v2010.10.26 - Convex Optimization

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 359Because there are n direction matrices W i to find, it can be advantageousto invoke a known closed-form solution for each from page 658. What makesthis combinatorial problem more tractable are relatively small semidefiniteconstraints in (794). (confer (790)) When a permutation A of vector B exists,number of iterations can be as small as 1. But this combinatorial Procrustesproblem can be made even more challenging when vector A has repeatedentries.solution via cardinality constraintNow the idea is to force solution at a vertex of permutation polyhedron (100)by finding a solution of desired sparsity. Because permutation matrix X isn-sparse by assumption, this combinatorial Procrustes problem may insteadbe formulated as a compressed sensing problem with convex iteration oncardinality of vectorized X (4.5.1): given nonzero vectors A, BminimizeX∈R n×n ‖A − XB‖ 1 + w〈X , Y 〉subject to X T 1 = 1X1 = 1X ≥ 0where direction vector Y is an optimal solution tominimizeY ∈R n×n 〈X ⋆ , Y 〉subject to 0 ≤ Y ≤ 11 T Y 1 = n 2 − n(525)(797)each a linear program. In this circumstance, use of closed-form solution fordirection vector Y is discouraged. When vector A is a permutation of B ,both linear programs have objectives that converge to 0. When vectors A andB are permutations and no entries of A are repeated, optimal solution X ⋆can be found as soon as the first iteration.In any case, X ⋆ = Ξ is a permutation matrix.4.6.0.0.4 Exercise. Combinatorial Procrustes constraints.Assume that the objective of semidefinite program (794) is 0 at optimality.Prove that the constraints in program (794) are necessary and sufficientto produce a permutation matrix as optimal solution. Alternatively andequivalently, prove those constraints necessary and sufficient to optimallyproduce a nonnegative orthogonal matrix.