From 3d6ec3f4f138a6278be4f393b491aa1ac2bbffa8 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Wed, 19 Dec 2018 20:13:34 -0500 Subject: [PATCH] mjo/cone/completely_positive.py: add completely_positive_operators_gens(). Add a new function to return (as matrices) the generators of the completely-positive cone of K, where K is some other given cone. --- mjo/cone/completely_positive.py | 54 +++++++++++++++++++++++++++++++++ 1 file changed, 54 insertions(+) diff --git a/mjo/cone/completely_positive.py b/mjo/cone/completely_positive.py index 8638a63..15be558 100644 --- a/mjo/cone/completely_positive.py +++ b/mjo/cone/completely_positive.py @@ -195,3 +195,57 @@ def is_extreme_completely_positive(A): # factorization into `$XX^{T}$` may not be unique! raise ValueError('Unable to determine extremity of ``A``.') + + +def completely_positive_operators_gens(K): + r""" + Return a list of generators (matrices) for the completely-positive + cone of ``K``. + + INPUT: + + - ``K`` -- a closed convex rational polyhedral cone. + + OUTPUT: + + A list of matrices, the conic hull of which is the + completely-positive cone of ``K``. + + SETUP:: + + sage: from mjo.cone.completely_positive import ( + ....: completely_positive_operators_gens, + ....: is_completely_positive ) + sage: from mjo.cone.nonnegative_orthant import nonnegative_orthant + sage: from mjo.matrix_vector import isomorphism + + EXAMPLES:: + + sage: K = nonnegative_orthant(2) + sage: completely_positive_operators_gens(K) + [ + [1 0] [0 0] + [0 0], [0 1] + ] + sage: all( is_completely_positive(M) + ....: for M in completely_positive_operators_gens(K) ) + True + + TESTS: + + The completely-positive cone of ``K`` is subdual:: + + sage: K = random_cone(max_ambient_dim=8, max_rays=10) + sage: cp_gens = completely_positive_operators_gens(K) + sage: n = K.lattice_dim() + sage: M = MatrixSpace(QQ, n, n) + sage: (p, p_inv) = isomorphism(M) + sage: L = ToricLattice(n**2) + sage: cp_cone = Cone( (p(m) for m in cp_gens), lattice=L ) + sage: copos_cone = Cone(cp_cone.dual().rays(), lattice=L ) + sage: all( x in copos_cone for x in cp_cone ) + True + + """ + return [ x.tensor_product(x) for x in K ] + -- 2.44.2