sage: set_random_seed()
sage: K = random_cone(max_ambient_dim=8)
sage: (P,S) = motzkin_decomposition(K)
- sage: x = K.random_element()
+ sage: x = K.random_element(ring=QQ)
sage: P.contains(x) or S.contains(x)
True
sage: x.is_zero() or (P.contains(x) != S.contains(x))
sage: S.lineality() == S.dim()
True
+ A strictly convex cone should be equal to its strictly convex component::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=8, strictly_convex=True)
+ sage: (P,_) = motzkin_decomposition(K)
+ sage: K.is_equivalent(P)
+ True
+
The generators of the components are obtained from orthogonal
projections of the original generators [Stoer-Witzgall]_::
sage: S.is_equivalent(expected_S)
True
"""
- # The lines() method only gives us one generator for each line,
- # so we negate the result and combine everything for the full set.
- S = Cone([p*l for p in [1,-1] for l in K.lines()], K.lattice())
+ # The lines() method only returns one generator per line. For a true
+ # line, we also need a generator pointing in the opposite direction.
+ S_gens = [ direction*gen for direction in [1,-1] for gen in K.lines() ]
+ S = Cone(S_gens, K.lattice(), check=False)
# Since ``S`` is a subspace, the rays of its dual generate its
# orthogonal complement.
- P = K.intersection( Cone(S.dual(), K.lattice()) )
+ S_perp = Cone(S.dual(), K.lattice(), check=False)
+ P = K.intersection(S_perp)
return (P,S)
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,)])
[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,)])
[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:
- A positive operator on a cone should send its generators into the cone::
+ Each positive operator generator should send the generators of the
+ cone into the cone::
sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=5)
+ sage: K = random_cone(max_ambient_dim=4)
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: all([ K.contains(P*x) for P in pi_of_K for x in K ])
+ True
+
+ Each positive operator generator should send a random element of the
+ cone into the cone::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: all([ K.contains(P*K.random_element(QQ)) for P in pi_of_K ])
+ True
+
+ A random element of the positive operator cone should send the
+ generators of the cone into the cone::
+
+ 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: P = matrix(K.lattice_dim(), pi_cone.random_element(QQ).list())
+ sage: all([ K.contains(P*x) for x in K ])
+ True
+
+ A random element of the positive operator cone should send a random
+ element of the cone into the cone::
+
+ 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: P = matrix(K.lattice_dim(), pi_cone.random_element(QQ).list())
+ sage: K.contains(P*K.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=5)
+ 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: actual = Cone([p.list() for p in pi_of_K], lattice=L).dim()
+ 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 lineality of the cone of positive operators is given by the
- corollary in my paper::
+ 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=5)
+ 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: actual = Cone([p.list() for p in pi_of_K], lattice=L).lineality()
+ 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 cone ``K`` is proper if and only if the cone of positive
- operators on ``K`` is proper::
+ 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=5)
+ 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)
+ 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
"""
# Matrices are not vectors in Sage, so we have to convert them
# to vectors explicitly before we can find a basis. We need these
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()))
+ check = True
+ if K.is_solid() or K.is_strictly_convex():
+ # The lineality space of either ``K`` or ``K.dual()`` is
+ # trivial and it's easy to show that our generating set is
+ # minimal. I would love a proof that this works when ``K`` is
+ # neither pointed nor solid.
+ #
+ # Note that in that case we can get *duplicates*, since the
+ # tensor product of (x,s) is the same as that of (-x,-s).
