X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=2d84337fda1cb5275add8171ba8e982559dc1853;hb=a2ad3efc39da8dbcc497c0ac861e1df200c6de5e;hp=55d9a06d55313c94f71be29783171c96233ccc31;hpb=42c5246c19d1af0c1f687ea122dbd79c5b42b7bc;p=sage.d.git

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index 55d9a06..2d84337 100644
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -8,55 +8,303 @@ addsitedir(abspath('../../'))
 from sage.all import *
 
 
-def project_span(K):
+def drop_dependent(vs):
     r"""
-    Project ``K`` into its own span.
+    Return the largest linearly-independent subset of ``vs``.
+    """
+    if len(vs) == 0:
+        # ...for lazy enough definitions of linearly-independent
+        return vs
+
+    result = []
+    old_V = VectorSpace(vs[0].parent().base_field(), 0)
+
+    for v in vs:
+        new_V = span(result + [v])
+        if new_V.dimension() > old_V.dimension():
+            result.append(v)
+            old_V = new_V
+
+    return result
+
+
+def basically_the_same(K1,K2):
+    r"""
+    ``True`` if ``K1`` and ``K2`` are basically the same, and ``False``
+    otherwise.
+    """
+    if K1.lattice_dim() != K2.lattice_dim():
+        return False
+
+    if K1.nrays() != K2.nrays():
+        return False
+
+    if K1.dim() != K2.dim():
+        return False
+
+    if lineality(K1) != lineality(K2):
+        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 iso_space(K):
+    r"""
+    Construct the space `W \times W^{\perp}` isomorphic to the ambient space
+    of ``K`` where `W` is equal to the span of ``K``.
+    """
+    V = K.lattice().vector_space()
+
+    # Create the space W \times W^{\perp} isomorphic to V.
+    # First we get an orthogonal (but not normal) basis...
+    M = matrix(V.base_field(), K.rays())
+    W_basis = drop_dependent(K.rays())
+
+    W = V.subspace_with_basis(W_basis)
+    W_perp = W.complement()
+
+    return W.cartesian_product(W_perp)
+
+
+def ips_iso(K):
+    r"""
+    Construct the IPS isomorphism and its inverse from our paper.
+
+    Given a cone ``K``, the returned isomorphism will split its ambient
+    vector space `V` into a cartesian product `W \times W^{\perp}` where
+    `W` equals the span of ``K``.
+    """
+    V = K.lattice().vector_space()
+    V_iso = iso_space(K)
+    (W, W_perp) = V_iso.cartesian_factors()
+
+    # A space equivalent to V, but using our basis.
+    V_user = V.subspace_with_basis( W.basis() + W_perp.basis() )
+
+    def phi(v):
+        # Write v in terms of our custom basis, where the first dim(W)
+        # coordinates are for the W-part of the basis.
+        cs = V_user.coordinates(v)
+
+        w1 = sum([ V_user.basis()[idx]*cs[idx]
+                    for idx in range(0, W.dimension()) ])
+        w2 = sum([ V_user.basis()[idx]*cs[idx]
+                    for idx in range(W.dimension(), V.dimension()) ])
+
+        return V_iso( (w1, w2) )
+
+
+    def phi_inv( pair ):
+        # Crash if the arguments are in the wrong spaces.
+        V_iso(pair)
+
+        #w = sum([ sub_w[idx]*W.basis()[idx] for idx in range(0,m) ])
+        #w_prime = sum([ sub_w_prime[idx]*W_perp.basis()[idx]
+        #             for idx in range(0,n-m) ])
+
+        return sum( pair.cartesian_factors() )
+
+
+    return (phi,phi_inv)
+
+
+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: restrict_span(K) == K
         True
 
         sage: K2 = Cone([(1,0)])
-        sage: project_span(K2).rays()
+        sage: restrict_span(K2).rays()
         N(1)
         in 1-d lattice N
         sage: K3 = Cone([(1,0,0)])
-        sage: project_span(K3).rays()
+        sage: restrict_span(K3).rays()
         N(1)
         in 1-d lattice N
-        sage: project_span(K2) == project_span(K3)
+        sage: restrict_span(K2) == restrict_span(K3)
         True
 
     TESTS:
 
     The projected cone should always be solid::
 
-        sage: K = random_cone(max_dim = 10)
-        sage: K_S = project_span(K)
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8)
+        sage: K_S = restrict_span(K)
         sage: K_S.is_solid()
         True
 
+    And the resulting cone should live in a space having the same
+    dimension as the space we restricted it to::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8)
+        sage: K_S = restrict_span(K, K.dual() )
+        sage: K_S.lattice_dim() == K.dual().dim()
+        True
+
+    This function should not affect the dimension of a cone::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8)
+        sage: K.dim() == restrict_span(K).dim()
+        True
+
+    Nor should it affect the lineality of a cone::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8)
+        sage: lineality(K) == lineality(restrict_span(K))
+        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: lineality(K) >= lineality(restrict_span(K))
+        True
+        sage: lineality(K) >= lineality(restrict_span(K, K.dual()))
+        True
+
     If we do this according to our paper, then the result is proper::
 
-        sage: K = random_cone(max_dim = 10)
-        sage: K_S = project_span(K)
-        sage: P = project_span(K_S.dual()).dual()
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=False)
+        sage: K_S = restrict_span(K)
+        sage: P = restrict_span(K_S.dual()).dual()
+        sage: P.is_proper()
+        True
+        sage: P = restrict_span(K_S, K_S.dual())
+        sage: P.is_proper()
+        True
+
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=False)
+        sage: K_S = restrict_span(K)
+        sage: P = restrict_span(K_S.dual()).dual()
+        sage: P.is_proper()
+        True
+        sage: P = restrict_span(K_S, K_S.dual())
+        sage: P.is_proper()
+        True
+
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=True)
+        sage: K_S = restrict_span(K)
+        sage: P = restrict_span(K_S.dual()).dual()
+        sage: P.is_proper()
+        True
+        sage: P = restrict_span(K_S, K_S.dual())
         sage: P.is_proper()
         True
 
