v2010.10.26 - Convex Optimization

2.3. HULLS 692.3.2.0.3 Exercise. Convex hull of outer product.Describe the interior of a Fantope.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} (97)Find the convex hull of nonsymmetric matrices bounded under some norm:{UV T | U ∈ R m×k , V ∈ R n×k , ‖UV T ‖ ≤ 1} (98)2.3.2.0.4 Example. Permutation polyhedron. [201] [317] [261]A permutation matrix Ξ is formed by interchanging rows and columns ofidentity matrix I . Since Ξ is square and Ξ T Ξ = I , the set of all permutationmatrices is a proper subset of the nonconvex manifold of orthogonal matrices(B.5). In fact, the only orthogonal matrices having all nonnegative entriesare permutations of the identity:Ξ −1 = Ξ T , Ξ ≥ 0 (99)And the only positive semidefinite permutation matrix is the identity.[333,6.5 prob.20]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}(100)where Π i is a linear operator here representing the i th permutation. Thispolyhedral hull, whose n! vertices are the permutation matrices, is alsoknown as the set of doubly stochastic matrices. The permutation matricesare the minimal cardinality (fewest nonzero entries) doubly stochasticmatrices. The only orthogonal matrices belonging to this polyhedron arethe permutation matrices.It is remarkable that n! permutation matrices can be described as theextreme points (2.6.0.0.1) of a bounded polyhedron, of affine dimension(n−1) 2 , that is itself described by 2n equalities (2n−1 linearly independentequality constraints in n 2 nonnegative variables). By Carathéodory’stheorem, conversely, any doubly stochastic matrix can be described as a