from sage.all import *
+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 project_span(K, K2 = None):
+def span_iso(K):
r"""
- Return a "copy" of ``K`` embeded in a lower-dimensional space.
+ 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::
- By default, we will project ``K`` into the subspace spanned by its
- rays. However, if ``K2`` is not ``None``, we will project into the
- space spanned by the rays of ``K2`` instead.
+ sage: K = random_cone(max_dim=10)
+ sage: (phi, phi_inv) = span_iso(K)
+ sage: phi_inv(phi(K)) == K
+ True
- EXAMPLES::
+ The image of ``K`` under the isomorphism should have full dimension::
- sage: K = Cone([(1,0,0), (0,1,0)])
- sage: project_span(K)
- 2-d cone in 2-d lattice N
- sage: project_span(K).rays()
- N(1, 0),
- N(0, 1)
- in 2-d lattice N
+ sage: K = random_cone(max_dim=10)
+ sage: (phi, phi_inv) = span_iso(K)
+ sage: phi(K).dim() == phi(K).lattice_dim()
+ True
- sage: K = Cone([(1,0,0), (0,1,0)])
- sage: K2 = Cone([(0,1)])
- sage: project_span(K, K2).rays()
- N(1)
- in 1-d lattice N
+ The isomorphism should be an inner product space isomorphism, and
+ thus it should preserve dual cones (and commute with the "dual"
+ operation). But beware the automatic renaming of the dual lattice.
+ OH AND YOU HAVE TO SORT THE CONES::
+
+ sage: K = random_cone(max_dim=10, strictly_convex=False, solid=True)
+ sage: L = K.lattice()
+ sage: rename_lattice(L, 'L')
+ sage: (phi, phi_inv) = span_iso(K)
+ sage: sorted(phi_inv( phi(K).dual() )) == sorted(K.dual())
+ True
+
+ We may need to isomorph twice to make sure we stop moving down to
+ smaller spaces. (Once you've done this on a cone and its dual, the
+ result should be proper.) OH AND YOU HAVE TO SORT THE CONES::
+
+ sage: K = random_cone(max_dim=10, strictly_convex=False, solid=False)
+ sage: L = K.lattice()
+ sage: rename_lattice(L, 'L')
+ sage: (phi, phi_inv) = span_iso(K)
+ sage: K_S = phi(K)
+ sage: (phi_dual, phi_dual_inv) = span_iso(K_S.dual())
+ sage: J_T = phi_dual(K_S.dual()).dual()
+ sage: phi_inv(phi_dual_inv(J_T)) == K
+ True
"""
- # Allow us to use a second cone to generate the subspace into
- # which we're "projecting."
- if K2 is None:
- K2 = K
+ 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())
- # Use these to generate the new cone.
- cs1 = K.rays().matrix().columns()
+ # 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)
- # And use these to figure out which indices to drop.
- cs2 = K2.rays().matrix().columns()
+ return Cone(newrays, lattice=L)
- perp_idxs = []
+ 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())
- for idx in range(0, len(cs2)):
- if cs2[idx].is_zero():
- perp_idxs.append(idx)
- solid_cols = [ cs1[idx] for idx in range(0,len(cs1))
- if not idx in perp_idxs
- and not idx >= len(cs2) ]
+ return (phi, phi_inverse)
- m = matrix(solid_cols)
- L = ToricLattice(len(m.rows()))
- J = Cone(m.transpose(), lattice=L)
- return J
def discrete_complementarity_set(K):
sage: sum(map(abs, l))
0
- Try the formula in my paper::
-
- sage: K = random_cone(max_dim=15, max_rays=25)
- sage: actual = lyapunov_rank(K)
- sage: K_S = project_span(K)
- sage: J_T1 = project_span(K, K_S.dual())
- sage: J_T2 = project_span(K_S.dual()).dual()
- sage: J_T2 = Cone(J_T2.rays(), lattice=J_T1.lattice())
- sage: J_T1 == J_T2
- True
- sage: J_T = J_T1
- 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
-
"""
V = K.lattice().vector_space()
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, phi1_inv) = span_iso(K)
+ sage: K_S = phi1(K)
+ sage: (phi2, phi2_inv) = span_iso(K_S.dual())
+ sage: J_T = phi2(K_S.dual()).dual()
+ sage: phi1_inv(phi2_inv(J_T)) == K
+ True
+ 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
+
"""
return len(LL(K))
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