X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=48b4061748281e0ebcb3ef19e9872be824666ba5;hb=af3e2ce56ad6561c5c9b1b6cf3df22d690550618;hp=60f9c34ec8bc271d65812859f51ca77636c8cbbc;hpb=874e3ce831e0b1901b3c280a32ffe18e36f54959;p=sage.d.git

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index 60f9c34..48b4061 100644
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -7,114 +7,208 @@ addsitedir(abspath('../../'))
 
 from sage.all import *
 
-def project_span(K):
+
+def _basically_the_same(K1, K2):
     r"""
-    Project ``K`` into its own span.
+    Test whether or not ``K1`` and ``K2`` are "basically the same."
+
+    This is a hack to get around the fact that it's difficult to tell
+    when two cones are linearly isomorphic. We have a proposition that
+    equates two cones, but represented over `\mathbb{Q}`, they are
+    merely linearly isomorphic (not equal). So rather than test for
+    equality, we test a list of properties that should be preserved
+    under an invertible linear transformation.
+
+    OUTPUT:
+
+    ``True`` if ``K1`` and ``K2`` are basically the same, and ``False``
+    otherwise.
+
+    EXAMPLES:
+
+    Any proper cone with three generators in `\mathbb{R}^{3}` is
+    basically the same as the nonnegative orthant::
+
+        sage: K1 = Cone([(1,0,0), (0,1,0), (0,0,1)])
+        sage: K2 = Cone([(1,2,3), (3, 18, 4), (66, 51, 0)])
+        sage: _basically_the_same(K1, K2)
+        True
+
+    Negating a cone gives you another cone that is basically the same::
+
+        sage: K = Cone([(0,2,-5), (-6, 2, 4), (0, 51, 0)])
+        sage: _basically_the_same(K, -K)
+        True
+
+    TESTS:
+
+    Any cone is basically the same as itself::
+
+        sage: K = random_cone(max_ambient_dim = 8)
+        sage: _basically_the_same(K, K)
+        True
+
+    After applying an invertible matrix to the rows of a cone, the
+    result should be basically the same as the cone we started with::
+
+        sage: K1 = random_cone(max_ambient_dim = 8)
+        sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular')
+        sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice())
+        sage: _basically_the_same(K1, K2)
+        True
+
+    """
+    if K1.lattice_dim() != K2.lattice_dim():
+        return False
+
+    if K1.nrays() != K2.nrays():
+        return False
+
+    if K1.dim() != K2.dim():
+        return False
+
+    if K1.lineality() != K2.lineality():
+        return False
+
+    if K1.is_solid() != K2.is_solid():
+        return False
+
+    if K1.is_strictly_convex() != K2.is_strictly_convex():
+        return False
+
+    if len(LL(K1)) != len(LL(K2)):
+        return False
+
+    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
+
+    if len(K1.facets()) != len(K2.facets()):
+        return False
+
+    return True
+
+
+
+def _rho(K, K2=None):
+    r"""
+    Restrict ``K`` into its own span, or the span of another cone.
+
+    INPUT:
+
+    - ``K2`` -- another cone whose lattice has the same rank as this
+                cone.
+
+    OUTPUT:
+
+    A new cone in a sublattice.
 
     EXAMPLES::
 
         sage: K = Cone([(1,)])
-        sage: project_span(K) == K
+        sage: _rho(K) == K
         True
 
         sage: K2 = Cone([(1,0)])
-        sage: project_span(K2).rays()
+        sage: _rho(K2).rays()
         N(1)
         in 1-d lattice N
         sage: K3 = Cone([(1,0,0)])
-        sage: project_span(K3).rays()
+        sage: _rho(K3).rays()
         N(1)
         in 1-d lattice N
-        sage: project_span(K2) == project_span(K3)
+        sage: _rho(K2) == _rho(K3)
         True
 
     TESTS:
 
     The projected cone should always be solid::
 
-        sage: K = random_cone()
-        sage: K_S = project_span(K)
+        sage: set_random_seed()
+        sage: K = random_cone(max_ambient_dim = 8)
+        sage: K_S = _rho(K)
         sage: K_S.is_solid()
         True
 
