from sage.all import *
-def rename_lattice(L,s):
+
+def drop_dependent(vs):
r"""
- Change all names of the given lattice to ``s``.
+ Return the largest linearly-independent subset of ``vs``.
"""
- L._name = s
- L._dual_name = s
- L._latex_name = s
- L._latex_dual_name = s
+ 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 span_iso(K):
+def basically_the_same(K1,K2):
r"""
- Return an isomorphism (and its inverse) that will send ``K`` into a
- lower-dimensional space isomorphic to its span (and back).
+ ``True`` if ``K1`` and ``K2`` are basically the same, and ``False``
+ otherwise.
+ """
+ if K1.lattice_dim() != K2.lattice_dim():
+ return False
- EXAMPLES:
+ 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:
- The inverse composed with the isomorphism should be the identity::
+ - ``K2`` -- another cone whose lattice has the same rank as this cone.
- sage: K = random_cone(max_dim=10)
- sage: (phi, phi_inv) = span_iso(K)
- sage: phi_inv(phi(K)) == K
+ OUTPUT:
+
+ A new cone in a sublattice.
+
+ EXAMPLES::
+
+ sage: K = Cone([(1,)])
+ sage: rho(K) == K
True
- The image of ``K`` under the isomorphism should have full dimension::
+ 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
+
+ TESTS:
- sage: K = random_cone(max_dim=10)
- sage: (phi, phi_inv) = span_iso(K)
- sage: phi(K).dim() == phi(K).lattice_dim()
+ The projected cone should always be solid::
+
+ 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: 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
"""
- phi_domain = K.sublattice().vector_space()
- phi_codo = VectorSpace(phi_domain.base_field(), phi_domain.dimension())
+ 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)
+
+ # 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() ]
+
+ L = ToricLattice(K2.dim())
+ return Cone(W_rays, lattice=L)
- # S goes from the new space to the cone space.
- S = linear_transformation(phi_codo, phi_domain, phi_domain.basis())
- # phi goes from the cone space to the new space.
- def phi(J_orig):
- r"""
- Takes a cone ``J`` and sends it into the new space.
- """
- newrays = map(S.inverse(), J_orig.rays())
- L = None
- if len(newrays) == 0:
- L = ToricLattice(0)
- return Cone(newrays, lattice=L)
+def lineality(K):
+ r"""
+ Compute the lineality of this cone.
+
+ The lineality of a cone is the dimension of the largest linear
+ subspace contained in that cone.
+
+ OUTPUT:
+
+ A nonnegative integer; the dimension of the largest subspace
+ contained within this cone.
+
+ REFERENCES:
+
+ .. [Rockafellar] R.T. Rockafellar. Convex Analysis. Princeton
+ University Press, Princeton, 1970.
+
+ EXAMPLES:
+
+ The lineality of the nonnegative orthant is zero, since it clearly
+ contains no lines::
+
+ sage: K = Cone([(1,0,0), (0,1,0), (0,0,1)])
+ sage: lineality(K)
+ 0
+
+ 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: K = Cone([(1,0,0), (-1,0,0), (0,1,0), (0,0,1)])
+ sage: lineality(K)
+ 1
+
+ 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: K = Cone([(1,0), (-1,0), (0,1), (0,-1)])
+ sage: lineality(K)
+ 2
- def phi_inverse(J_sub):
- r"""
- The inverse to phi which goes from the new space to the cone space.
