return all([(L*x).inner_product(s) == 0
for (x,s) in K.discrete_complementarity_set()])
-
-def positive_operator_gens(K1, K2 = None):
- r"""
- Compute generators of the cone of positive operators on this cone. A
- linear operator on a cone is positive if the image of the cone under
- the operator is a subset of the cone. This concept can be extended
- to two cones, where the image of the first cone under a positive
- operator is a subset of the second cone.
-
- INPUT:
-
- - ``K2`` -- (default: ``K1``) the codomain cone; the image of this
- cone under the returned operators is a subset of ``K2``.
-
- OUTPUT:
-
- A list of `m`-by-``n`` matrices where ``m == K2.lattice_dim()`` and
- ``n == K1.lattice_dim()``. Each matrix ``P`` in the list should have
- the property that ``P*x`` is an element of ``K2`` whenever ``x`` is
- an element of ``K1``. Moreover, any nonnegative linear combination of
- these matrices shares the same property.
-
- .. SEEALSO::
-
- :meth:`cross_positive_operator_gens`, :meth:`Z_operator_gens`,
-
- REFERENCES:
-
- .. [Tam]
- B.-S. Tam.
- Some results of polyhedral cones and simplicial cones.
- Linear and Multilinear Algebra, 4:4 (1977) 281--284.
-
- EXAMPLES:
-
- 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]
- ]
-
- The trivial cone in a trivial space has no positive operators::
-
- sage: K = Cone([], ToricLattice(0))
- sage: positive_operator_gens(K)
- []
-
- Every operator is positive on the trivial cone::
-
- sage: K = Cone([(0,)])
- sage: positive_operator_gens(K)
- [[1], [-1]]
-
- sage: K = Cone([(0,0)])
- sage: K.is_trivial()
- 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]
- ]
-
- 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]
- ]
-
- A non-obvious application is to find the positive operators on the
- right half-plane::
-
- sage: K = Cone([(1,0),(0,1),(0,-1)])
- sage: positive_operator_gens(K)
- [
- [1 0] [0 0] [ 0 0] [0 0] [ 0 0]
- [0 0], [1 0], [-1 0], [0 1], [ 0 -1]
- ]
-
- TESTS:
-
- Each positive operator generator should send the generators of one
- cone into the other cone::
-
- sage: set_random_seed()
- sage: K1 = random_cone(max_ambient_dim=4)
- sage: K2 = random_cone(max_ambient_dim=4)
- sage: pi_K1_K2 = positive_operator_gens(K1,K2)
- sage: all([ K2.contains(P*x) for P in pi_K1_K2 for x in K1 ])
- True
-
- Each positive operator generator should send a random element of one
- cone into the other cone::
-
- sage: set_random_seed()
- sage: K1 = random_cone(max_ambient_dim=4)
- sage: K2 = random_cone(max_ambient_dim=4)
- sage: pi_K1_K2 = positive_operator_gens(K1,K2)
- sage: all([ K2.contains(P*K1.random_element(QQ)) for P in pi_K1_K2 ])
- True
-
- A random element of the positive operator cone should send the
- generators of one cone into the other cone::
-
- sage: set_random_seed()
- sage: K1 = random_cone(max_ambient_dim=4)
- sage: K2 = random_cone(max_ambient_dim=4)
- sage: pi_K1_K2 = positive_operator_gens(K1,K2)
- sage: L = ToricLattice(K1.lattice_dim() * K2.lattice_dim())
- sage: pi_cone = Cone([ g.list() for g in pi_K1_K2 ],
- ....: lattice=L,
- ....: check=False)
- sage: P = matrix(K2.lattice_dim(),
- ....: K1.lattice_dim(),
- ....: pi_cone.random_element(QQ).list())
- sage: all([ K2.contains(P*x) for x in K1 ])
- True
-
- A random element of the positive operator cone should send a random
- element of one cone into the other cone::
-
- sage: set_random_seed()
- sage: K1 = random_cone(max_ambient_dim=4)
- sage: K2 = random_cone(max_ambient_dim=4)
- sage: pi_K1_K2 = positive_operator_gens(K1,K2)
