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``.
+ """
+ result = []
+ m = matrix(vs).echelon_form()
+ for idx in range(0, m.nrows()):
+ if not m[idx].is_zero():
+ result.append(m[idx])
+
+ 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 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(max_dim = 10)
- sage: K_S = project_span(K)
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8)
+ sage: K_S = rho(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 = rho(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() == 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: lineality(K) == lineality(rho(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(rho(K))
+ True
+ sage: lineality(K) >= lineality(rho(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 = rho(K)
+ sage: P = rho(K_S.dual()).dual()
+ sage: P.is_proper()
+ True
+ sage: P = rho(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 = rho(K)
+ sage: P = rho(K_S.dual()).dual()
+ sage: P.is_proper()
+ True
+ sage: P = rho(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 = rho(K)
+ sage: P = rho(K_S.dual()).dual()
+ sage: P.is_proper()
+ True
+ sage: P = rho(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 = rho(K)
+ sage: P = rho(K_S.dual()).dual()
sage: P.is_proper()
True
+ sage: P = rho(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 = rho(K, K.dual())
+ sage: K_star_W_star = rho(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 = rho(K, K.dual())
+ sage: K_star_W_star = rho(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 = rho(K, K.dual())
+ sage: K_star_W_star = rho(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 = rho(K, K.dual())
+ sage: K_star_W_star = rho(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
+
+ # First we project K onto the span of K2. This can be done with
+ # cones (i.e. without converting to vector spaces), but it's
+ # annoying to deal with lattice mismatches.
+ span_K2 = Cone(K2.rays() + (-K2).rays(), lattice=K.lattice())
+ K = K.intersection(span_K2)
+
+ V = K.lattice().vector_space()
+
+ # Create the space W \times W^{\perp} isomorphic to V.
+ # First we get an orthogonal (but not normal) basis...
+ W_basis = drop_dependent(K2.rays())
+ W = V.subspace_with_basis(W_basis)
- newL = None
- if len(vecs) == 0:
- newL = ToricLattice(0)
+ # 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 Cone(vecs, lattice=newL)
+ L = ToricLattice(K2.dim())
+ return Cone(W_rays, lattice=L)
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
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
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.
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
"""
[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
[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
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)
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
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
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
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
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 = rho(K)
+ sage: P = rho(K_S.dual()).dual()
sage: l = lineality(K)
sage: c = codim(K)
sage: expected = lyapunov_rank(P) + K.dim()*(l + c) + c**2
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()
if m < n:
# K is not solid, project onto its span.
- K = project_span(K)
+ 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()
+ # 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
projects / sage.d.git / blobdiff