- """
- newrays = map(S, J_sub.rays())
- return Cone(newrays, lattice=K.lattice())
+ Per the definition, the lineality of the trivial cone in a trivial
+ space is zero::
+ sage: K = Cone([], lattice=ToricLattice(0))
+ sage: lineality(K)
+ 0
+
+ TESTS:
+
+ The lineality of a cone should be an integer between zero and the
+ dimension of the ambient space, inclusive::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8)
+ sage: l = lineality(K)
+ sage: l in ZZ
+ True
+ sage: (0 <= l) and (l <= K.lattice_dim())
+ True
- return (phi, phi_inverse)
+ A strictly convex cone should have lineality zero::
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8, strictly_convex = True)
+ sage: lineality(K)
+ 0
+
+ """
+ return K.linear_subspace().dimension()
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
"""
[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
sage: lyapunov_rank(octant)
3
+ The full space `\mathbb{R}^{n}` has Lyapunov rank `n^{2}`
+ [Orlitzky/Gowda]_::
+
+ sage: R5 = VectorSpace(QQ, 5)
+ sage: gs = R5.basis() + [ -r for r in R5.basis() ]
+ sage: K = Cone(gs)
+ sage: lyapunov_rank(K)
+ 25
+
The `L^{3}_{1}` cone is known to have a Lyapunov rank of one
[Rudolf et al.]_::
sage: lyapunov_rank(L3infty)
1
- The Lyapunov rank should be additive on a product of cones
+ A single ray in `n` dimensions should have Lyapunov rank `n^{2} - n
+ + 1` [Orlitzky/Gowda]_::
+
+ sage: K = Cone([(1,0,0,0,0)])
+ sage: lyapunov_rank(K)
+ 21
+ sage: K.lattice_dim()**2 - K.lattice_dim() + 1
+ 21
+
+ A subspace (of dimension `m`) in `n` dimensions should have a
+ Lyapunov rank of `n^{2} - m\left(n - m)` [Orlitzky/Gowda]_::
+
+ sage: e1 = (1,0,0,0,0)
+ sage: neg_e1 = (-1,0,0,0,0)
+ sage: e2 = (0,1,0,0,0)
+ sage: neg_e2 = (0,-1,0,0,0)
+ 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)
+ 19
+
+ The Lyapunov rank should be additive on a product of proper cones
[Rudolf et al.]_::
sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)])
TESTS:
- The Lyapunov rank should be additive on a product of cones
+ The Lyapunov rank should be additive on a product of proper cones
[Rudolf et al.]_::
- sage: K1 = random_cone(max_dim=10, max_rays=10)
- sage: K2 = random_cone(max_dim=10, max_rays=10)
+ 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=15, solid=False, strictly_convex=False)
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim=8)
sage: actual = lyapunov_rank(K)
- sage: (phi1, _) = span_iso(K)
- sage: K_S = phi1(K)
- sage: (phi2, _) = span_iso(K_S.dual())
- sage: J_T = phi2(K_S.dual()).dual()
- sage: l = K.linear_subspace().dimension()
- sage: codim = K.lattice_dim() - K.dim()
- sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2
+ 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
sage: actual == expected
True
- Repeat the previous test with different ``random_cone()`` params::
+ The Lyapunov rank of a proper cone is just the dimension of ``LL(K)``::
- sage: K = random_cone(max_dim=15, solid=False, strictly_convex=True)
- sage: actual = lyapunov_rank(K)
- sage: (phi1, _) = span_iso(K)
- sage: K_S = phi1(K)
- sage: (phi2, _) = span_iso(K_S.dual())
- sage: J_T = phi2(K_S.dual()).dual()
- sage: l = K.linear_subspace().dimension()
- sage: codim = K.lattice_dim() - K.dim()
- sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2
- sage: actual == expected
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True)
+ sage: lyapunov_rank(K) == len(LL(K))
True
- sage: K = random_cone(max_dim=15, solid=True, strictly_convex=False)
- sage: actual = lyapunov_rank(K)
- sage: (phi1, _) = span_iso(K)
- sage: K_S = phi1(K)
- sage: (phi2, _) = span_iso(K_S.dual())
- sage: J_T = phi2(K_S.dual()).dual()
- sage: l = K.linear_subspace().dimension()
- sage: codim = K.lattice_dim() - K.dim()
- sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2
- sage: actual == expected
+ 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: K = random_cone(max_dim=15, solid=True, strictly_convex=True)
- sage: actual = lyapunov_rank(K)
- sage: (phi1, _) = span_iso(K)
- sage: K_S = phi1(K)
- sage: (phi2, _) = span_iso(K_S.dual())
- sage: J_T = phi2(K_S.dual()).dual()
- sage: l = K.linear_subspace().dimension()
- sage: codim = K.lattice_dim() - K.dim()
- sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2
- sage: actual == expected
+ ::
+
+ 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: K = random_cone(max_dim=15)
- sage: actual = lyapunov_rank(K)
- sage: (phi1, _) = span_iso(K)
- sage: K_S = phi1(K)
- sage: (phi2, _) = span_iso(K_S.dual())
- sage: J_T = phi2(K_S.dual()).dual()
- sage: l = K.linear_subspace().dimension()
- sage: codim = K.lattice_dim() - K.dim()
- sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2
- sage: actual == expected
+ ::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim=8, strictly_convex=False, solid=False)
+ sage: lyapunov_rank(K) == len(LL(K))
True
"""
- return len(LL(K))
+ K_orig = K
+ beta = 0
+
+ m = K.dim()
+ n = K.lattice_dim()
+ l = lineality(K)
+
+ if m < n:
+ # K is not solid, project onto 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.
+ # 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
+
+ beta += len(LL(K))
+ return beta
projects / sage.d.git / blobdiff