+ 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, n)
- return [ M(v.list()) for v in pi_cone.rays() ]
+ return [ M(v.list()) for v in pi_cone ]
def Z_transformation_gens(K):
sage: Z_transformation_gens(K)
[]
+ Every operator is a Z-transformation on the ambient vector space::
+
+ sage: K = Cone([(1,),(-1,)])
+ sage: K.is_full_space()
+ True
+ sage: Z_transformation_gens(K)
+ [[-1], [1]]
+
+ sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
+ sage: K.is_full_space()
+ True
+ sage: Z_transformation_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 Z-transformations on the
+ right half-plane::
+
+ sage: K = Cone([(1,0),(0,1),(0,-1)])
+ sage: Z_transformation_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]
+ ]
+
Z-transformations on a subspace are Lyapunov-like and vice-versa::
sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
The Z-property is possessed by every Z-transformation::
sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=6)
+ sage: K = random_cone(max_ambient_dim=4)
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::
+ The lineality space of the cone of Z-transformations is the space of
+ Lyapunov-like transformations::
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(max_ambient_dim=4)
+ sage: L = ToricLattice(K.lattice_dim()**2)
+ sage: Z_cone = Cone([ z.list() for z in Z_transformation_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: Z_cone.linear_subspace() == lls
True
- And thus, the lineality of Z is the Lyapunov rank::
+ The lineality of the Z-transformations on a cone is the Lyapunov
+ rank of that cone::
sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=6)
+ sage: K = random_cone(max_ambient_dim=4)
sage: Z_of_K = Z_transformation_gens(K)
sage: L = ToricLattice(K.lattice_dim()**2)
- sage: z_cone = Cone([ z.list() for z in Z_of_K ], lattice=L)
- sage: z_cone.lineality() == K.lyapunov_rank()
+ sage: Z_cone = Cone([ z.list() for z in Z_of_K ],
+ ....: lattice=L,
+ ....: check=False)
+ sage: Z_cone.lineality() == K.lyapunov_rank()
True
- The lineality spaces of pi-star and Z-star are equal:
+ The lineality spaces of the duals of the positive operator and
+ Z-transformation cones are equal. From this it follows that the
+ dimensions of the Z-transformation cone and positive operator cone
+ are equal::
sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=5)
+ sage: K = random_cone(max_ambient_dim=4)
sage: pi_of_K = positive_operator_gens(K)
sage: Z_of_K = Z_transformation_gens(K)
sage: L = ToricLattice(K.lattice_dim()**2)
- sage: pi_star = Cone([p.list() for p in pi_of_K], lattice=L).dual()
- sage: z_star = Cone([ z.list() for z in Z_of_K], lattice=L).dual()
+ sage: pi_cone = Cone([p.list() for p in pi_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: Z_cone = Cone([ z.list() for z in Z_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: pi_cone.dim() == Z_cone.dim()
+ True
+ sage: pi_star = pi_cone.dual()
+ sage: z_star = Z_cone.dual()
sage: pi_star.linear_subspace() == z_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: Z_of_K = Z_transformation_gens(K)
+ sage: Z_cone = Cone([z.list() for z in Z_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = Z_cone.dim()
+ sage: actual == n^2
+ True
+ sage: K = K.dual()
+ sage: K.is_full_space()
+ True
+ sage: Z_of_K = Z_transformation_gens(K)
+ sage: Z_cone = Cone([z.list() for z in Z_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = Z_cone.dim()
+ sage: actual == n^2
+ True
+ sage: K = Cone([(1,0),(0,1),(0,-1)])
+ sage: Z_of_K = Z_transformation_gens(K)
+ sage: Z_cone = Cone([z.list() for z in Z_of_K], check=False)
+ sage: Z_cone.dim() == 3
+ True
+
+ The Z-transformations 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: Z_of_pK = Z_transformation_gens(pK)
+ sage: actual = Cone([t.list() for t in Z_of_pK],
+ ....: lattice=L,
+ ....: check=False)
+ sage: Z_of_K = Z_transformation_gens(K)
+ sage: expected = Cone([(p*t*p.inverse()).list() for t in Z_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual.is_equivalent(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
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()))
+ check = True
+ if K.is_solid() or K.is_strictly_convex():
+ # The lineality space of either ``K`` or ``K.dual()`` is
+ # trivial and it's easy to show that our generating set is
+ # minimal. I would love a proof that this works when ``K`` is
+ # neither pointed nor solid.
+ #
+ # Note that in that case we can get *duplicates*, since the
+ # tensor product of (x,s) is the same as that of (-x,-s).
+ 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()
# 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() ]
+ return [ -M(v.list()) for v in Sigma_cone ]
def Z_cone(K):
gens = Z_transformation_gens(K)
- L = None
- if len(gens) == 0:
- L = ToricLattice(0)
- return Cone([ g.list() for g in gens ], lattice=L)
+ 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 = None
- if len(gens) == 0:
- L = ToricLattice(0)
- return Cone([ g.list() for g in gens ], lattice=L)
+ L = ToricLattice(K.lattice_dim()**2)
+ return Cone([ g.list() for g in gens ], lattice=L, check=False)
projects / sage.d.git / blobdiff