- EXAMPLES:
-
- The discrete complementarity set of the nonnegative orthant consists
- of pairs of standard basis vectors::
-
- sage: K = Cone([(1,0),(0,1)])
- sage: discrete_complementarity_set(K)
- [((1, 0), (0, 1)), ((0, 1), (1, 0))]
-
- If the cone consists of a single ray, the second components of the
- discrete complementarity set should generate the orthogonal
- complement of that ray::
-
- sage: K = Cone([(1,0)])
- sage: discrete_complementarity_set(K)
- [((1, 0), (0, 1)), ((1, 0), (0, -1))]
- sage: K = Cone([(1,0,0)])
- sage: discrete_complementarity_set(K)
- [((1, 0, 0), (0, 1, 0)),
- ((1, 0, 0), (0, -1, 0)),
- ((1, 0, 0), (0, 0, 1)),
- ((1, 0, 0), (0, 0, -1))]
-
- When the cone is the entire space, its dual is the trivial cone, so
- the discrete complementarity set is empty::
-
- sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
- sage: discrete_complementarity_set(K)
- []
-
- Likewise when this cone is trivial (its dual is the entire space)::
-
- sage: L = ToricLattice(0)
- sage: K = Cone([], ToricLattice(0))
- sage: discrete_complementarity_set(K)
- []
-
- TESTS:
-
- The complementarity set of the dual can be obtained by switching the
- components of the complementarity set of the original cone::
-
- sage: set_random_seed()
- sage: K1 = random_cone(max_ambient_dim=6)
- sage: K2 = K1.dual()
- sage: expected = [(x,s) for (s,x) in discrete_complementarity_set(K2)]
- sage: actual = discrete_complementarity_set(K1)
- sage: sorted(actual) == sorted(expected)
+ sage: n = ZZ.random_element().abs()
+ sage: K = Cone([[0] * n], ToricLattice(n))
+ sage: K.is_trivial()
+ True
+ sage: L = ToricLattice(n^2)
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: pi_cone = Cone([p.list() for p in pi_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = pi_cone.dim()
+ sage: actual == n^2
+ True
+ sage: K = K.dual()
+ sage: K.is_full_space()
+ True
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: pi_cone = Cone([p.list() for p in pi_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = pi_cone.dim()
+ sage: actual == n^2
+ True
+ sage: K = Cone([(1,0),(0,1),(0,-1)])
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: actual = Cone([p.list() for p in pi_of_K], check=False).dim()
+ sage: actual == 3