X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=e9d0f1e643e3a0d40d8a754faf551ac317e36f37;hb=d27a3f72ac6b898534615de9111ace4082a4c55e;hp=6ade5e628f1035c99294048c7fb55b4b9c1204d9;hpb=7d2f3fba7f494158dbce5f7a3eca1d15ee7f577e;p=sage.d.git

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index 6ade5e6..e9d0f1e 100644
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -8,124 +8,231 @@ addsitedir(abspath('../../'))
 from sage.all import *
 
 
-def random_cone(min_dim=0, max_dim=None, min_rays=0, max_rays=None):
+def basically_the_same(K1,K2):
     r"""
-    Generate a random rational convex polyhedral cone.
+    ``True`` if ``K1`` and ``K2`` are basically the same, and ``False``
+    otherwise. This is intended as a lazy way to check whether or not
+    ``K1`` and ``K2`` are linearly isomorphic (i.e. ``A(K1) == K2`` for
+    some invertible linear transformation ``A``).
+    """
+    if K1.lattice_dim() != K2.lattice_dim():
+        return False
 
-    Lower and upper bounds may be provided for both the dimension of the
-    ambient space and the number of generating rays of the cone. If a
-    lower bound is left unspecified, it defaults to zero. Unspecified
-    upper bounds will be chosen randomly.
+    if K1.nrays() != K2.nrays():
+        return False
 
-    INPUT:
+    if K1.dim() != K2.dim():
+        return False
 
-    - ``min_dim`` (default: zero) -- A nonnegative integer representing the
-                                     minimum dimension of the ambient lattice.
+    if K1.lineality() != K2.lineality():
+        return False
 
-    - ``max_dim`` (default: random) -- A nonnegative integer representing
-                                       the maximum dimension of the ambient
-                                       lattice.
+    if K1.is_solid() != K2.is_solid():
+        return False
 
-    - ``min_rays`` (default: zero) -- A nonnegative integer representing the
-                                      minimum number of generating rays of the
-                                      cone.
+    if K1.is_strictly_convex() != K2.is_strictly_convex():
+        return False
 
-    - ``max_rays`` (default: random) -- A nonnegative integer representing the
-                                        maximum number of generating rays of the
-                                        cone.
+    if len(LL(K1)) != len(LL(K2)):
+        return False
 
-    OUTPUT:
+    C_of_K1 = discrete_complementarity_set(K1)
+    C_of_K2 = discrete_complementarity_set(K2)
+    if len(C_of_K1) != len(C_of_K2):
+        return False
 
-    A new, randomly generated cone.
+    if len(K1.facets()) != len(K2.facets()):
+        return False
+
+    return True
 
-    EXAMPLES:
 
-    If we set the lower/upper bounds to zero, then our result is
-    predictable::
 
-        sage: random_cone(0,0,0,0)
-        0-d cone in 0-d lattice N
+def rho(K, K2=None):
+    r"""
+    Restrict ``K`` into its own span, or the span of another cone.
+
+    INPUT:
 
-    In fact, as long as we ask for zero rays, we should be able to predict
-    the output when ``min_dim == max_dim``::
+    - ``K2`` -- another cone whose lattice has the same rank as this
+                cone.
+
+    OUTPUT:
 
-        sage: random_cone(min_dim=4, max_dim=4, min_rays=0, max_rays=0)
-        0-d cone in 4-d lattice N
+    A new cone in a sublattice.
+
+    EXAMPLES::
+
+        sage: K = Cone([(1,)])
+        sage: rho(K) == K
+        True
+
+        sage: K2 = Cone([(1,0)])
+        sage: rho(K2).rays()
+        N(1)
+        in 1-d lattice N
+        sage: K3 = Cone([(1,0,0)])
+        sage: rho(K3).rays()
+        N(1)
+        in 1-d lattice N
+        sage: rho(K2) == rho(K3)
+        True
 
     TESTS:
 
-    It's hard to test the output of a random process, but we can at
-    least make sure that we get a cone back::
+    The projected cone should always be solid::
 
-        sage: from sage.geometry.cone import is_Cone # long time
-        sage: K = random_cone() # long time
-        sage: is_Cone(K)        # long time
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8)
+        sage: K_S = rho(K)
+        sage: K_S.is_solid()
         True
 