- sage: L = ToricLattice(K1.lattice_dim() * K2.lattice_dim())
- sage: pi_cone = Cone([ g.list() for g in pi_K1_K2 ],
- ....: lattice=L,
- ....: check=False)
- sage: P = matrix(K2.lattice_dim(),
- ....: K1.lattice_dim(),
- ....: pi_cone.random_element(QQ).list())
- sage: K2.contains(P*K1.random_element(ring=QQ))
- True
-
- The lineality space of the dual of the cone of positive operators
- can be computed from the lineality spaces of the cone and its dual::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: pi_of_K = positive_operator_gens(K)
- sage: L = ToricLattice(K.lattice_dim()**2)
- sage: pi_cone = Cone([ g.list() for g in pi_of_K ],
- ....: lattice=L,
- ....: check=False)
- sage: actual = pi_cone.dual().linear_subspace()
- sage: U1 = [ vector((s.tensor_product(x)).list())
- ....: for x in K.lines()
- ....: for s in K.dual() ]
- sage: U2 = [ vector((s.tensor_product(x)).list())
- ....: for x in K
- ....: for s in K.dual().lines() ]
- sage: expected = pi_cone.lattice().vector_space().span(U1 + U2)
- sage: actual == expected
- True
-
- The lineality of the dual of the cone of positive operators
- is known from its lineality space::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: n = K.lattice_dim()
- sage: m = K.dim()
- sage: l = K.lineality()
- sage: pi_of_K = positive_operator_gens(K)
- sage: L = ToricLattice(n**2)
- sage: pi_cone = Cone([p.list() for p in pi_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: actual = pi_cone.dual().lineality()
- sage: expected = l*(m - l) + m*(n - m)
- sage: actual == expected
- True
-
- The dimension of the cone of positive operators is given by the
- corollary in my paper::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: n = K.lattice_dim()
- sage: m = K.dim()
- sage: l = K.lineality()
- sage: pi_of_K = positive_operator_gens(K)
- sage: L = ToricLattice(n**2)
- sage: pi_cone = Cone([p.list() for p in pi_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: actual = pi_cone.dim()
- sage: expected = n**2 - l*(m - l) - (n - m)*m
- sage: actual == expected
- True
-
- The trivial cone, full space, and half-plane all give rise to the
- expected dimensions::
-
- sage: n = ZZ.random_element().abs()
- sage: K = Cone([[0] * n], ToricLattice(n))
- sage: K.is_trivial()
- True
- sage: L = ToricLattice(n^2)
- sage: pi_of_K = positive_operator_gens(K)
- sage: pi_cone = Cone([p.list() for p in pi_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: actual = pi_cone.dim()
- sage: actual == n^2
- True
- sage: K = K.dual()
- sage: K.is_full_space()
- True
- sage: pi_of_K = positive_operator_gens(K)
- sage: pi_cone = Cone([p.list() for p in pi_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: actual = pi_cone.dim()
- sage: actual == n^2
- True
- sage: K = Cone([(1,0),(0,1),(0,-1)])
- sage: pi_of_K = positive_operator_gens(K)
- sage: actual = Cone([p.list() for p in pi_of_K], check=False).dim()
- sage: actual == 3
- True
-
- The lineality of the cone of positive operators follows from the
- description of its generators::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: n = K.lattice_dim()
- sage: pi_of_K = positive_operator_gens(K)
- sage: L = ToricLattice(n**2)
- sage: pi_cone = Cone([p.list() for p in pi_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: actual = pi_cone.lineality()
- sage: expected = n**2 - K.dim()*K.dual().dim()
- sage: actual == expected
- True
-
- The trivial cone, full space, and half-plane all give rise to the
- expected linealities::
-
- sage: n = ZZ.random_element().abs()
- sage: K = Cone([[0] * n], ToricLattice(n))
