from sage.all import *
-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:
-
- M. Orlitzky. The Lyapunov rank of an improper cone.
- http://www.optimization-online.org/DB_HTML/2015/10/5135.html
-
- EXAMPLES:
-
- The identity is always Lyapunov-like in a nontrivial space::
-
- sage: set_random_seed()
- sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 8)
- sage: L = identity_matrix(K.lattice_dim())
- sage: is_lyapunov_like(L,K)
- True
-
- As is the "zero" transformation::
-
- sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 8)
- sage: R = K.lattice().vector_space().base_ring()
- sage: L = zero_matrix(R, K.lattice_dim())
- sage: is_lyapunov_like(L,K)
- True
-
- Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like
- on ``K``::
-
- sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 6)
- sage: all([ is_lyapunov_like(L,K) for L in K.lyapunov_like_basis() ])
- True
-
- """
- return all([(L*x).inner_product(s) == 0
- for (x,s) in K.discrete_complementarity_set()])
-
-
-def random_element(K):
- r"""
- Return a random element of ``K`` from its ambient vector space.
-
- ALGORITHM:
-
- The cone ``K`` is specified in terms of its generators, so that
- ``K`` is equal to the convex conic combination of those generators.
- To choose a random element of ``K``, we assign random nonnegative
- coefficients to each generator of ``K`` and construct a new vector
- from the scaled rays.
-
- A vector, rather than a ray, is returned so that the element may
- have non-integer coordinates. Thus the element may have an
- arbitrarily small norm.
-
- EXAMPLES:
-
- A random element of the trivial cone is zero::
-
- sage: set_random_seed()
- sage: K = Cone([], ToricLattice(0))
- sage: random_element(K)
- ()
- sage: K = Cone([(0,)])
- sage: random_element(K)
- (0)
- sage: K = Cone([(0,0)])
- sage: random_element(K)
- (0, 0)
- sage: K = Cone([(0,0,0)])
- sage: random_element(K)
- (0, 0, 0)
-
- TESTS:
-
- Any cone should contain an element of itself::
-
- sage: set_random_seed()
- sage: K = random_cone(max_rays = 8)
- sage: K.contains(random_element(K))
- True
-
- """
- V = K.lattice().vector_space()
- F = V.base_ring()
- coefficients = [ F.random_element().abs() for i in range(K.nrays()) ]
- vector_gens = map(V, K.rays())
- scaled_gens = [ coefficients[i]*vector_gens[i]
- for i in range(len(vector_gens)) ]
-
- # Make sure we return a vector. Without the coercion, we might
- # return ``0`` when ``K`` has no rays.
- v = V(sum(scaled_gens))
- return v
-
-
-def positive_operator_gens(K):
- r"""
- Compute generators of the cone of positive operators on this cone.
-
- OUTPUT:
-
- A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``.
- Each matrix ``P`` in the list should have the property that ``P*x``
- is an element of ``K`` whenever ``x`` is an element of
- ``K``. Moreover, any nonnegative linear combination of these
- matrices shares the same property.
-
- EXAMPLES:
-
- The trivial cone in a trivial space has no positive operators::
-
- sage: K = Cone([], ToricLattice(0))
- sage: positive_operator_gens(K)
- []
-
- Positive operators on the nonnegative orthant are nonnegative matrices::
-
- sage: K = Cone([(1,)])
- sage: positive_operator_gens(K)
- [[1]]
-
- sage: K = Cone([(1,0),(0,1)])
- sage: positive_operator_gens(K)
- [
- [1 0] [0 1] [0 0] [0 0]
- [0 0], [0 0], [1 0], [0 1]
- ]
-
- Every operator is positive on the ambient vector space::
-
- sage: K = Cone([(1,),(-1,)])
- sage: K.is_full_space()
- True
- sage: positive_operator_gens(K)
- [[1], [-1]]
-
- sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
- sage: K.is_full_space()
- True
- sage: positive_operator_gens(K)
- [
- [1 0] [-1 0] [0 1] [ 0 -1] [0 0] [ 0 0] [0 0] [ 0 0]
- [0 0], [ 0 0], [0 0], [ 0 0], [1 0], [-1 0], [0 1], [ 0 -1]
- ]
-
- TESTS:
-
- A positive operator on a cone should send its generators into the cone::
-
- sage: K = random_cone(max_ambient_dim = 6)
- sage: pi_of_K = positive_operator_gens(K)
- sage: all([K.contains(p*x) for p in pi_of_K for x in K.rays()])
- True
-
- The dimension of the cone of positive operators is given by the
- corollary in my paper::
-
- sage: K = random_cone(max_ambient_dim = 6)
- sage: n = K.lattice_dim()
- sage: m = K.dim()
- sage: l = K.lineality()
- sage: pi_of_K = positive_operator_gens(K)
- sage: actual = Cone([p.list() for p in pi_of_K]).dim()
- sage: expected = n**2 - l*(n - l) - (n - m)*m
- sage: actual == expected
- True
-
- """
- # Matrices are not vectors in Sage, so we have to convert them
- # to vectors explicitly before we can find a basis. We need these
- # two values to construct the appropriate "long vector" space.
