v2009.01.01 - Convex Optimization

2.3. HULLS 63
2.3.2.0.3 Exercise. Convex hull of factorized matrices.
Find the convex hull of nonorthogonal projection matrices (E.1.1):
{UV T | U ∈ R N×k , V ∈ R N×k , V T U = I} (89)
Find the convex hull of nonsymmetric matrices bounded under some norm:
{UV T | U ∈ R m×k , V ∈ R n×k , ‖UV T ‖ ≤ 1} (90)
2.3.2.0.4 Example. Convex hull of permutation matrices. [175] [275] [225]
A permutation matrix Ξ is formed by interchanging rows and columns of
the identity I . Since Ξ is square and Ξ T Ξ = I , the set of all permutation
matrices is a proper subset of the nonconvex manifold of orthogonal matrices
(B.5). In fact, the only orthogonal matrices having all nonnegative entries
are permutations of the identity.
Regarding the permutation matrices as a set of points in Euclidean space,
its convex hull is a bounded polyhedron (2.12) described (Birkhoff, 1946)
conv{Ξ = Π i (I ∈ S n )∈ R n×n , i=1... n!} = {X ∈ R n×n | X T 1=1, X1=1, X ≥ 0}
(91)
where Π i is a linear operator representing the i th permutation. This
polyhedral hull, whose n! vertices are the permutation matrices, is known as
the set of doubly stochastic matrices. The only orthogonal matrices belonging
to this polyhedron are the permutation matrices. It is remarkable that
n! permutation matrices can be described as the extreme points of a bounded
polyhedron, of affine dimension (n−1) 2 , that is itself described by only
2n equalities (2n−1 linearly independent equality constraints in nonnegative
variables). By Carathéodory’s theorem, conversely, any doubly stochastic
matrix can be described as a convex combination of at most (n−1) 2 +1
permutation matrices. [176,8.7] This polyhedron, then, can be a device for
relaxing an integer, combinatorial, or Boolean optimization problem. 2.16 [58]
[244,3.1]
2.16 Dantzig first showed in 1951 that, by this device, the so-called assignment problem
can be formulated as a linear program. [274]