- sage: K.is_trivial()
- True
- sage: L = ToricLattice(n^2)
- sage: pi_of_K = positive_operator_gens(K)
- sage: pi_cone = Cone([p.list() for p in pi_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: actual = pi_cone.lineality()
- sage: actual == n^2
- True
- sage: K = K.dual()
- sage: K.is_full_space()
- True
- sage: pi_of_K = positive_operator_gens(K)
- sage: pi_cone = Cone([p.list() for p in pi_of_K], lattice=L)
- sage: pi_cone.lineality() == n^2
- True
- sage: K = Cone([(1,0),(0,1),(0,-1)])
- sage: pi_of_K = positive_operator_gens(K)
- sage: pi_cone = Cone([p.list() for p in pi_of_K], check=False)
- sage: actual = pi_cone.lineality()
- sage: actual == 2
- True
-
- A cone is proper if and only if its cone of positive operators
- is proper::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: pi_of_K = positive_operator_gens(K)
- sage: L = ToricLattice(K.lattice_dim()**2)
- sage: pi_cone = Cone([p.list() for p in pi_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: K.is_proper() == pi_cone.is_proper()
- True
-
- The positive operators of a permuted cone can be obtained by
- conjugation::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: L = ToricLattice(K.lattice_dim()**2)
- sage: p = SymmetricGroup(K.lattice_dim()).random_element().matrix()
- sage: pK = Cone([ p*k for k in K ], K.lattice(), check=False)
- sage: pi_of_pK = positive_operator_gens(pK)
- sage: actual = Cone([t.list() for t in pi_of_pK],
- ....: lattice=L,
- ....: check=False)
- sage: pi_of_K = positive_operator_gens(K)
- sage: expected = Cone([(p*t*p.inverse()).list() for t in pi_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: actual.is_equivalent(expected)
- True
-
- A transformation is positive on a cone if and only if its adjoint is
- positive on the dual of that cone::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: F = K.lattice().vector_space().base_field()
- sage: n = K.lattice_dim()
- sage: L = ToricLattice(n**2)
- sage: W = VectorSpace(F, n**2)
- sage: pi_of_K = positive_operator_gens(K)
- sage: pi_of_K_star = positive_operator_gens(K.dual())
- sage: pi_cone = Cone([p.list() for p in pi_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: pi_star = Cone([p.list() for p in pi_of_K_star],
- ....: lattice=L,
- ....: check=False)
- sage: M = MatrixSpace(F, n)
- sage: L = M(pi_cone.random_element(ring=QQ).list())
- sage: pi_star.contains(W(L.transpose().list()))
- True
-
- sage: L = W.random_element()
- sage: L_star = W(M(L.list()).transpose().list())
- sage: pi_cone.contains(L) == pi_star.contains(L_star)
- True
-
- The Lyapunov rank of the positive operator cone is the product of
- the Lyapunov ranks of the associated cones if they're all proper::
-
- sage: K1 = random_cone(max_ambient_dim=4,
- ....: strictly_convex=True,
- ....: solid=True)
- sage: K2 = random_cone(max_ambient_dim=4,
- ....: strictly_convex=True,
- ....: solid=True)
- sage: pi_K1_K2 = positive_operator_gens(K1,K2)
- sage: L = ToricLattice(K1.lattice_dim() * K2.lattice_dim())
- sage: pi_cone = Cone([ g.list() for g in pi_K1_K2 ],
- ....: lattice=L,
- ....: check=False)
- sage: beta1 = K1.lyapunov_rank()
- sage: beta2 = K2.lyapunov_rank()
- sage: pi_cone.lyapunov_rank() == beta1*beta2
- True
-
- The Lyapunov-like operators on a proper polyhedral positive operator
- cone can be computed from the Lyapunov-like operators on the cones
- with respect to which the operators are positive::
-
- sage: K1 = random_cone(max_ambient_dim=4,
- ....: strictly_convex=True,
- ....: solid=True)
- sage: K2 = random_cone(max_ambient_dim=4,