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=True)
+        sage: K_S = restrict_span(K)
+        sage: P = restrict_span(K_S.dual()).dual()
+        sage: P.is_proper()
+        True
+        sage: P = restrict_span(K_S, K_S.dual())
+        sage: P.is_proper()
+        True
+
+    Test the proposition in our paper concerning the duals, where the
+    subspace `W` is the span of `K^{*}`::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8, solid=False, strictly_convex=False)
+        sage: K_W = restrict_span(K, K.dual())
+        sage: K_star_W_star = restrict_span(K.dual()).dual()
+        sage: basically_the_same(K_W, K_star_W_star)
+        True
+
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8, solid=True, strictly_convex=False)
+        sage: K_W = restrict_span(K, K.dual())
+        sage: K_star_W_star = restrict_span(K.dual()).dual()
+        sage: basically_the_same(K_W, K_star_W_star)
+        True
+
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8, solid=False, strictly_convex=True)
+        sage: K_W = restrict_span(K, K.dual())
+        sage: K_star_W_star = restrict_span(K.dual()).dual()
+        sage: basically_the_same(K_W, K_star_W_star)
+        True
+
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8, solid=True, strictly_convex=True)
+        sage: K_W = restrict_span(K, K.dual())
+        sage: K_star_W_star = restrict_span(K.dual()).dual()
+        sage: basically_the_same(K_W, K_star_W_star)
+        True
+
     """
-    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() ]
+    if K2 is None:
+        K2 = K
+
+    phi,_ = ips_iso(K2)
+    (W, W_perp) = iso_space(K2).cartesian_factors()
+
+    ray_pairs = [ phi(r) for r in K.rays() ]
 
-    newL = None
-    if len(vecs) == 0:
-        newL = ToricLattice(0)
+    # Shouldn't matter?
+    #
+    #if any([ w2 != W_perp.zero() for (_, w2) in ray_pairs ]):
+    #    msg = 'Cone has nonzero components in W-perp!'
+    #    raise ValueError(msg)
+
+    # Represent the cone in terms of a basis for W, i.e. with smaller
+    # vectors.
+    ws = [ W.coordinate_vector(w1) for (w1, _) in ray_pairs ]
+
+    L = ToricLattice(W.dimension())
 
-    return Cone(vecs, lattice=newL)
+    return Cone(ws, lattice=L)
 
 
 
@@ -112,7 +360,8 @@ def lineality(K):
     The lineality 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: set_random_seed()
+        sage: K = random_cone(max_dim = 8)
         sage: l = lineality(K)
         sage: l in ZZ
         True
@@ -121,7 +370,8 @@ def lineality(K):
 
     A strictly convex cone should have lineality zero::
 
-        sage: K = random_cone(max_dim = 10, strictly_convex = True)
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8, strictly_convex = True)
         sage: lineality(K)
         0
 
@@ -129,91 +379,6 @@ def lineality(K):
     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.
-
-    .. seealso::
-
-        :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: 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: 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: K = Cone([(1,0), (-1,0), (0,1), (0,-1)])
-        sage: codim(K)
-        0
-
-    And if the cone is trivial in any space, then its codimension is
-    equal to the dimension of the ambient space::
-
-        sage: K = Cone([], lattice=ToricLattice(0))
-        sage: codim(K)
-        0
-
-        sage: K = Cone([(0,)])
-        sage: codim(K)
-        1
-
-        sage: K = Cone([(0,0)])
-        sage: codim(K)
-        2
-
-    TESTS:
-
-    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())
-        True
-
-    A solid cone should have codimension zero::
-
-        sage: K = random_cone(max_dim = 10, solid = True)
-        sage: codim(K)
-        0
-
-    The codimension of a cone is equal to the lineality of its dual::
-
-        sage: K = random_cone(max_dim = 10, solid = True)
-        sage: codim(K) == lineality(K.dual())
-        True
-
-    """
-    return (K.lattice_dim() - K.dim())
-
-
 def discrete_complementarity_set(K):
     r"""
     Compute the discrete complementarity set of this cone.
@@ -268,11 +433,12 @@ 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 +512,47 @@ 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 +654,8 @@ 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 +688,11 @@ 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()*(K.lattice_dim() - K.dim())
+        sage: K.lattice_dim()**2 - K.dim()*codim(K)
         19
 
     The Lyapunov rank should be additive on a product of proper cones
@@ -535,8 +724,9 @@ 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 +734,36 @@ 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 +773,8 @@ 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 +785,8 @@ 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
@@ -574,23 +795,48 @@ 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: K_S = restrict_span(K)
+        sage: P = restrict_span(K_S.dual()).dual()
         sage: l = lineality(K)
-        sage: codim = K.lattice_dim() - K.dim()
-        sage: expected = lyapunov_rank(P) + K.dim()*(l + codim) + codim**2
+        sage: c = codim(K)
+        sage: expected = lyapunov_rank(P) + 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
 
     """
+    K_orig = K
     beta = 0
 
     m = K.dim()
@@ -599,14 +845,17 @@ def lyapunov_rank(K):
 
     if m < n:
         # K is not solid, project onto its span.
-        K = project_span(K)
+        K = restrict_span(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()
+        # Uses a proposition from our paper, i.e. this is
+        # equivalent to K = restrict_span(K.dual()).dual()
+        #K = restrict_span(intersect_span(K,K.dual()), K.dual())
+        K = restrict_span(K, K.dual())
 
         # Lemma 3
         beta += m * l