-    Ensure that an exception is raised when either lower bound is greater
-    than its respective upper bound::
+    And the resulting cone should live in a space having the same
+    dimension as the space we restricted it to::
 
-        sage: random_cone(min_dim=5, max_dim=2)
-        Traceback (most recent call last):
-        ...
-        ValueError: max_dim must be greater than or equal to min_dim.
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8)
+        sage: K_S = rho(K, K.dual() )
+        sage: K_S.lattice_dim() == K.dual().dim()
+        True
 
-        sage: random_cone(min_rays=5, max_rays=2)
-        Traceback (most recent call last):
-        ...
-        ValueError: max_rays must be greater than or equal to min_rays.
+    This function should not affect the dimension of a cone::
 
-    """
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8)
+        sage: K.dim() == rho(K).dim()
+        True
+
+    Nor should it affect the lineality of a cone::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8)
+        sage: K.lineality() == rho(K).lineality()
+        True
+
+    No matter which space we restrict to, the lineality should not
+    increase::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8)
+        sage: K.lineality() >= rho(K).lineality()
+        True
+        sage: K.lineality() >= rho(K, K.dual()).lineality()
+        True
+
+    If we do this according to our paper, then the result is proper::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=False)
+        sage: K_S = rho(K)
+        sage: K_SP = rho(K_S.dual()).dual()
+        sage: K_SP.is_proper()
+        True
+        sage: K_SP = rho(K_S, K_S.dual())
+        sage: K_SP.is_proper()
+        True
+
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=False)
+        sage: K_S = rho(K)
+        sage: K_SP = rho(K_S.dual()).dual()
+        sage: K_SP.is_proper()
+        True
+        sage: K_SP = rho(K_S, K_S.dual())
+        sage: K_SP.is_proper()
+        True
+
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=True)
+        sage: K_S = rho(K)
+        sage: K_SP = rho(K_S.dual()).dual()
+        sage: K_SP.is_proper()
+        True
+        sage: K_SP = rho(K_S, K_S.dual())
+        sage: K_SP.is_proper()
+        True
+
+    ::
 
-    # Catch obvious mistakes so that we can generate clear error
-    # messages.
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=True)
+        sage: K_S = rho(K)
+        sage: K_SP = rho(K_S.dual()).dual()
+        sage: K_SP.is_proper()
+        True
+        sage: K_SP = rho(K_S, K_S.dual())
+        sage: K_SP.is_proper()
+        True
+
+    Test the proposition in our paper concerning the duals and
+    restrictions. Generate a random cone, then create a subcone of
+    it. The operation of dual-taking should then commute with rho::
+
+        sage: set_random_seed()
+        sage: J = random_cone(max_dim = 8, solid=False, strictly_convex=False)
+        sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
+        sage: K_W = rho(K, J)
+        sage: K_star_W_star = rho(K.dual(), J).dual()
+        sage: basically_the_same(K_W, K_star_W_star)
+        True
 
-    if min_dim < 0:
-        raise ValueError('min_dim must be nonnegative.')
+    ::
 
-    if min_rays < 0:
-        raise ValueError('min_rays must be nonnegative.')
+        sage: set_random_seed()
+        sage: J = random_cone(max_dim = 8, solid=True, strictly_convex=False)
+        sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
+        sage: K_W = rho(K, J)
+        sage: K_star_W_star = rho(K.dual(), J).dual()
+        sage: basically_the_same(K_W, K_star_W_star)
+        True
 
-    if max_dim is not None:
-        if max_dim < 0:
-            raise ValueError('max_dim must be nonnegative.')
-        if (min_dim > max_dim):
-            raise ValueError('max_dim must be greater than or equal to min_dim.')
+    ::
 
-    if max_rays is not None:
-        if max_rays < 0:
-            raise ValueError('max_rays must be nonnegative.')
-        if (min_rays > max_rays):
-            raise ValueError('max_rays must be greater than or equal to min_rays.')
+        sage: set_random_seed()
+        sage: J = random_cone(max_dim = 8, solid=False, strictly_convex=True)
+        sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
+        sage: K_W = rho(K, J)
+        sage: K_star_W_star = rho(K.dual(), J).dual()
+        sage: basically_the_same(K_W, K_star_W_star)
+        True
 
