+
+
+
+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_rays = 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_rays = 5)
+ 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_rays = 5)
+ 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_operators(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_operators(K)
+ []
+
+ Positive operators on the nonnegative orthant are nonnegative matrices::
+
+ sage: K = Cone([(1,)])
+ sage: positive_operators(K)
+ [[1]]
+
+ sage: K = Cone([(1,0),(0,1)])
+ sage: positive_operators(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_operators(K)
+ [[1], [-1]]
+
+ sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
+ sage: K.is_full_space()
+ True
+ sage: positive_operators(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_operators(K)
+ sage: all([K.contains(p*x) for p in pi_of_K for x in K.rays()])
+ True
+
+ """
+ # 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.
+ V = K.lattice().vector_space()
+ W = VectorSpace(V.base_ring(), V.dimension()**2)
+
+ tensor_products = [ s.tensor_product(x) for x in K for s in K.dual() ]
+
+ # Turn our matrices into long vectors...
+ vectors = [ W(m.list()) for m in tensor_products ]
+
+ # Create the *dual* cone of the positive operators, expressed as
+ # long vectors..
+ L = ToricLattice(W.dimension())
+ pi_dual = Cone(vectors, lattice=L)
+
+ # Now compute the desired cone from its dual...
+ pi_cone = pi_dual.dual()
+
+ # And finally convert its rays back to matrix representations.
+ M = MatrixSpace(V.base_ring(), V.dimension())
+
+ return [ M(v.list()) for v in pi_cone.rays() ]
+
+
+def Z_transformations(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_transformations(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_transformations(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_transformations(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_transformations(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_transformations(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_transformations(K) ])
+ sage: z_cone.linear_subspace() == lls
+ True
+
+ """
+ # 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.
+ V = K.lattice().vector_space()
+ W = VectorSpace(V.base_ring(), V.dimension()**2)
+
+ C_of_K = K.discrete_complementarity_set()
+ tensor_products = [ s.tensor_product(x) for (x,s) in C_of_K ]
+
+ # Turn our matrices into long vectors...
+ vectors = [ W(m.list()) for m in tensor_products ]
+
+ # Create the *dual* cone of the cross-positive operators,
+ # expressed as long vectors..
+ L = ToricLattice(W.dimension())
+ Sigma_dual = Cone(vectors, lattice=L)
+
+ # 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(V.base_ring(), V.dimension())
+
+ return [ -M(v.list()) for v in Sigma_cone.rays() ]