from sage.all import *
+def project_span(K):
+ r"""
+ Project ``K`` into its own span.
+
+ EXAMPLES::
+
+ sage: K = Cone([(1,)])
+ sage: project_span(K) == K
+ True
+
+ 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
+
+ TESTS:
+
+ The projected cone should always be solid::
+
+ sage: K = random_cone()
+ sage: K_S = project_span(K)
+ sage: K_S.is_solid()
+ True
+
+ If we do this according to our paper, then the result is proper::
+
+ sage: K = random_cone()
+ sage: K_S = project_span(K)
+ sage: P = project_span(K_S.dual()).dual()
+ sage: P.is_proper()
+ True
+
+ """
+ F = K.lattice().base_field()
+ Q = K.lattice().quotient(K.sublattice_complement())
+ vecs = [ vector(F, reversed(list(Q(r)))) for r in K.rays() ]
+
+ L = None
+ if len(vecs) == 0:
+ L = ToricLattice(0)
+
+ return Cone(vecs, lattice=L)
+
+
+def rename_lattice(L,s):
+ r"""
+ Change all names of the given lattice to ``s``.
+ """
+ L._name = s
+ L._dual_name = s
+ L._latex_name = s
+ L._latex_dual_name = s
+
+def span_iso(K):
+ r"""
+ Return an isomorphism (and its inverse) that will send ``K`` into a
+ lower-dimensional space isomorphic to its span (and back).
+
+ EXAMPLES:
+
+ The inverse composed with the isomorphism should be the identity::
+
+ sage: K = random_cone(max_dim=10)
+ sage: (phi, phi_inv) = span_iso(K)
+ sage: phi_inv(phi(K)) == K
+ True
+
+ The image of ``K`` under the isomorphism should have full dimension::
+
+ sage: K = random_cone(max_dim=10)
+ sage: (phi, phi_inv) = span_iso(K)
+ sage: phi(K).dim() == phi(K).lattice_dim()
+ True
+
+ """
+ phi_domain = K.sublattice().vector_space()
+ phi_codo = VectorSpace(phi_domain.base_field(), phi_domain.dimension())
+
+ # 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 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())
+
+
+ return (phi, phi_inverse)
+
+
def discrete_complementarity_set(K):
r"""
cone and Lyapunov-like transformations, Mathematical Programming, 147
(2014) 155-170.
+ .. [Orlitzky/Gowda] M. Orlitzky and M. S. Gowda. The Lyapunov Rank of an
+ Improper Cone. Work in-progress.
+
.. [Rudolf et al.] G. Rudolf, N. Noyan, D. Papp, and F. Alizadeh, Bilinear
optimality constraints for the cone of positive polynomials,
Mathematical Programming, Series B, 129 (2011) 5-31.
sage: b == n-1
False
+ 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: b = lyapunov_rank(K)
+ sage: n = K.lattice_dim()
+ sage: b == n-1
+ False
+
+ 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: 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
+ True
+
+ Repeat the previous test with different ``random_cone()`` params::
+
+ 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
+ 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
+ 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
+ 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
+ True
+
+ And test with the project_span function::
+
+ sage: K = random_cone(max_dim=15)
+ 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: actual == expected
+ True
+
"""
return len(LL(K))
projects / sage.d.git / blobdiff