-    If we do this according to our paper, then the result is proper::
+    And the resulting cone should live in a space having the same
+    dimension as the space we restricted it to::
 
-        sage: K = random_cone()
-        sage: K_S = project_span(K)
-        sage: P = project_span(K_S.dual()).dual()
-        sage: P.is_proper()
+        sage: set_random_seed()
+        sage: K = random_cone(max_ambient_dim = 8)
+        sage: K_S = _rho(K, K.dual() )
+        sage: K_S.lattice_dim() == K.dual().dim()
         True
 
-    """
-    F = K.lattice().base_field()
-    Q = K.lattice().quotient(K.sublattice_complement())
-    vecs = [ vector(F, reversed(list(Q(r)))) for r in K.rays() ]
+    This function should not affect the dimension of a cone::
 
-    L = None
-    if len(vecs) == 0:
-        L = ToricLattice(0)
-
-    return Cone(vecs, lattice=L)
+        sage: set_random_seed()
+        sage: K = random_cone(max_ambient_dim = 8)
+        sage: K.dim() == _rho(K).dim()
+        True
 
+    Nor should it affect the lineality of a cone::
 
-def rename_lattice(L,s):
-    r"""
-    Change all names of the given lattice to ``s``.
-    """
-    L._name = s
-    L._dual_name = s
-    L._latex_name = s
-    L._latex_dual_name = s
+        sage: set_random_seed()
+        sage: K = random_cone(max_ambient_dim = 8)
+        sage: K.lineality() == _rho(K).lineality()
+        True
 
-def span_iso(K):
-    r"""
-    Return an isomorphism (and its inverse) that will send ``K`` into a
-    lower-dimensional space isomorphic to its span (and back).
+    No matter which space we restrict to, the lineality should not
+    increase::
 
-    EXAMPLES:
+        sage: set_random_seed()
+        sage: K = random_cone(max_ambient_dim = 8)
+        sage: K.lineality() >= _rho(K).lineality()
+        True
+        sage: K.lineality() >= _rho(K, K.dual()).lineality()
+        True
 
-    The inverse composed with the isomorphism should be the identity::
+    If we do this according to our paper, then the result is proper::
 
-        sage: K = random_cone(max_dim=10)
-        sage: (phi, phi_inv) = span_iso(K)
-        sage: phi_inv(phi(K)) == K
+        sage: set_random_seed()
+        sage: K = random_cone(max_ambient_dim = 8)
+        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
 
-    The image of ``K`` under the isomorphism should have full dimension::
+    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: K = random_cone(max_dim=10)
-        sage: (phi, phi_inv) = span_iso(K)
-        sage: phi(K).dim() == phi(K).lattice_dim()
+        sage: set_random_seed()
+        sage: J = random_cone(max_ambient_dim = 8)
+        sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
+        sage: K_W_star = _rho(K, J).dual()
+        sage: K_star_W = _rho(K.dual(), J)
+        sage: _basically_the_same(K_W_star, K_star_W)
         True
 
     """
-    phi_domain = K.sublattice().vector_space()
-    phi_codo = VectorSpace(phi_domain.base_field(), phi_domain.dimension())
-
-    # S goes from the new space to the cone space.
-    S = linear_transformation(phi_codo, phi_domain, phi_domain.basis())
-
-    # phi goes from the cone space to the new space.
-    def phi(J_orig):
-        r"""
-        Takes a cone ``J`` and sends it into the new space.
-        """
-        newrays = map(S.inverse(), J_orig.rays())
-        L = None
-        if len(newrays) == 0:
-            L = ToricLattice(0)
+    if K2 is None:
+        K2 = K
 
-        return Cone(newrays, lattice=L)
+    # 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)
 
-    def phi_inverse(J_sub):
-        r"""
-        The inverse to phi which goes from the new space to the cone space.
-        """
-        newrays = map(S, J_sub.rays())
-        return Cone(newrays, lattice=K.lattice())
+    # 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()
 
+    # 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() ]
 
-    return (phi, phi_inverse)
+    L = ToricLattice(K2.dim())
+    return Cone(W_rays, lattice=L)
 
 
 
@@ -122,21 +216,29 @@ def discrete_complementarity_set(K):
     r"""
     Compute the discrete complementarity set of this cone.
 
