v2009.01.01 - Convex Optimization

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 311
and
minimize
W ∈ S 2n+1 〈G ⋆ , W 〉
subject to 0 ≼ W ≼ I
trW = 2n
(708)
which has an optimal solution W that is known in closed form (p.599). These
two problems are iterated until convergence and a rank-1 G matrix is found.
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 seems
always to be found in one or few iterations. This problem formulation is
extensible, 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 of
vector B , solution to orthogonal Procrustes problem
minimize ‖A − XB‖ F
X∈R n×n
(1593)
subject to X T = X −1
is not necessarily a permutation matrix Ξ even though an optimal objective
value of 0 is found by the known analytical solution (C.3). The simplest
method of solution finds permutation matrix X ⋆ = Ξ simply by sorting
vector B with respect to A .
Instead of sorting, we design two different convex problems each of whose
optimal solution is a permutation matrix: one design is based on rank
constraint, the other on cardinality. Because permutation matrices are sparse
by definition, we depart from a traditional Procrustes problem by instead
demanding a vector 1-norm which is known to produce solutions more sparse
than 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 (91) of doubly stochastic matrices.
That means: