X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=e9d0f1e643e3a0d40d8a754faf551ac317e36f37;hb=d27a3f72ac6b898534615de9111ace4082a4c55e;hp=ff7d195d134c15943dbb75c1f26b741bb4a0afba;hpb=10142e85f34c47fa35df002f519d1d58a79a74f4;p=sage.d.git

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index ff7d195..e9d0f1e 100644
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -8,210 +8,231 @@ addsitedir(abspath('../../'))
 from sage.all import *
 
 
-def project_span(K):
+def basically_the_same(K1,K2):
     r"""
-    Project ``K`` into its own span.
-
-    EXAMPLES::
-
-        sage: K = Cone([(1,)])
-        sage: project_span(K) == K
-        True
+    ``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
 
-        sage: K2 = Cone([(1,0)])
-        sage: project_span(K2).rays()
-        N(1)
-        in 1-d lattice N
-        sage: K3 = Cone([(1,0,0)])
-        sage: project_span(K3).rays()
-        N(1)
-        in 1-d lattice N
-        sage: project_span(K2) == project_span(K3)
-        True
+    if K1.nrays() != K2.nrays():
+        return False
 
-    TESTS:
+    if K1.dim() != K2.dim():
+        return False
 
-    The projected cone should always be solid::
+    if K1.lineality() != K2.lineality():
+        return False
 
-        sage: K = random_cone(max_dim = 10)
-        sage: K_S = project_span(K)
-        sage: K_S.is_solid()
-        True
+    if K1.is_solid() != K2.is_solid():
+        return False
 
-    If we do this according to our paper, then the result is proper::
+    if K1.is_strictly_convex() != K2.is_strictly_convex():
+        return False
 
-        sage: K = random_cone(max_dim = 10)
-        sage: K_S = project_span(K)
-        sage: P = project_span(K_S.dual()).dual()
-        sage: P.is_proper()
-        True
+    if len(LL(K1)) != len(LL(K2)):
+        return False
 
-    """
-    L = K.lattice()
-    F = L.base_field()
-    Q = L.quotient(K.sublattice_complement())
-    vecs = [ vector(F, reversed(list(Q(r)))) for r in K.rays() ]
+    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
 
-    newL = None
-    if len(vecs) == 0:
-        newL = ToricLattice(0)
+    if len(K1.facets()) != len(K2.facets()):
+        return False
 
-    return Cone(vecs, lattice=newL)
+    return True
 
 
 
-def lineality(K):
+def rho(K, K2=None):
     r"""
-    Compute the lineality of this cone.
+    Restrict ``K`` into its own span, or the span of another cone.
 
-    The lineality of a cone is the dimension of the largest linear
-    subspace contained in that cone.
+    INPUT:
+
+    - ``K2`` -- another cone whose lattice has the same rank as this
+                cone.
 
     OUTPUT:
 
-    A nonnegative integer; the dimension of the largest subspace
-    contained within this cone.
+    A new cone in a sublattice.
 
-    REFERENCES:
+    EXAMPLES::
 
-    .. [Rockafellar] R.T. Rockafellar. Convex Analysis. Princeton
-       University Press, Princeton, 1970.
+        sage: K = Cone([(1,)])
+        sage: rho(K) == K
+        True
 
-    EXAMPLES:
+        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
 
-    The lineality of the nonnegative orthant is zero, since it clearly
-    contains no lines::
+    TESTS:
 
-        sage: K = Cone([(1,0,0), (0,1,0), (0,0,1)])
-        sage: lineality(K)
-        0
+    The projected cone should always be solid::
 
-    However, if we add another ray so that the entire `x`-axis belongs
-    to the cone, then the resulting cone will have lineality one::
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8)
+        sage: K_S = rho(K)
+        sage: K_S.is_solid()
+        True
 
-        sage: K = Cone([(1,0,0), (-1,0,0), (0,1,0), (0,0,1)])
-        sage: lineality(K)
-        1
+    And the resulting cone should live in a space having the same
+    dimension as the space we restricted it to::
 
-    If our cone is all of `\mathbb{R}^{2}`, then its lineality is equal
-    to the dimension of the ambient space (i.e. two)::
+        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: K = Cone([(1,0), (-1,0), (0,1), (0,-1)])
-        sage: lineality(K)
-        2
+    This function should not affect the dimension of a cone::
 
-    Per the definition, the lineality of the trivial cone in a trivial
-    space is zero::
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8)
+        sage: K.dim() == rho(K).dim()
+        True
 
-        sage: K = Cone([], lattice=ToricLattice(0))
-        sage: lineality(K)
-        0
+    Nor should it affect the lineality of a cone::
 
-    TESTS:
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8)
+        sage: K.lineality() == rho(K).lineality()
+        True
 
-    The lineality of a cone should be an integer between zero and the
-    dimension of the ambient space, inclusive::
+    No matter which space we restrict to, the lineality should not
+    increase::
 
-        sage: K = random_cone(max_dim = 10)
-        sage: l = lineality(K)
-        sage: l in ZZ
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8)
+        sage: K.lineality() >= rho(K).lineality()
         True
-        sage: (0 <= l) and (l <= K.lattice_dim())
+        sage: K.lineality() >= rho(K, K.dual()).lineality()
         True
 