-    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.
+    The complementarity set of a cone is the set of all orthogonal pairs
+    `(x,s)` such that `x` is in the cone, and `s` is in its dual. The
+    discrete complementarity set is a subset of the complementarity set
+    where `x` and `s` are required to be generators of their respective
+    cones.
+
+    For polyhedral cones, the discrete complementarity set is always
+    finite.
 
     OUTPUT:
 
     A list of pairs `(x,s)` such that,
 
-      * `x` is in this cone.
+      * Both `x` and `s` are vectors (not rays).
       * `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.
 
+    REFERENCES:
+
+    .. [Orlitzky/Gowda] M. Orlitzky and M. S. Gowda. The Lyapunov Rank of an
+       Improper Cone. Work in-progress.
+
     EXAMPLES:
 
     The discrete complementarity set of the nonnegative orthant consists
@@ -167,24 +269,43 @@ def discrete_complementarity_set(K):
         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: K1 = random_cone(max_dim=10, max_rays=10)
+        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: actual == expected
+        sage: sorted(actual) == sorted(expected)
         True
 
+    The pairs in the discrete complementarity set are in fact
+    complementary::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_ambient_dim=6)
+        sage: dcs = discrete_complementarity_set(K)
+        sage: sum([x.inner_product(s).abs() for (x,s) in dcs])
+        0
+
     """
     V = K.lattice().vector_space()
 
-    # Convert the rays to vectors so that we can compute inner
-    # products.
+    # Convert rays to vectors so that we can compute inner products.
     xs = [V(x) for x in K.rays()]
+
+    # We also convert the generators of the dual cone so that we
+    # return pairs of vectors and not (vector, ray) pairs.
     ss = [V(s) for s in K.dual().rays()]
 
     return [(x,s) for x in xs for s in ss if x.inner_product(s) == 0]
@@ -250,24 +371,47 @@ def LL(K):
         [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: K = random_cone(max_dim=8, max_rays=10)
+        sage: set_random_seed()
+        sage: K = random_cone(max_ambient_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_ambient_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]
+    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
@@ -365,6 +509,15 @@ def lyapunov_rank(K):
         sage: lyapunov_rank(octant)
         3
 
+    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.]_::
 
@@ -378,7 +531,30 @@ def lyapunov_rank(K):
         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)])
@@ -404,19 +580,34 @@ 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_ambient_dim=8,
+        ....:                  strictly_convex=True,
+        ....:                  solid=True)
+        sage: K2 = random_cone(max_ambient_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 Lyapunov rank is invariant under a linear isomorphism
+    [Orlitzky/Gowda]_::
+
+        sage: K1 = random_cone(max_ambient_dim = 8)
+        sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular')
+        sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice())
+        sage: lyapunov_rank(K1) == lyapunov_rank(K2)
+        True
+
     The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
     itself [Rudolf et al.]_::
 
-        sage: K = random_cone(max_dim=10, max_rays=10)
+        sage: set_random_seed()
+        sage: K = random_cone(max_ambient_dim=8)
         sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
         True
 
@@ -426,7 +617,10 @@ def lyapunov_rank(K):
     trivial cone in a trivial space as well. However, in zero dimensions,
     the Lyapunov rank of the trivial cone will be zero::
 
-        sage: K = random_cone(max_dim=10, strictly_convex=True, solid=True)
+        sage: set_random_seed()
+        sage: K = random_cone(max_ambient_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
@@ -437,7 +631,8 @@ def lyapunov_rank(K):
     In fact [Orlitzky/Gowda]_, no closed convex polyhedral cone can have
     Lyapunov rank `n-1` in `n` dimensions::
 
-        sage: K = random_cone(max_dim=10)
+        sage: set_random_seed()
+        sage: K = random_cone(max_ambient_dim=8)
         sage: b = lyapunov_rank(K)
         sage: n = K.lattice_dim()
         sage: b == n-1
@@ -446,79 +641,57 @@ def lyapunov_rank(K):
     The calculation of the Lyapunov rank of an improper cone can be
     reduced to that of a proper cone [Orlitzky/Gowda]_::
 