+    ::
 
-    def random_min_max(l,u):
-        r"""
-        We need to handle two cases for the upper bounds, and we need to do
-        the same thing for max_dim/max_rays. So we consolidate the logic here.
-        """
-        if u is None:
-            # The upper bound is unspecified; return a random integer
-            # in [l,infinity).
-            return l + ZZ.random_element().abs()
-        else:
-            # We have an upper bound, and it's greater than or equal
-            # to our lower bound. So we generate a random integer in
-            # [0,u-l], and then add it to l to get something in
-            # [l,u]. To understand the "+1", check the
-            # ZZ.random_element() docs.
-            return l + ZZ.random_element(u - l + 1)
+        sage: set_random_seed()
+        sage: J = random_cone(max_dim = 8, solid=True, strictly_convex=True)
+        sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
+        sage: K_W = rho(K, J)
+        sage: K_star_W_star = rho(K.dual(), J).dual()
+        sage: basically_the_same(K_W, K_star_W_star)
+        True
+
+    """
+    if K2 is None:
+        K2 = K
 
+    # First we project K onto the span of K2. This will explode if the
+    # rank of ``K2.lattice()`` doesn't match ours.
+    span_K2 = Cone(K2.rays() + (-K2).rays(), lattice=K.lattice())
+    K = K.intersection(span_K2)
 
-    d = random_min_max(min_dim, max_dim)
-    r = random_min_max(min_rays, max_rays)
+    # Cheat a little to get the subspace span(K2). The paper uses the
+    # rays of K2 as a basis, but everything is invariant under linear
+    # isomorphism (i.e. a change of basis), and this is a little
+    # faster.
+    W = span_K2.linear_subspace()
 
-    L = ToricLattice(d)
-    rays = [L.random_element() for i in range(0,r)]
+    # We've already intersected K with the span of K2, so every
+    # generator of K should belong to W now.
+    W_rays = [ W.coordinate_vector(r) for r in K.rays() ]
+
+    L = ToricLattice(K2.dim())
+    return Cone(W_rays, lattice=L)
 
-    # The lattice parameter is required when no rays are given, so we
-    # pass it just in case.
-    return Cone(rays, lattice=L)
 
 
 def discrete_complementarity_set(K):
@@ -141,9 +248,7 @@ def discrete_complementarity_set(K):
 
     A list of pairs `(x,s)` such that,
 
-      * `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.
 
@@ -182,11 +287,12 @@ def discrete_complementarity_set(K):
     The complementarity set of the dual can be obtained by switching the
     components of the complementarity set of the original cone::
 
-        sage: K1 = random_cone(max_dim=10, max_rays=10)
+        sage: set_random_seed()
+        sage: K1 = random_cone(max_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: actual == expected
+        sage: sorted(actual) == sorted(expected)
         True
 
     """
@@ -200,6 +306,135 @@ def discrete_complementarity_set(K):
     return [(x,s) for x in xs for s in ss if x.inner_product(s) == 0]
 
 
+def LL(K):
+    r"""
+    Compute the space `\mathbf{LL}` of all Lyapunov-like transformations
+    on this cone.
+
+    OUTPUT:
+
+    A list of matrices forming a basis for the space of all
+    Lyapunov-like transformations on the given cone.
+
+    EXAMPLES:
+
+    The trivial cone has no Lyapunov-like transformations::
+
+        sage: L = ToricLattice(0)
+        sage: K = Cone([], lattice=L)
+        sage: LL(K)
+        []
+
+    The Lyapunov-like transformations on the nonnegative orthant are
+    simply diagonal matrices::
+
+        sage: K = Cone([(1,)])
+        sage: LL(K)
+        [[1]]
+
+        sage: K = Cone([(1,0),(0,1)])
+        sage: LL(K)
+        [
+        [1 0]  [0 0]
+        [0 0], [0 1]
+        ]
+
+        sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
+        sage: LL(K)
+        [
+        [1 0 0]  [0 0 0]  [0 0 0]
+        [0 0 0]  [0 1 0]  [0 0 0]
+        [0 0 0], [0 0 0], [0 0 1]
+        ]
+
+    Only the identity matrix is Lyapunov-like on the `L^{3}_{1}` and
+    `L^{3}_{\infty}` cones [Rudolf et al.]_::
+
+        sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)])
+        sage: LL(L31)
+        [
+        [1 0 0]
+        [0 1 0]
+        [0 0 1]
+        ]
+
+        sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)])
+        sage: LL(L3infty)
+        [
+        [1 0 0]
+        [0 1 0]
+        [0 0 1]
+        ]
+
+    If our cone is the entire space, then every transformation on it is
+    Lyapunov-like::
+
+        sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)])
+        sage: M = MatrixSpace(QQ,2)
+        sage: M.basis() == LL(K)
+        True
+
+    TESTS:
+
+    The inner product `\left< L\left(x\right), s \right>` is zero for
+    every pair `\left( x,s \right)` in the discrete complementarity set
+    of the cone::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8)
+        sage: C_of_K = discrete_complementarity_set(K)
+        sage: l = [ (L*x).inner_product(s) for (x,s) in C_of_K for L in LL(K) ]
+        sage: sum(map(abs, l))
+        0
+
+    The Lyapunov-like transformations on a cone and its dual are related
+    by transposition, but we're not guaranteed to compute transposed
+    elements of `LL\left( K \right)` as our basis for `LL\left( K^{*}
+    \right)`
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8)
+        sage: LL2 = [ L.transpose() for L in LL(K.dual()) ]
+        sage: V = VectorSpace( K.lattice().base_field(), K.lattice_dim()^2)
+        sage: LL1_vecs = [ V(m.list()) for m in LL(K) ]
+        sage: LL2_vecs = [ V(m.list()) for m in LL2   ]
+        sage: V.span(LL1_vecs) == V.span(LL2_vecs)
+        True
+
+    """
+    V = K.lattice().vector_space()
+
+    C_of_K = discrete_complementarity_set(K)
+
+    tensor_products = [ s.tensor_product(x) 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
+    # many are linearly-indepenedent.
+    #
+    # The space W has the same base ring as V, but dimension
+    # dim(V)^2. So it has the same dimension as the space of linear
+    # transformations on V. In other words, it's just the right size
+    # to create an isomorphism between it and our matrices.
+    W = VectorSpace(V.base_ring(), V.dimension()**2)
+
+    # Turn our matrices into long vectors...
+    vectors = [ W(m.list()) for m in tensor_products ]
+
+    # Vector space representation of Lyapunov-like matrices
+    # (i.e. vec(L) where L is Luapunov-like).
+    LL_vector = W.span(vectors).complement()
+
+    # Now construct an ambient MatrixSpace in which to stick our
+    # transformations.
+    M = MatrixSpace(V.base_ring(), V.dimension())
+
+    matrix_basis = [ M(v.list()) for v in LL_vector.basis() ]
+
+    return matrix_basis
+
+
+
 def lyapunov_rank(K):
     r"""
     Compute the Lyapunov (or bilinearity) rank of this cone.
@@ -243,17 +478,21 @@ def lyapunov_rank(K):
 
     REFERENCES:
 
-    1. M.S. Gowda and J. Tao. On the bilinearity rank of a proper cone
-       and Lyapunov-like transformations, Mathematical Programming, 147
+    .. [Gowda/Tao] M.S. Gowda and J. Tao. On the bilinearity rank of a proper
+       cone and Lyapunov-like transformations, Mathematical Programming, 147
        (2014) 155-170.
 
-    2. G. Rudolf, N. Noyan, D. Papp, and F. Alizadeh, Bilinear
+    .. [Orlitzky/Gowda] M. Orlitzky and M. S. Gowda. The Lyapunov Rank of an
+       Improper Cone. Work in-progress.
+
+    .. [Rudolf et al.] G. Rudolf, N. Noyan, D. Papp, and F. Alizadeh, Bilinear
        optimality constraints for the cone of positive polynomials,
        Mathematical Programming, Series B, 129 (2011) 5-31.
 
     EXAMPLES:
 
-    The nonnegative orthant in `\mathbb{R}^{n}` always has rank `n`::
+    The nonnegative orthant in `\mathbb{R}^{n}` always has rank `n`
+    [Rudolf et al.]_::
 
         sage: positives = Cone([(1,)])
         sage: lyapunov_rank(positives)
@@ -261,23 +500,57 @@ def lyapunov_rank(K):
         sage: quadrant = Cone([(1,0), (0,1)])
         sage: lyapunov_rank(quadrant)
         2
-        sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)])
+       sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)])
         sage: lyapunov_rank(octant)
         3
 
-    The `L^{3}_{1}` cone is known to have a Lyapunov rank of one::
+    The full space `\mathbb{R}^{n}` has Lyapunov rank `n^{2}`
+    [Orlitzky/Gowda]_::
+
+        sage: R5 = VectorSpace(QQ, 5)
+        sage: gs = R5.basis() + [ -r for r in R5.basis() ]
+        sage: K = Cone(gs)
+        sage: lyapunov_rank(K)
+        25
+
+    The `L^{3}_{1}` cone is known to have a Lyapunov rank of one
+    [Rudolf et al.]_::
 
         sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)])
         sage: lyapunov_rank(L31)
         1
 
-    Likewise for the `L^{3}_{\infty}` cone::
+    Likewise for the `L^{3}_{\infty}` cone [Rudolf et al.]_::
 
         sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)])
         sage: lyapunov_rank(L3infty)
         1
 
-    The Lyapunov rank should be additive on a product of cones::
+    A single ray in `n` dimensions should have Lyapunov rank `n^{2} - n
+    + 1` [Orlitzky/Gowda]_::
+
+        sage: K = Cone([(1,0,0,0,0)])
+        sage: lyapunov_rank(K)
+        21
+        sage: K.lattice_dim()**2 - K.lattice_dim() + 1
+        21
+
+    A subspace (of dimension `m`) in `n` dimensions should have a
+    Lyapunov rank of `n^{2} - m\left(n - m)` [Orlitzky/Gowda]_::
+
+        sage: e1 = (1,0,0,0,0)
+        sage: neg_e1 = (-1,0,0,0,0)
+        sage: e2 = (0,1,0,0,0)
+        sage: neg_e2 = (0,-1,0,0,0)
+        sage: z = (0,0,0,0,0)
+        sage: K = Cone([e1, neg_e1, e2, neg_e2, z, z, z])
+        sage: lyapunov_rank(K)
+        19
+        sage: K.lattice_dim()**2 - K.dim()*K.codim()
+        19
+
+    The Lyapunov rank should be additive on a product of proper cones
+    [Rudolf et al.]_::
 
         sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)])
         sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)])
@@ -285,8 +558,8 @@ def lyapunov_rank(K):
         sage: lyapunov_rank(K) == lyapunov_rank(L31) + lyapunov_rank(octant)
         True
 
-    Two isomorphic cones should have the same Lyapunov rank. The cone
-    ``K`` in the following example is isomorphic to the nonnegative
+    Two isomorphic cones should have the same Lyapunov rank [Rudolf et al.]_.
+    The cone ``K`` in the following example is isomorphic to the nonnegative
     octant in `\mathbb{R}^{3}`::
 
         sage: K = Cone([(1,2,3), (-1,1,0), (1,0,6)])
@@ -294,7 +567,7 @@ def lyapunov_rank(K):
         3
 
     The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
-    itself::
+    itself [Rudolf et al.]_::
 
         sage: K = Cone([(2,2,4), (-1,9,0), (2,0,6)])
         sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
@@ -302,44 +575,151 @@ def lyapunov_rank(K):
 
     TESTS:
 
-    The Lyapunov rank should be additive on a product of cones::
+    The Lyapunov rank should be additive on a product of proper cones
+    [Rudolf et al.]_::
 
-        sage: K1 = random_cone(max_dim=10, max_rays=10)
-        sage: K2 = random_cone(max_dim=10, max_rays=10)
+        sage: set_random_seed()
+        sage: K1 = random_cone(max_dim=8, strictly_convex=True, solid=True)
+        sage: K2 = random_cone(max_dim=8, strictly_convex=True, solid=True)
         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::
+    itself [Rudolf et al.]_::
 
-        sage: K = random_cone(max_dim=10, max_rays=10)
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8)
         sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
         True
 
