v2009.01.01 - Convex Optimization

314 CHAPTER 4. SEMIDEFINITE PROGRAMMING
constraints in (711). (confer (707)) When a permutation A of vector B exists,
number of iterations can be as small as 1. But this combinatorial Procrustes
problem can be made even more challenging when vector A has repeated
entries.
solution via cardinality constraint
Now the idea is to force solution at a vertex of permutation polyhedron (91)
by finding a solution of desired sparsity. Because permutation matrix X is
n-sparse by assumption, this combinatorial Procrustes problem may instead
be formulated as a compressed sensing problem with convex iteration on
cardinality of vectorized X (4.5.1): given nonzero vectors A, B
minimize
X∈R n×n ‖A − XB‖ 1 + w〈X , Y 〉
subject to X T 1 = 1
X1 = 1
X ≥ 0
(714)
where direction vector Y is an optimal solution to
minimize
Y ∈R n×n 〈X ⋆ , Y 〉
subject to 0 ≤ Y ≤ 1
1 T Y 1 = n 2 − n
(467)
each a linear program. In this circumstance, use of closed-form solution for
direction vector Y is discouraged. When vector A is a permutation of B ,
both linear programs have objectives that converge to 0. When vectors A and
B 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 (711) is 0 at optimality.
Prove that the constraints in program (711) are necessary and sufficient
to produce a permutation matrix as optimal solution. Alternatively and
equivalently, prove those constraints necessary and sufficient to optimally
produce a nonnegative orthogonal matrix.