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."
- EXAMPLES::
+ 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 _restrict_to_space(K, W):
+ r"""
+ Restrict this cone a subspace of its ambient space.
+
+ INPUT:
+
+ - ``W`` -- The subspace into which this cone will be restricted.
+
+ OUTPUT:
+
+ A new cone in a sublattice corresponding to ``W``.
+
+ EXAMPLES:
+
+ When this cone is solid, restricting it into its own span should do
+ nothing::
sage: K = Cone([(1,)])
- sage: project_span(K) == K
+ sage: _restrict_to_space(K, K.span()) == K
True
+ A single ray restricted into its own span gives the same output
+ regardless of the ambient space::
+
sage: K2 = Cone([(1,0)])
- sage: project_span(K2).rays()
+ sage: K2_S = _restrict_to_space(K2, K2.span()).rays()
+ sage: K2_S
N(1)
in 1-d lattice N
sage: K3 = Cone([(1,0,0)])
- sage: project_span(K3).rays()
+ sage: K3_S = _restrict_to_space(K3, K3.span()).rays()
+ sage: K3_S
N(1)
in 1-d lattice N
- sage: project_span(K2) == project_span(K3)
+ sage: K2_S == K3_S
True
TESTS:
The projected cone should always be solid::
- sage: K = random_cone(max_dim = 10)
- sage: K_S = project_span(K)
- sage: K_S.is_solid()
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim = 8)
+ sage: _restrict_to_space(K, K.span()).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_ambient_dim = 8)
+ sage: K_P = _restrict_to_space(K, K.dual().span())
+ sage: K_P.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_ambient_dim = 8)
+ sage: K.dim() == _restrict_to_space(K,K.span()).dim()
+ True
+
+ Nor should it affect the lineality of a cone::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim = 8)
+ sage: K.lineality() == _restrict_to_space(K, K.span()).lineality()
+ True
+
+ No matter which space we restrict to, the lineality should not
+ increase::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim = 8)
+ sage: S = K.span(); P = K.dual().span()
+ sage: K.lineality() >= _restrict_to_space(K,S).lineality()
+ True
+ sage: K.lineality() >= _restrict_to_space(K,P).lineality()
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: P.is_proper()
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim = 8)
+ sage: K_S = _restrict_to_space(K, K.span())
+ sage: K_SP = _restrict_to_space(K_S.dual(), K_S.dual().span()).dual()
+ sage: K_SP.is_proper()
+ True
+ sage: K_SP = _restrict_to_space(K_S, K_S.dual().span())
+ sage: K_SP.is_proper()
+ True
+
+ 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
+ _restrict_to_space::
+
+ 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 = _restrict_to_space(K, J.span()).dual()
+ sage: K_star_W = _restrict_to_space(K.dual(), J.span())
+ sage: _basically_the_same(K_W_star, K_star_W)
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() ]
+ # First we want to intersect ``K`` with ``W``. The easiest way to
+ # do this is via cone intersection, so we turn the subspace ``W``
+ # into a cone.
+ W_cone = Cone(W.basis() + [-b for b in W.basis()], lattice=K.lattice())
+ K = K.intersection(W_cone)
- 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.
+ K_W_rays = [ W.coordinate_vector(r) for r in K.rays() ]
- return Cone(vecs, lattice=newL)
+ L = ToricLattice(W.dimension())
+ return Cone(K_W_rays, lattice=L)
def discrete_complementarity_set(K):
r"""
- Compute the discrete complementarity set of this cone.
+ Compute a 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.
+ A discrete complementarity set of `K` is the set of all orthogonal
+ pairs `(x,s)` such that `x \in G_{1}` and `s \in G_{2}` for some
+ generating sets `G_{1}` of `K` and `G_{2}` of its dual. Polyhedral
+ convex cones are input in terms of their generators, so "the" (this
+ particular) discrete complementarity set corresponds to ``G1
+ == K.rays()`` and ``G2 == K.dual().rays()``.
OUTPUT:
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.
+ * Both `x` and `s` are vectors (not rays).
+ * `x` is one of ``K.rays()``.
+ * `s` is one of ``K.dual().rays()``.
* `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
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]
[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
def lyapunov_rank(K):
r"""
- Compute the Lyapunov (or bilinearity) rank of this cone.
+ Compute the Lyapunov rank (or bilinearity rank) of this cone.
The Lyapunov rank of a cone can be thought of in (mainly) two ways:
An integer representing the Lyapunov rank of the cone. If the
dimension of the ambient vector space is `n`, then the Lyapunov rank
will be between `1` and `n` inclusive; however a rank of `n-1` is
- not possible (see the first reference).
-
- .. note::
-
- In the references, the cones are always assumed to be proper. We
- do not impose this restriction.
-
- .. seealso::
-
- :meth:`is_proper`
+ not possible (see [Orlitzky/Gowda]_).
ALGORITHM:
[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_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
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
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
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_ambient_dim=8)
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: K_S = _restrict_to_space(K, K.span())
+ sage: K_SP = _restrict_to_space(K_S.dual(), K_S.dual().span()).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)``::
+ The Lyapunov rank of any 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_ambient_dim=8)
sage: lyapunov_rank(K) == len(LL(K))
True
+ We can make an imperfect cone perfect by adding a slack variable
+ (a Theorem in [Orlitzky/Gowda]_)::
+
+ 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
+
"""
beta = 0
m = K.dim()
n = K.lattice_dim()
- l = K.linear_subspace().dimension()
+ 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 = _restrict_to_space(K, K.span())
- # Lemma 2
- beta += m*(n - m) + (n - m)**2
+ # Non-solid reduction lemma.
+ beta += (n - m)*n
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 = _restrict_to_space(K, K.dual().span())
- # Lemma 3
- beta += m * l
+ # Non-pointed reduction lemma.
+ beta += l * m
beta += len(LL(K))
return beta
+
+
+
+def is_lyapunov_like(L,K):
+ r"""
+ Determine whether or not ``L`` is Lyapunov-like on ``K``.
+
+ We say that ``L`` is Lyapunov-like on ``K`` if `\left\langle
+ L\left\lparenx\right\rparen,s\right\rangle = 0` for all pairs
+ `\left\langle x,s \right\rangle` in the complementarity set of
+ ``K``. It is known [Orlitzky]_ that this property need only be
+ checked for generators of ``K`` and its dual.
+
+ INPUT:
+
+ - ``L`` -- A linear transformation or matrix.
+
+ - ``K`` -- A polyhedral closed convex cone.
+
+ OUTPUT:
+
+ ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``,
+ and ``False`` otherwise.
+
+ .. WARNING::
+
+ If this function returns ``True``, then ``L`` is Lyapunov-like
+ on ``K``. However, if ``False`` is returned, that could mean one
+ of two things. The first is that ``L`` is definitely not
+ Lyapunov-like on ``K``. The second is more of an "I don't know"
+ answer, returned (for example) if we cannot prove that an inner
+ product is zero.
+
+ REFERENCES:
+
+ .. [Orlitzky] M. Orlitzky. The Lyapunov rank of an
+ improper cone (preprint).
+
+ EXAMPLES:
+
+ todo.
+
+ TESTS:
+
+ todo.
+
+ """
+ return all([(L*x).inner_product(s) == 0
+ for (x,s) in discrete_complementarity_set(K)])
projects / sage.d.git / blobdiff