v2009.01.01 - Convex Optimization

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 313
column-sum of X must also be unity. It is this combination of nonnegativity,
sum, and sum square constraints that extracts the permutation matrices:
(Figure 84) given nonzero vectors A, B
minimize
X∈R n×n , G i ∈ S n+1
subject to
∑
‖A − XB‖ 1 + w n 〈G i , W i 〉
i=1
[ ] ⎫
Gi (1:n, 1:n) X(:, i) ⎬
G i =
X(:, i) T ≽ 0
1
⎭ ,
trG i = 2
i=1... n
(711)
X T 1 = 1
X1 = 1
X ≥ 0
where w ≈10 positively weights the rank regularization term. Optimal
solutions G ⋆ i are key to finding direction matrices W i for the next iteration
of semidefinite programs (711) (712):
⎫
minimize 〈G ⋆
W i ∈ S n+1 i , W i 〉 ⎪⎬
subject to 0 ≼ W i ≼ I , i=1... n (712)
trW i = n
⎪⎭
Direction matrices thus found lead toward rank-1 matrices G ⋆ i on subsequent
iterations. Constraint on trace of G ⋆ i normalizes the i th column of X ⋆ to unity
because (confer p.367)
[ ] X
G ⋆ i =
⋆ (:, i) [X ⋆ (:, i) T 1]
1
(713)
at convergence. Binary-valued X ⋆ column entries result from the further
sum constraint X1=1. Columnar orthogonality is a consequence of the
further transpose-sum constraint X T 1=1 in conjunction with nonnegativity
constraint X ≥ 0 ; but we leave proof of orthogonality an exercise. The
optimal objective value is 0 for both semidefinite programs when vectors A
and B are related by permutation. In any case, optimal solution X ⋆ becomes
a permutation matrix Ξ .
Because there are n direction matrices W i to find, it can be advantageous
to invoke a known closed-form solution for each from page 599. What makes
this combinatorial problem more tractable are relatively small semidefinite