-        sage: K = random_cone(max_dim=15, solid=False, strictly_convex=False)
+        sage: set_random_seed()
+        sage: K = random_cone(max_ambient_dim=8)
         sage: actual = lyapunov_rank(K)
-        sage: (phi1, _) = span_iso(K)
-        sage: K_S = phi1(K)
-        sage: (phi2, _) = span_iso(K_S.dual())
-        sage: J_T = phi2(K_S.dual()).dual()
-        sage: l = K.linear_subspace().dimension()
-        sage: codim = K.lattice_dim() - K.dim()
-        sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2
+        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
 
-    Repeat the previous test with different ``random_cone()`` params::
+    The Lyapunov rank of any cone is just the dimension of ``LL(K)``::
 
-        sage: K = random_cone(max_dim=15, solid=False, strictly_convex=True)
-        sage: actual = lyapunov_rank(K)
-        sage: (phi1, _) = span_iso(K)
-        sage: K_S = phi1(K)
-        sage: (phi2, _) = span_iso(K_S.dual())
-        sage: J_T = phi2(K_S.dual()).dual()
-        sage: l = K.linear_subspace().dimension()
-        sage: codim = K.lattice_dim() - K.dim()
-        sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2
-        sage: actual == expected
+        sage: set_random_seed()
+        sage: K = random_cone(max_ambient_dim=8)
+        sage: lyapunov_rank(K) == len(LL(K))
         True
 
-        sage: K = random_cone(max_dim=15, solid=True, strictly_convex=False)
-        sage: actual = lyapunov_rank(K)
-        sage: (phi1, _) = span_iso(K)
-        sage: K_S = phi1(K)
-        sage: (phi2, _) = span_iso(K_S.dual())
-        sage: J_T = phi2(K_S.dual()).dual()
-        sage: l = K.linear_subspace().dimension()
-        sage: codim = K.lattice_dim() - K.dim()
-        sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2
-        sage: actual == expected
-        True
+    Test Theorem 3 in [Orlitzky/Gowda]_::
 
-        sage: K = random_cone(max_dim=15, solid=True, strictly_convex=True)
-        sage: actual = lyapunov_rank(K)
-        sage: (phi1, _) = span_iso(K)
-        sage: K_S = phi1(K)
-        sage: (phi2, _) = span_iso(K_S.dual())
-        sage: J_T = phi2(K_S.dual()).dual()
-        sage: l = K.linear_subspace().dimension()
-        sage: codim = K.lattice_dim() - K.dim()
-        sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2
-        sage: actual == expected
+        sage: set_random_seed()
+        sage: K = random_cone(max_ambient_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
 
-        sage: K = random_cone(max_dim=15)
-        sage: actual = lyapunov_rank(K)
-        sage: (phi1, _) = span_iso(K)
-        sage: K_S = phi1(K)
-        sage: (phi2, _) = span_iso(K_S.dual())
-        sage: J_T = phi2(K_S.dual()).dual()
-        sage: l = K.linear_subspace().dimension()
-        sage: codim = K.lattice_dim() - K.dim()
-        sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2
-        sage: actual == expected
-        True
+    """
+    beta = 0
 
-    And test with the project_span function::
+    m = K.dim()
+    n = K.lattice_dim()
+    l = K.lineality()
 
-        sage: K = random_cone(max_dim=15)
-        sage: actual = lyapunov_rank(K)
-        sage: K_S = project_span(K)
-        sage: P = project_span(K_S.dual()).dual()
-        sage: l = K.linear_subspace().dimension()
-        sage: codim = K.lattice_dim() - K.dim()
-        sage: expected = lyapunov_rank(P) + K.dim()*(l + codim) + codim**2
-        sage: actual == expected
-        True
+    if m < n:
+        # K is not solid, restrict to its span.
+        K = _rho(K)
 
-    """
-    return len(LL(K))
+        # Lemma 2
+        beta += m*(n - m) + (n - m)**2
+
+    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())
+
+        # Lemma 3
+        beta += m * l
+
+    beta += len(LL(K))
+    return beta