- F = K.lattice().base_field()
- n = K.lattice_dim()
-
- tensor_products = [ s.tensor_product(x) for x in K for s in K.dual() ]
-
- # Convert those tensor products to long vectors.
- W = VectorSpace(F, n**2)
- vectors = [ W(tp.list()) for tp in tensor_products ]
-
- # Create the *dual* cone of the positive operators, expressed as
- # long vectors..
- pi_dual = Cone(vectors, ToricLattice(W.dimension()))
-
- # Now compute the desired cone from its dual...
- pi_cone = pi_dual.dual()
-
- # And finally convert its rays back to matrix representations.
- M = MatrixSpace(F, n)
- return [ M(v.list()) for v in pi_cone.rays() ]
-
-
-def Z_transformation_gens(K):
- r"""
- Compute generators of the cone of Z-transformations on this cone.
-
- OUTPUT:
-
- A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``.
- Each matrix ``L`` in the list should have the property that
- ``(L*x).inner_product(s) <= 0`` whenever ``(x,s)`` is an element the
- discrete complementarity set of ``K``. Moreover, any nonnegative
- linear combination of these matrices shares the same property.
-
- EXAMPLES:
-
- Z-transformations on the nonnegative orthant are just Z-matrices.
- That is, matrices whose off-diagonal elements are nonnegative::
-
- sage: K = Cone([(1,0),(0,1)])
- sage: Z_transformation_gens(K)
- [
- [ 0 -1] [ 0 0] [-1 0] [1 0] [ 0 0] [0 0]
- [ 0 0], [-1 0], [ 0 0], [0 0], [ 0 -1], [0 1]
- ]
- sage: K = Cone([(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)])
- sage: all([ z[i][j] <= 0 for z in Z_transformation_gens(K)
- ....: for i in range(z.nrows())
- ....: for j in range(z.ncols())
- ....: if i != j ])
- True
-
- The trivial cone in a trivial space has no Z-transformations::
-
- sage: K = Cone([], ToricLattice(0))
- sage: Z_transformation_gens(K)
- []
-
- Z-transformations on a subspace are Lyapunov-like and vice-versa::
-
- sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
- sage: K.is_full_space()
- True
- sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
- sage: zs = span([ vector(z.list()) for z in Z_transformation_gens(K) ])
- sage: zs == lls
- True
-
- TESTS:
-
- The Z-property is possessed by every Z-transformation::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 6)
- sage: Z_of_K = Z_transformation_gens(K)
- sage: dcs = K.discrete_complementarity_set()
- sage: all([(z*x).inner_product(s) <= 0 for z in Z_of_K
- ....: for (x,s) in dcs])
- True
-
- The lineality space of Z is LL::
-
- sage: set_random_seed()
- sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 6)
- sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
- sage: z_cone = Cone([ z.list() for z in Z_transformation_gens(K) ])
- sage: z_cone.linear_subspace() == lls
- True
-
- """
- # Matrices are not vectors in Sage, so we have to convert them
- # to vectors explicitly before we can find a basis. We need these
- # two values to construct the appropriate "long vector" space.
- F = K.lattice().base_field()
- n = K.lattice_dim()
-
- # These tensor products contain generators for the dual cone of
- # the cross-positive transformations.
- tensor_products = [ s.tensor_product(x)
- for (x,s) in K.discrete_complementarity_set() ]
-
- # Turn our matrices into long vectors...
- W = VectorSpace(F, n**2)
- vectors = [ W(m.list()) for m in tensor_products ]
-
- # Create the *dual* cone of the cross-positive operators,
- # expressed as long vectors..
- Sigma_dual = Cone(vectors, lattice=ToricLattice(W.dimension()))
-
- # Now compute the desired cone from its dual...
- Sigma_cone = Sigma_dual.dual()
-
- # And finally convert its rays back to matrix representations.
- # But first, make them negative, so we get Z-transformations and
- # not cross-positive ones.
- M = MatrixSpace(F, n)
- return [ -M(v.list()) for v in Sigma_cone.rays() ]
+def LL_cone(K):
+ gens = K.lyapunov_like_basis()
+ L = ToricLattice(K.lattice_dim()**2)
+ return Cone(( g.list() for g in gens ), lattice=L, check=False)
+
+def Sigma_cone(K):
+ gens = K.cross_positive_operators_gens()
+ L = ToricLattice(K.lattice_dim()**2)
+ return Cone(( g.list() for g in gens ), lattice=L, check=False)
+
+def Z_cone(K):
+ gens = K.Z_operators_gens()
+ L = ToricLattice(K.lattice_dim()**2)
+ return Cone(( g.list() for g in gens ), lattice=L, check=False)
+
+def pi_cone(K1, K2=None):
+ if K2 is None:
+ K2 = K1
+ gens = K1.positive_operators_gens(K2)
+ L = ToricLattice(K1.lattice_dim()*K2.lattice_dim())
+ return Cone(( g.list() for g in gens ), lattice=L, check=False)
projects / sage.d.git / blobdiff