+    Make sure we exercise the non-strictly-convex/non-solid case::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8, strictly_convex=False, solid=False)
+        sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+        True
+
+    Let's check the other permutations as well, just to be sure::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8, strictly_convex=False, solid=True)
+        sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+        True
+
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8, strictly_convex=True, solid=False)
+        sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+        True
+
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True)
+        sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+        True
+
+    The Lyapunov rank of a proper polyhedral cone in `n` dimensions can
+    be any number between `1` and `n` inclusive, excluding `n-1`
+    [Gowda/Tao]_. By accident, the `n-1` restriction will hold for the
+    trivial cone in a trivial space as well. However, in zero dimensions,
+    the Lyapunov rank of the trivial cone will be zero::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True)
+        sage: b = lyapunov_rank(K)
+        sage: n = K.lattice_dim()
+        sage: (n == 0 or 1 <= b) and b <= n
+        True
+        sage: b == n-1
+        False
+
+    In fact [Orlitzky/Gowda]_, no closed convex polyhedral cone can have
+    Lyapunov rank `n-1` in `n` dimensions::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8)
+        sage: b = lyapunov_rank(K)
+        sage: n = K.lattice_dim()
+        sage: b == n-1
+        False
+
+    The calculation of the Lyapunov rank of an improper cone can be
+    reduced to that of a proper cone [Orlitzky/Gowda]_::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8)
+        sage: actual = lyapunov_rank(K)
+        sage: K_S = rho(K)
+        sage: K_SP = rho(K_S.dual()).dual()
+        sage: l = K.lineality()
+        sage: c = K.codim()
+        sage: expected = lyapunov_rank(K_SP) + K.dim()*(l + c) + c**2
+        sage: actual == expected
+        True
+
+    The Lyapunov rank of a proper cone is just the dimension of ``LL(K)``::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True)
+        sage: lyapunov_rank(K) == len(LL(K))
+        True
+
+    In fact the same can be said of any cone. These additional tests
+    just increase our confidence that the reduction scheme works::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8, strictly_convex=True, solid=False)
+        sage: lyapunov_rank(K) == len(LL(K))
+        True
+
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8, strictly_convex=False, solid=True)
+        sage: lyapunov_rank(K) == len(LL(K))
+        True
+
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8, strictly_convex=False, solid=False)
+        sage: lyapunov_rank(K) == len(LL(K))
+        True
+
+    Test Theorem 3 in [Orlitzky/Gowda]_::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True)
+        sage: L = ToricLattice(K.lattice_dim() + 1)
+        sage: K = Cone([ r.list() + [0] for r in K.rays() ], lattice=L)
+        sage: lyapunov_rank(K) >= K.lattice_dim()
+        True
+
     """
-    V = K.lattice().vector_space()
+    beta = 0
 
-    C_of_K = discrete_complementarity_set(K)
+    m = K.dim()
+    n = K.lattice_dim()
+    l = K.lineality()
 
-    matrices = [x.tensor_product(s) for (x,s) in C_of_K]
+    if m < n:
+        # K is not solid, restrict to its span.
+        K = rho(K)
 
-    # Sage doesn't think matrices are vectors, so we have to convert
-    # our matrices to vectors explicitly before we can figure out how
-    # many are linearly-indepenedent.
-    #
-    # The space W has the same base ring as V, but dimension
-    # dim(V)^2. So it has the same dimension as the space of linear
-    # transformations on V. In other words, it's just the right size
-    # to create an isomorphism between it and our matrices.
-    W = VectorSpace(V.base_ring(), V.dimension()**2)
+        # Lemma 2
+        beta += m*(n - m) + (n - m)**2
 
-    def phi(m):
-        r"""
-        Convert a matrix to a vector isomorphically.
-        """
-        return W(m.list())
+    if l > 0:
+        # K is not pointed, restrict to the span of its dual. Uses a
+        # proposition from our paper, i.e. this is equivalent to K =
+        # rho(K.dual()).dual().
+        K = rho(K, K.dual())
 
-    vectors = [phi(m) for m in matrices]
+        # Lemma 3
+        beta += m * l
 
-    return (W.dimension() - W.span(vectors).rank())
+    beta += len(LL(K))
+    return beta