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 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
+
+ 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
+
+ """
+ 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"""
OUTPUT:
- A ``MatrixSpace`` object `M` such that every matrix `L \in M` is
- Lyapunov-like on this cone.
+ A list of matrices forming a basis for the space of all
+ Lyapunov-like transformations on the given cone.
+
+ EXAMPLES:
+
+ The trivial cone has no Lyapunov-like transformations::
+
+ sage: L = ToricLattice(0)
+ sage: K = Cone([], lattice=L)
+ sage: LL(K)
+ []
+
+ The Lyapunov-like transformations on the nonnegative orthant are
+ simply diagonal matrices::
+
+ sage: K = Cone([(1,)])
+ sage: LL(K)
+ [[1]]
+
+ sage: K = Cone([(1,0),(0,1)])
+ sage: LL(K)
+ [
+ [1 0] [0 0]
+ [0 0], [0 1]
+ ]
+
+ sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
+ sage: LL(K)
+ [
+ [1 0 0] [0 0 0] [0 0 0]
+ [0 0 0] [0 1 0] [0 0 0]
+ [0 0 0], [0 0 0], [0 0 1]
+ ]
+
+ Only the identity matrix is Lyapunov-like on the `L^{3}_{1}` and
+ `L^{3}_{\infty}` cones [Rudolf et al.]_::
+
+ sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)])
+ sage: LL(L31)
+ [
+ [1 0 0]
+ [0 1 0]
+ [0 0 1]
+ ]
+
+ sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)])
+ sage: LL(L3infty)
+ [
+ [1 0 0]
+ [0 1 0]
+ [0 0 1]
+ ]
+
+ 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: 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
"""
V = K.lattice().vector_space()
C_of_K = discrete_complementarity_set(K)
- matrices = [x.tensor_product(s) 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
W = VectorSpace(V.base_ring(), V.dimension()**2)
# Turn our matrices into long vectors...
- vectors = [ W(m.list()) for m in matrices ]
+ vectors = [ W(m.list()) for m in tensor_products ]
# Vector space representation of Lyapunov-like matrices
# (i.e. vec(L) where L is Luapunov-like).
# transformations.
M = MatrixSpace(V.base_ring(), V.dimension())
- matrices = [ M(v.list()) for v in LL_vector.basis() ]
+ matrix_basis = [ M(v.list()) for v in LL_vector.basis() ]
- return matrices
+ return matrix_basis
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.
The Lyapunov rank of a proper polyhedral cone in `n` dimensions can
be any number between `1` and `n` inclusive, excluding `n-1`
- [Gowda/Tao]_ (by accident, this holds for the trivial cone in a
- trivial space as well)::
+ [Gowda/Tao]_. By accident, the `n-1` restriction will hold for the
+ 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: b = lyapunov_rank(K)
sage: n = K.lattice_dim()
- sage: 1 <= b and b <= n
+ sage: (n == 0 or 1 <= b) and b <= n
True
sage: b == n-1
False
- """
- V = K.lattice().vector_space()
+ In fact [Orlitzky/Gowda]_, no closed convex polyhedral cone can have
+ Lyapunov rank `n-1` in `n` dimensions::
- C_of_K = discrete_complementarity_set(K)
-
- matrices = [x.tensor_product(s) 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
- # many are linearly-indepenedent.
- #
- # The space W has the same base ring as V, but dimension
- # dim(V)^2. So it has the same dimension as the space of linear
- # transformations on V. In other words, it's just the right size
- # to create an isomorphism between it and our matrices.
- W = VectorSpace(V.base_ring(), V.dimension()**2)
+ sage: K = random_cone(max_dim=10)
+ sage: b = lyapunov_rank(K)
+ sage: n = K.lattice_dim()
+ sage: b == n-1
+ False
- def phi(m):
- r"""
- Convert a matrix to a vector isomorphically.
- """
- return W(m.list())
+ The calculation of the Lyapunov rank of an improper cone can be
+ reduced to that of a proper cone [Orlitzky/Gowda]_::
- vectors = [phi(m) for m in matrices]
+ 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 (W.dimension() - W.span(vectors).rank())
+ """
+ return len(LL(K))
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