- ....: strictly_convex=True,
- ....: solid=True)
- sage: pi_K1_K2 = positive_operator_gens(K1,K2)
- sage: F = K1.lattice().base_field()
- sage: m = K1.lattice_dim()
- sage: n = K2.lattice_dim()
- sage: L = ToricLattice(m*n)
- sage: M1 = MatrixSpace(F, m, m)
- sage: M2 = MatrixSpace(F, n, n)
- sage: LL_K1 = [ M1(x.list()) for x in K1.dual().lyapunov_like_basis() ]
- sage: LL_K2 = [ M2(x.list()) for x in K2.lyapunov_like_basis() ]
- sage: tps = [ s.tensor_product(x) for x in LL_K1 for s in LL_K2 ]
- sage: W = VectorSpace(F, (m**2)*(n**2))
- sage: expected = span(F, [ W(x.list()) for x in tps ])
- sage: pi_cone = Cone([p.list() for p in pi_K1_K2],
- ....: lattice=L,
- ....: check=False)
- sage: LL_pi = pi_cone.lyapunov_like_basis()
- sage: actual = span(F, [ W(x.list()) for x in LL_pi ])
- sage: actual == expected
- True
-
- """
- if K2 is None:
- K2 = K1
-
- # 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 = K1.lattice().base_field()
- n = K1.lattice_dim()
- m = K2.lattice_dim()
-
- tensor_products = [ s.tensor_product(x) for x in K1 for s in K2.dual() ]
-
- # Convert those tensor products to long vectors.
- W = VectorSpace(F, n*m)
- vectors = [ W(tp.list()) for tp in tensor_products ]
-
- check = True
- if K1.is_proper() and K2.is_proper():
- # All of the generators involved are extreme vectors and
- # therefore minimal. If this cone is neither solid nor
- # strictly convex, then the tensor product of ``s`` and ``x``
- # is the same as that of ``-s`` and ``-x``. However, as a
- # /set/, ``tensor_products`` may still be minimal.
- check = False
-
- # Create the dual cone of the positive operators, expressed as
- # long vectors.
- pi_dual = Cone(vectors, ToricLattice(W.dimension()), check=check)
-
- # 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, m, n)
- return [ M(v.list()) for v in pi_cone ]
-
-
-def cross_positive_operator_gens(K):
- r"""
- Compute generators of the cone of cross-positive operators 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 of
- this cone's :meth:`discrete_complementarity_set`. Moreover, any
- conic (nonnegative linear) combination of these matrices shares the
- same property.
-
- .. SEEALSO::
-
- :meth:`positive_operator_gens`, :meth:`Z_operator_gens`,
-
- EXAMPLES:
-
- Cross-positive operators on the nonnegative orthant are negations
- of Z-matrices; that is, matrices whose off-diagonal elements are
- nonnegative::
-
- sage: K = Cone([(1,0),(0,1)])
- sage: cross_positive_operator_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([ c[i][j] >= 0 for c in cross_positive_operator_gens(K)
- ....: for i in range(c.nrows())
- ....: for j in range(c.ncols())
- ....: if i != j ])
- True
-
- The trivial cone in a trivial space has no cross-positive operators::
-
- sage: K = Cone([], ToricLattice(0))
- sage: cross_positive_operator_gens(K)
- []
-
- Every operator is a cross-positive operator on the ambient vector
- space::
-
- sage: K = Cone([(1,),(-1,)])
- sage: K.is_full_space()
- True
- sage: cross_positive_operator_gens(K)
- [[1], [-1]]
-
- sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
- sage: K.is_full_space()
- True
- sage: cross_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]
- ]
-
- A non-obvious application is to find the cross-positive operators
- on the right half-plane::
-
- sage: K = Cone([(1,0),(0,1),(0,-1)])
- sage: cross_positive_operator_gens(K)
- [
- [1 0] [-1 0] [0 0] [ 0 0] [0 0] [ 0 0]
- [0 0], [ 0 0], [1 0], [-1 0], [0 1], [ 0 -1]
- ]
-
