X must also be unity. It is this combination of nonnegativity,sum, and sum square constraints that extracts the permutation matrices:(Figure 105) given nonzero vectors A, BminimizeX∈R n×n , G i ∈ S n+1subject to∑‖A − XB‖ 1 + w n 〈G i , W i 〉[ i=1 ] ⎫Gi (1:n, 1:n) X(:, i) ⎬G i =X(:, i) T ≽ 01⎭ ,trG i = 2i=1... n(794)X T 1 = 1X1 = 1X ≥ 0where w ≈10 positively weights the rank regularization term. Optimalsolutions G ⋆ i are key to finding direction matrices W i for the next iterationof semidefinite programs (794) (795):minimizeW i ∈ S n+1 〈G ⋆ i , W i 〉subject to 0 ≼ W i ≼ ItrW i = n⎫⎪⎬, i=1... n (795)⎪⎭Direction matrices thus found lead toward rank-1 matrices G ⋆ i on subsequentiterations. Constraint on trace of G ⋆ i normalizes the i th column of X ⋆ to unitybecause (confer p.419)[ ] XG ⋆ i =⋆ (:, i) [ X ⋆ (:, i) T 1]1(796)at convergence. Binary-valued X ⋆ column entries result from the furthersum constraint X1=1. Columnar orthogonality is a consequence of thefurther transpose-sum constraint X T 1=1 in conjunction with nonnegativityconstraint X ≥ 0 ; but we leave proof of orthogonality an exercise. Theoptimal objective value is 0 for both semidefinite programs when vectors Aand B are related by permutation. In any case, optimal solution X ⋆ becomesa permutation matrix Ξ .