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):
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.
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()*(K.lattice_dim() - K.dim())
+ sage: K.lattice_dim()**2 - K.dim()*K.codim()
19
The Lyapunov rank should be additive on a product of proper cones
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: l = lineality(K)
- sage: codim = K.lattice_dim() - K.dim()
- sage: expected = lyapunov_rank(P) + 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
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
projects / sage.d.git / blobdiff