- Cross-positive operators 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: cs = span([ vector(c.list()) for c in cross_positive_operator_gens(K) ])
- sage: cs == lls
- True
-
- TESTS:
-
- The cross-positive property is possessed by every cross-positive
- operator::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: Sigma_of_K = cross_positive_operator_gens(K)
- sage: dcs = K.discrete_complementarity_set()
- sage: all([(c*x).inner_product(s) >= 0 for c in Sigma_of_K
- ....: for (x,s) in dcs])
- True
-
- The lineality space of the cone of cross-positive operators is the
- space of Lyapunov-like operators::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: L = ToricLattice(K.lattice_dim()**2)
- sage: Sigma_cone = Cone([ c.list() for c in cross_positive_operator_gens(K) ],
- ....: lattice=L,
- ....: check=False)
- sage: ll_basis = [ vector(l.list()) for l in K.lyapunov_like_basis() ]
- sage: lls = L.vector_space().span(ll_basis)
- sage: Sigma_cone.linear_subspace() == lls
- True
-
- The lineality of the cross-positive operators on a cone is the
- Lyapunov rank of that cone::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: Sigma_of_K = cross_positive_operator_gens(K)
- sage: L = ToricLattice(K.lattice_dim()**2)
- sage: Sigma_cone = Cone([ c.list() for c in Sigma_of_K ],
- ....: lattice=L,
- ....: check=False)
- sage: Sigma_cone.lineality() == K.lyapunov_rank()
- True
-
- The lineality spaces of the duals of the positive and cross-positive
- operator cones are equal. From this it follows that the dimensions of
- the cross-positive operator cone and positive operator cone are equal::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: pi_of_K = positive_operator_gens(K)
- sage: Sigma_of_K = cross_positive_operator_gens(K)
- sage: L = ToricLattice(K.lattice_dim()**2)
- sage: pi_cone = Cone([p.list() for p in pi_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: Sigma_cone = Cone([ c.list() for c in Sigma_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: pi_cone.dim() == Sigma_cone.dim()
- True
- sage: pi_star = pi_cone.dual()
- sage: sigma_star = Sigma_cone.dual()
- sage: pi_star.linear_subspace() == sigma_star.linear_subspace()
- True
-
- The trivial cone, full space, and half-plane all give rise to the
- expected dimensions::
-
- sage: n = ZZ.random_element().abs()
- sage: K = Cone([[0] * n], ToricLattice(n))
- sage: K.is_trivial()
- True
- sage: L = ToricLattice(n^2)
- sage: Sigma_of_K = cross_positive_operator_gens(K)
- sage: Sigma_cone = Cone([c.list() for c in Sigma_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: actual = Sigma_cone.dim()
- sage: actual == n^2
- True
- sage: K = K.dual()
- sage: K.is_full_space()
- True
- sage: Sigma_of_K = cross_positive_operator_gens(K)
- sage: Sigma_cone = Cone([ c.list() for c in Sigma_of_K ],
- ....: lattice=L,
- ....: check=False)
- sage: actual = Sigma_cone.dim()
- sage: actual == n^2
- True
- sage: K = Cone([(1,0),(0,1),(0,-1)])
- sage: Sigma_of_K = cross_positive_operator_gens(K)
- sage: Sigma_cone = Cone([ c.list() for c in Sigma_of_K ], check=False)
- sage: Sigma_cone.dim() == 3
- True
-
- The cross-positive operators of a permuted cone can be obtained by
- conjugation::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: L = ToricLattice(K.lattice_dim()**2)
- sage: p = SymmetricGroup(K.lattice_dim()).random_element().matrix()
- sage: pK = Cone([ p*k for k in K ], K.lattice(), check=False)
- sage: Sigma_of_pK = cross_positive_operator_gens(pK)
- sage: actual = Cone([t.list() for t in Sigma_of_pK],
- ....: lattice=L,
- ....: check=False)
- sage: Sigma_of_K = cross_positive_operator_gens(K)
