from sage.all import *
-def random_cone(min_dim=None, max_dim=None, min_rays=None, max_rays=None):
+def discrete_complementarity_set(K):
r"""
- Generate a random rational convex polyhedral cone.
+ Compute the discrete complementarity set of this cone.
- Lower and upper bounds may be provided for both the dimension of the
- ambient space and the number of generating rays of the cone. Any
- parameters left unspecified will be chosen randomly.
+ The complementarity set of this cone is the set of all orthogonal
+ pairs `(x,s)` such that `x` is in this cone, and `s` is in its
+ dual. The discrete complementarity set restricts `x` and `s` to be
+ generators of their respective cones.
- INPUT:
-
- - ``min_dim`` (default: random) -- The minimum dimension of the ambient
- lattice.
-
- - ``max_dim`` (default: random) -- The maximum dimension of the ambient
- lattice.
+ OUTPUT:
- - ``min_rays`` (default: random) -- The minimum number of generating rays
- of the cone.
+ A list of pairs `(x,s)` such that,
- - ``max_rays`` (default: random) -- The maximum number of generating rays
- of the cone.
+ * `x` is in this cone.
+ * `x` is a generator of this cone.
+ * `s` is in this cone's dual.
+ * `s` is a generator of this cone's dual.
+ * `x` and `s` are orthogonal.
- OUTPUT:
+ EXAMPLES:
- A new, randomly generated cone.
+ The discrete complementarity set of the nonnegative orthant consists
+ of pairs of standard basis vectors::
- TESTS:
+ sage: K = Cone([(1,0),(0,1)])
+ sage: discrete_complementarity_set(K)
+ [((1, 0), (0, 1)), ((0, 1), (1, 0))]
- It's hard to test the output of a random process, but we can at
- least make sure that we get a cone back::
+ 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: from sage.geometry.cone import is_Cone
- sage: K = random_cone()
- sage: is_Cone(K) # long time
- True
+ 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::
- def random_min_max(l,u):
- r"""
- We need to handle four cases to prevent us from doing
- something stupid like having an upper bound that's lower than
- our lower bound. And we would need to repeat all of that logic
- for the dimension/rays, so we consolidate it here.
- """
- if l is None and u is None:
- # They're both random, just return a random nonnegative
- # integer.
- return ZZ.random_element().abs()
+ sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
+ sage: discrete_complementarity_set(K)
+ []
- if l is not None and u is not None:
- # Both were specified. Again, just make up a number and
- # return it. If the user wants to give us u < l then he
- # can have an exception.
- return ZZ.random_element(l,u)
+ TESTS:
- if l is not None and u is None:
- # In this case, we're generating the upper bound randomly
- # GIVEN A LOWER BOUND. So we add a random nonnegative
- # integer to the given lower bound.
- u = l + ZZ.random_element().abs()
- return ZZ.random_element(l,u)
+ The complementarity set of the dual can be obtained by switching the
+ components of the complementarity set of the original cone::
- # Here we must be in the only remaining case, where we are
- # given an upper bound but no lower bound. We might as well
- # use zero.
- return ZZ.random_element(0,u)
+ sage: K1 = random_cone(max_dim=10, max_rays=10)
+ sage: K2 = K1.dual()
+ sage: expected = [(x,s) for (s,x) in discrete_complementarity_set(K2)]
+ sage: actual = discrete_complementarity_set(K1)
+ sage: actual == expected
+ True
- d = random_min_max(min_dim, max_dim)
- r = random_min_max(min_rays, max_rays)
+ """
+ V = K.lattice().vector_space()
- L = ToricLattice(d)
- rays = [L.random_element() for i in range(0,r)]
+ # Convert the rays to vectors so that we can compute inner
+ # products.
+ xs = [V(x) for x in K.rays()]
+ ss = [V(s) for s in K.dual().rays()]
- # We pass the lattice in case there are no rays.
- return Cone(rays, lattice=L)
+ return [(x,s) for x in xs for s in ss if x.inner_product(s) == 0]
def lyapunov_rank(K):
The Lyapunov rank should be additive on a product of cones::
- sage: K1 = random_cone(0,10,0,10)
- sage: K2 = random_cone(0,10,0,10)
+ sage: K1 = random_cone(max_dim=10, max_rays=10)
+ sage: K2 = random_cone(max_dim=10, max_rays=10)
sage: K = K1.cartesian_product(K2)
sage: lyapunov_rank(K) == lyapunov_rank(K1) + lyapunov_rank(K2)
True
The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
itself::
- sage: K = random_cone(0,10,0,10)
+ sage: K = random_cone(max_dim=10, max_rays=10)
sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
True
"""
V = K.lattice().vector_space()
- xs = [V(x) for x in K.rays()]
- ss = [V(s) for s in K.dual().rays()]
-
- # WARNING: This isn't really C(K), it only contains the pairs
- # (x,s) in C(K) where x,s are extreme in their respective cones.
- C_of_K = [(x,s) for x in xs for s in ss if x.inner_product(s) == 0]
+ C_of_K = discrete_complementarity_set(K)
- matrices = [x.column() * s.row() for (x,s) in C_of_K]
+ matrices = [x.tensor_product(s) for (x,s) in C_of_K]
# Sage doesn't think matrices are vectors, so we have to convert
# our matrices to vectors explicitly before we can figure out how
projects / sage.d.git / blobdiff