-    A strictly convex cone should have lineality zero::
-
-        sage: K = random_cone(max_dim = 10, strictly_convex = True)
-        sage: lineality(K)
-        0
-
-    """
-    return K.linear_subspace().dimension()
-
-
-def codim(K):
-    r"""
-    Compute the codimension of this cone.
-
-    The codimension of a cone is the dimension of the space of all
-    elements perpendicular to every element of the cone. In other words,
-    the codimension is the difference between the dimension of the
-    ambient space and the dimension of the cone itself.
-
-    OUTPUT:
-
-    A nonnegative integer representing the dimension of the space of all
-    elements perpendicular to this cone.
+    If we do this according to our paper, then the result is proper::
 
-    .. seealso::
+        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
 
-        :meth:`dim`, :meth:`lattice_dim`
+    ::
 
-    EXAMPLES:
-
-    The codimension of the nonnegative orthant is zero, since the span of
-    its generators equals the entire ambient space::
+        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: K = Cone([(1,0,0), (0,1,0), (0,0,1)])
-        sage: codim(K)
-        0
+    ::
 
-    However, if we remove a ray so that the entire cone is contained
-    within the `x-y`-plane, then the resulting cone will have
-    codimension one, because the `z`-axis is perpendicular to every
-    element of the cone::
+        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
 
-        sage: K = Cone([(1,0,0), (0,1,0)])
-        sage: codim(K)
-        1
+    ::
 
-    If our cone is all of `\mathbb{R}^{2}`, then its codimension is zero::
+        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
 
-        sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)])
-        sage: codim(K)
-        0
+    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::
 
-    And if the cone is trivial in any space, then its codimension is
-    equal to the dimension of the ambient space::
+        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
 
-        sage: K = Cone([], lattice=ToricLattice(0))
-        sage: codim(K)
-        0
+    ::
 
-        sage: K = Cone([(0,)])
-        sage: codim(K)
-        1
+        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
 
-        sage: K = Cone([(0,0)])
-        sage: codim(K)
-        2
+    ::
 
-    TESTS:
+        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
 
-    The codimension of a cone should be an integer between zero and
-    the dimension of the ambient space, inclusive::
+    ::
 
-        sage: K = random_cone(max_dim = 10)
-        sage: c = codim(K)
-        sage: c in ZZ
-        True
-        sage: (0 <= c) and (c <= K.lattice_dim())
+        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
 
-    A solid cone should have codimension zero::
+    """
+    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)
 
-        sage: K = random_cone(max_dim = 10, solid = True)
-        sage: codim(K)
-        0
+    # 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()
 
-    The codimension of a cone is equal to the lineality of its dual::
+    # 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() ]
 
-        sage: K = random_cone(max_dim = 10, solid = True)
-        sage: codim(K) == lineality(K.dual())
-        True
+    L = ToricLattice(K2.dim())
+    return Cone(W_rays, lattice=L)
 
-    """
-    return (K.lattice_dim() - K.dim())
 
 
 def discrete_complementarity_set(K):
@@ -227,9 +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.
 
@@ -268,11 +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
 
     """
@@ -346,24 +366,47 @@ 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_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]
+    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
@@ -465,8 +508,8 @@ def lyapunov_rank(K):
     [Orlitzky/Gowda]_::
 
         sage: R5 = VectorSpace(QQ, 5)
-        sage: gens = R5.basis() + [ -r for r in R5.basis() ]
-        sage: K = Cone(gens)
+        sage: gs = R5.basis() + [ -r for r in R5.basis() ]
+        sage: K = Cone(gs)
         sage: lyapunov_rank(K)
         25
 
@@ -499,11 +542,11 @@ def lyapunov_rank(K):
         sage: neg_e1 = (-1,0,0,0,0)
         sage: e2 = (0,1,0,0,0)
         sage: neg_e2 = (0,-1,0,0,0)
-        sage: zero = (0,0,0,0,0)
-        sage: K = Cone([e1, neg_e1, e2, neg_e2, zero, zero, zero])
+        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()*codim(K)
+        sage: K.lattice_dim()**2 - K.dim()*K.codim()
         19
 
     The Lyapunov rank should be additive on a product of proper cones
@@ -535,8 +578,9 @@ def lyapunov_rank(K):
     The Lyapunov rank should be additive on a product of proper cones
     [Rudolf et al.]_::
 
-        sage: K1 = random_cone(max_dim=10, strictly_convex=True, solid=True)
-        sage: K2 = random_cone(max_dim=10, strictly_convex=True, solid=True)
+        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
@@ -544,7 +588,36 @@ def lyapunov_rank(K):
     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_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
 
@@ -554,7 +627,8 @@ 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_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
@@ -565,7 +639,8 @@ 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_dim=8)
         sage: b = lyapunov_rank(K)
         sage: n = K.lattice_dim()
         sage: b == n-1
@@ -574,39 +649,74 @@ 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=10)
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8)
         sage: actual = lyapunov_rank(K)
-        sage: K_S = project_span(K)
-        sage: P = project_span(K_S.dual()).dual()
-        sage: l = lineality(K)
-        sage: c = codim(K)
-        sage: expected = lyapunov_rank(P) + K.dim()*(l + c) + c**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
 
     The Lyapunov rank of a proper cone is just the dimension of ``LL(K)``::
 
-        sage: K = random_cone(max_dim=10, strictly_convex=True, solid=True)
+        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
+
     """
     beta = 0
 
     m = K.dim()
     n = K.lattice_dim()
-    l = lineality(K)
+    l = K.lineality()
 
     if m < n:
-        # K is not solid, project onto its span.
-        K = project_span(K)
+        # K is not solid, restrict to its span.
+        K = rho(K)
 
         # Lemma 2
         beta += m*(n - m) + (n - m)**2
 
     if l > 0:
-        # K is not pointed, project its dual onto its span.
-        K = project_span(K.dual()).dual()
+        # 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