- sage: expected = Cone([ (p*t*p.inverse()).list() for t in Sigma_of_K ],
- ....: lattice=L,
- ....: check=False)
- sage: actual.is_equivalent(expected)
- True
-
- An operator is cross-positive on a cone if and only if its
- adjoint is cross-positive on the dual of that cone::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: F = K.lattice().vector_space().base_field()
- sage: n = K.lattice_dim()
- sage: L = ToricLattice(n**2)
- sage: W = VectorSpace(F, n**2)
- sage: Sigma_of_K = cross_positive_operator_gens(K)
- sage: Sigma_of_K_star = cross_positive_operator_gens(K.dual())
- sage: Sigma_cone = Cone([ p.list() for p in Sigma_of_K ],
- ....: lattice=L,
- ....: check=False)
- sage: Sigma_star = Cone([ p.list() for p in Sigma_of_K_star ],
- ....: lattice=L,
- ....: check=False)
- sage: M = MatrixSpace(F, n)
- sage: L = M(Sigma_cone.random_element(ring=QQ).list())
- sage: Sigma_star.contains(W(L.transpose().list()))
- True
-
- sage: L = W.random_element()
- sage: L_star = W(M(L.list()).transpose().list())
- sage: Sigma_cone.contains(L) == Sigma_star.contains(L_star)
- 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 operators.
- 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 ]
-
- check = True
- if K.is_proper():
- # All of the generators involved are extreme vectors and
- # therefore minimal. If this cone is neither solid nor
- # strictly convex, then the tensor product of ``s`` and ``x``
- # is the same as that of ``-s`` and ``-x``. However, as a
- # /set/, ``tensor_products`` may still be minimal.
- check = False
-
- # Create the dual cone of the cross-positive operators,
- # expressed as long vectors.
- Sigma_dual = Cone(vectors, lattice=ToricLattice(W.dimension()), check=check)
-
- # Now compute the desired cone from its dual...
- Sigma_cone = Sigma_dual.dual()
-
- # And finally convert its rays back to matrix representations.
- M = MatrixSpace(F, n)
- return [ M(v.list()) for v in Sigma_cone ]
-
-
-def Z_operator_gens(K):
- r"""
- Compute generators of the cone of Z-operators on this cone.
-
- The Z-operators on a cone generalize the Z-matrices over the
- nonnegative orthant. They are simply negations of the
- :meth:`cross_positive_operators`.
-
- 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 of
- this cone's :meth:`discrete_complementarity_set`. Moreover, any
- conic (nonnegative linear) combination of these matrices shares the
- same property.
-
- .. SEEALSO::
-
- :meth:`positive_operator_gens`, :meth:`cross_positive_operator_gens`,
-
- TESTS:
-
- The Z-property is possessed by every Z-operator::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: Z_of_K = Z_operator_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
- """
- return [ -cp for cp in cross_positive_operator_gens(K) ]
-
-
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 = cross_positive_operator_gens(K)
+ gens = K.cross_positive_operator_gens()
L = ToricLattice(K.lattice_dim()**2)
return Cone([ g.list() for g in gens ], lattice=L, check=False)
def Z_cone(K):
- gens = Z_operator_gens(K)
+ gens = K.Z_operator_gens()
L = ToricLattice(K.lattice_dim()**2)
return Cone([ g.list() for g in gens ], lattice=L, check=False)
-def pi_cone(K):
- gens = positive_operator_gens(K)
- L = ToricLattice(K.lattice_dim()**2)
+def pi_cone(K1, K2=None):
+ if K2 is None:
+ K2 = K1
+ gens = K1.positive_operator_gens(K2)
+ L = ToricLattice(K1.lattice_dim()*K2.lattice_dim())
return Cone([ g.list() for g in gens ], lattice=L, check=False)
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