from sage.all import *
-def is_lyapunov_like(L,K):
+def is_positive_on(L,K):
r"""
- Determine whether or not ``L`` is Lyapunov-like on ``K``.
+ Determine whether or not ``L`` is positive 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.
+ We say that ``L`` is positive on ``K`` if `L\left\lparen x
+ \right\rparen` belongs to ``K`` for all `x` in ``K``. This
+ property need only be checked for generators of ``K``.
INPUT:
OUTPUT:
- ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``,
+ ``True`` if it can be proven that ``L`` is positive on ``K``,
and ``False`` otherwise.
.. WARNING::
- If this function returns ``True``, then ``L`` is Lyapunov-like
+ If this function returns ``True``, then ``L`` is positive
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"
+ positive 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.
+ product is nonnegative.
- REFERENCES:
+ EXAMPLES:
- M. Orlitzky. The Lyapunov rank of an improper cone.
- http://www.optimization-online.org/DB_HTML/2015/10/5135.html
+ Positive operators on the nonnegative orthant are nonnegative
+ matrices::
- EXAMPLES:
+ sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
+ sage: L = random_matrix(QQ,3).apply_map(abs)
+ sage: is_positive_on(L,K)
+ True
- The identity is always Lyapunov-like in a nontrivial space::
+ TESTS:
+
+ The identity is always positive in a nontrivial space::
sage: set_random_seed()
- sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 8)
+ 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)
+ sage: is_positive_on(L,K)
True
As is the "zero" transformation::
- sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 8)
+ 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)
+ sage: is_positive_on(L,K)
True
- Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like
- on ``K``::
+ Everything in ``K.positive_operators_gens()`` should be
+ positive 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() ])
+ sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6)
+ sage: all([ is_positive_on(L,K)
+ ....: for L in K.positive_operators_gens() ])
+ True
+ sage: all([ is_positive_on(L.change_ring(SR),K)
+ ....: for L in K.positive_operators_gens() ])
True
"""
- return all([(L*x).inner_product(s) == 0
- for (x,s) in K.discrete_complementarity_set()])
+ if L.base_ring().is_exact():
+ # This could potentially be extended to other types of ``K``...
+ return all([ L*x in K for x in K ])
+ elif L.base_ring() is SR:
+ # Fall back to inequality-checking when the entries of ``L``
+ # might be symbolic.
+ return all([ s*(L*x) >= 0 for x in K for s in K.dual() ])
+ else:
+ # The only inexact ring that we're willing to work with is SR,
+ # since it can still be exact when working with symbolic
+ # constants like pi and e.
+ raise ValueError('base ring of operator L is neither SR nor exact')
+
+
+def is_cross_positive_on(L,K):
+ r"""
+ Determine whether or not ``L`` is cross-positive on ``K``.
+ We say that ``L`` is cross-positive on ``K`` if `\left\langle
+ L\left\lparenx\right\rparen,s\right\rangle \ge 0` for all pairs
+ `\left\langle x,s \right\rangle` in the complementarity set of
+ ``K``. This property need only be checked for generators of
+ ``K`` and its dual.
-def random_element(K):
- r"""
- Return a random element of ``K`` from its ambient vector space.
+ INPUT:
+
+ - ``L`` -- A linear transformation or matrix.
+
+ - ``K`` -- A polyhedral closed convex cone.
+
+ OUTPUT:
- ALGORITHM:
+ ``True`` if it can be proven that ``L`` is cross-positive on ``K``,
+ and ``False`` otherwise.
- 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.
+ .. WARNING::
- 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.
+ If this function returns ``True``, then ``L`` is cross-positive
+ on ``K``. However, if ``False`` is returned, that could mean one
+ of two things. The first is that ``L`` is definitely not
+ cross-positive 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 nonnegative.
EXAMPLES:
- A random element of the trivial cone is zero::
+ The identity is always cross-positive in a nontrivial space::
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)
+ sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
+ sage: L = identity_matrix(K.lattice_dim())
+ sage: is_cross_positive_on(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_cross_positive_on(L,K)
+ True
TESTS:
- Any cone should contain an element of itself::
+ Everything in ``K.cross_positive_operators_gens()`` should be
+ cross-positive on ``K``::
- sage: set_random_seed()
- sage: K = random_cone(max_rays = 8)
- sage: K.contains(random_element(K))
+ sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6)
+ sage: all([ is_cross_positive_on(L,K)
+ ....: for L in K.cross_positive_operators_gens() ])
+ True
+ sage: all([ is_cross_positive_on(L.change_ring(SR),K)
+ ....: for L in K.cross_positive_operators_gens() ])
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)) ]
+ if L.base_ring().is_exact() or L.base_ring() is SR:
+ return all([ s*(L*x) >= 0
+ for (x,s) in K.discrete_complementarity_set() ])
+ else:
+ # The only inexact ring that we're willing to work with is SR,
+ # since it can still be exact when working with symbolic
+ # constants like pi and e.
+ raise ValueError('base ring of operator L is neither SR nor exact')
- # 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):
+def is_Z_on(L,K):
r"""
- Compute generators of the cone of positive operators on this cone.
+ Determine whether or not ``L`` is a Z-operator on ``K``.
- OUTPUT:
+ We say that ``L`` is a Z-operator on ``K`` if `\left\langle
+ L\left\lparenx\right\rparen,s\right\rangle \le 0` for all pairs
+ `\left\langle x,s \right\rangle` in the complementarity set of
+ ``K``. It is known that this property need only be
+ checked for generators of ``K`` and its dual.
- 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.
+ A matrix is a Z-operator on ``K`` if and only if its negation is a
+ cross-positive operator on ``K``.
- EXAMPLES:
+ INPUT:
- The trivial cone in a trivial space has no positive operators::
+ - ``L`` -- A linear transformation or matrix.
- sage: K = Cone([], ToricLattice(0))
- sage: positive_operator_gens(K)
- []
+ - ``K`` -- A polyhedral closed convex cone.
- Positive operators on the nonnegative orthant are nonnegative matrices::
+ OUTPUT:
- sage: K = Cone([(1,)])
- sage: positive_operator_gens(K)
- [[1]]
+ ``True`` if it can be proven that ``L`` is a Z-operator on ``K``,
+ and ``False`` otherwise.
- 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]
- ]
+ .. WARNING::
- Every operator is positive on the ambient vector space::
+ If this function returns ``True``, then ``L`` is a Z-operator
+ on ``K``. However, if ``False`` is returned, that could mean one
+ of two things. The first is that ``L`` is definitely not
+ a Z-operator 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 nonnegative.
+
+ EXAMPLES:
- sage: K = Cone([(1,),(-1,)])
- sage: K.is_full_space()
+ The identity is always a Z-operator 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_Z_on(L,K)
True
- sage: positive_operator_gens(K)
- [[1], [-1]]
- sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
- sage: K.is_full_space()
+ 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_Z_on(L,K)
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::
+ Everything in ``K.Z_operators_gens()`` should be a Z-operator
+ on ``K``::
- 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()])
+ sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6)
+ sage: all([ is_Z_on(L,K)
+ ....: for L in K.Z_operators_gens() ])
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
+ sage: all([ is_Z_on(L.change_ring(SR),K)
+ ....: for L in K.Z_operators_gens() ])
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() ]
+ return is_cross_positive_on(-L,K)
- # 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()))
+def is_lyapunov_like_on(L,K):
+ r"""
+ Determine whether or not ``L`` is Lyapunov-like on ``K``.
- # Now compute the desired cone from its dual...
- pi_cone = pi_dual.dual()
+ 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``. This property need only be checked for generators of
+ ``K`` and its dual.
- # And finally convert its rays back to matrix representations.
- M = MatrixSpace(F, n)
- return [ M(v.list()) for v in pi_cone.rays() ]
+ INPUT:
+ - ``L`` -- A linear transformation or matrix.
-def Z_transformation_gens(K):
- r"""
- Compute generators of the cone of Z-transformations on this cone.
+ - ``K`` -- A polyhedral closed convex 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:
+ ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``,
+ and ``False`` otherwise.
- 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
+ .. WARNING::
- The trivial cone in a trivial space has no Z-transformations::
+ 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.
- sage: K = Cone([], ToricLattice(0))
- sage: Z_transformation_gens(K)
- []
+ EXAMPLES:
- Z-transformations on a subspace are Lyapunov-like and vice-versa::
+ Lyapunov-like operators on the nonnegative orthant are diagonal
+ matrices::
- 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
+ sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
+ sage: L = diagonal_matrix(random_vector(QQ,3))
+ sage: is_lyapunov_like_on(L,K)
True
TESTS:
- The Z-property is possessed by every Z-transformation::
+ The identity is always Lyapunov-like in a nontrivial space::
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])
+ sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
+ sage: L = identity_matrix(K.lattice_dim())
+ sage: is_lyapunov_like_on(L,K)
True
- The lineality space of Z is LL::
+ As is the "zero" transformation::
- 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
+ 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_on(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_on(L,K)
+ ....: for L in K.lyapunov_like_basis() ])
+ True
+ sage: all([ is_lyapunov_like_on(L.change_ring(SR),K)
+ ....: for L in K.lyapunov_like_basis() ])
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() ]
+ if L.base_ring().is_exact() or L.base_ring() is SR:
+ # The "fast method" of creating a vector space based on a
+ # ``lyapunov_like_basis`` is actually slower than this.
+ return all([ s*(L*x) == 0
+ for (x,s) in K.discrete_complementarity_set() ])
+ else:
+ # The only inexact ring that we're willing to work with is SR,
+ # since it can still be exact when working with symbolic
+ # constants like pi and e.
+ raise ValueError('base ring of operator L is neither SR nor exact')
+
+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