for (x,s) in K.discrete_complementarity_set()])
-def random_element(K):
+def motzkin_decomposition(K):
r"""
- Return a random element of ``K`` from its ambient vector space.
+ Return the pair of components in the Motzkin decomposition of this cone.
- ALGORITHM:
+ Every convex cone is the direct sum of a strictly convex cone and a
+ linear subspace [Stoer-Witzgall]_. Return a pair ``(P,S)`` of cones
+ such that ``P`` is strictly convex, ``S`` is a subspace, and ``K``
+ is the direct sum of ``P`` and ``S``.
- 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.
+ OUTPUT:
- 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.
+ An ordered pair ``(P,S)`` of closed convex polyhedral cones where
+ ``P`` is strictly convex, ``S`` is a subspace, and ``K`` is the
+ direct sum of ``P`` and ``S``.
- EXAMPLES:
+ REFERENCES:
- A random element of the trivial cone is zero::
+ .. [Stoer-Witzgall] J. Stoer and C. Witzgall. Convexity and
+ Optimization in Finite Dimensions I. Springer-Verlag, New
+ York, 1970.
- 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)
+ EXAMPLES:
- A random element of the nonnegative orthant should have all
- components nonnegative::
+ The nonnegative orthant is strictly convex, so it is its own
+ strictly convex component and its subspace component is trivial::
- sage: set_random_seed()
sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
- sage: all([ x >= 0 for x in random_element(K) ])
+ sage: (P,S) = motzkin_decomposition(K)
+ sage: K.is_equivalent(P)
+ True
+ sage: S.is_trivial()
+ True
+
+ Likewise, full spaces are their own subspace components::
+
+ sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
+ sage: K.is_full_space()
+ True
+ sage: (P,S) = motzkin_decomposition(K)
+ sage: K.is_equivalent(S)
+ True
+ sage: P.is_trivial()
True
TESTS:
- Any cone should contain a random element of itself::
+ A random point in the cone should belong to either the strictly
+ convex component or the subspace component. If the point is nonzero,
+ it cannot be in both::
sage: set_random_seed()
sage: K = random_cone(max_ambient_dim=8)
- sage: K.contains(random_element(K))
+ sage: (P,S) = motzkin_decomposition(K)
+ sage: x = K.random_element(ring=QQ)
+ sage: P.contains(x) or S.contains(x)
True
-
- A strictly convex cone contains no lines, and thus no negative
- multiples of any of its elements besides zero::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8, strictly_convex=True)
- sage: x = random_element(K)
- sage: x.is_zero() or not K.contains(-x)
+ sage: x.is_zero() or (P.contains(x) != S.contains(x))
True
- The sum of random elements of a cone lies in the cone::
+ The strictly convex component should always be strictly convex, and
+ the subspace component should always be a subspace::
sage: set_random_seed()
sage: K = random_cone(max_ambient_dim=8)
- sage: K.contains(sum([random_element(K) for i in range(10)]))
+ sage: (P,S) = motzkin_decomposition(K)
+ sage: P.is_strictly_convex()
+ True
+ sage: S.lineality() == S.dim()
True
- """
- V = K.lattice().vector_space()
- scaled_gens = [ V.base_field().random_element().abs()*V(r) for r in K ]
-
- # Make sure we return a vector. Without the coercion, we might
- # return ``0`` when ``K`` has no rays.
- return V(sum(scaled_gens))
-
-
-def motzkin_decomposition(K):
- """
- Every convex cone is the direct sum of a pointed cone and a linear
- subspace. Return a pair ``(P,S)`` of cones such that ``P`` is
- pointed, ``S`` is a subspace, and ``K`` is the direct sum of ``P``
- and ``S``.
-
- OUTPUT:
-
- An ordered pair ``(P,S)`` of closed convex polyhedral cones where
- ``P`` is pointed, ``S`` is a subspace, and ``K`` is the direct sum
- of ``P`` and ``S``.
-
- TESTS:
-
- A random point in the cone should belong to either the pointed
- subcone ``P`` or the subspace ``S``. If the point is nonzero, it
- should lie in one but not both of them::
+ The generators of the components are obtained from orthogonal
+ projections of the original generators [Stoer-Witzgall]_::
sage: set_random_seed()
sage: K = random_cone(max_ambient_dim=8)
sage: (P,S) = motzkin_decomposition(K)
- sage: x = random_element(K)
- sage: P.contains(x) or S.contains(x)
+ sage: A = S.linear_subspace().complement().matrix()
+ sage: proj_S_perp = A.transpose() * (A*A.transpose()).inverse() * A
+ sage: expected_P = Cone([ proj_S_perp*g for g in K ], K.lattice())
+ sage: P.is_equivalent(expected_P)
True
- sage: x.is_zero() or (P.contains(x) != S.contains(x))
+ sage: A = S.linear_subspace().matrix()
+ sage: proj_S = A.transpose() * (A*A.transpose()).inverse() * A
+ sage: expected_S = Cone([ proj_S*g for g in K ], K.lattice())
+ sage: S.is_equivalent(expected_S)
True
"""
- linspace_gens = [ copy(b) for b in K.linear_subspace().basis() ]
- linspace_gens += [ -b for b in linspace_gens ]
-
- S = Cone(linspace_gens, 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())
- # Since ``S`` is a subspace, its dual is its orthogonal complement
- # (albeit in the wrong lattice).
+ # Since ``S`` is a subspace, the rays of its dual generate its
+ # orthogonal complement.
S_perp = Cone(S.dual(), K.lattice())
P = K.intersection(S_perp)
return (P,S)
+
def positive_operator_gens(K):
r"""
Compute generators of the cone of positive operators on this cone.
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: pi_of_K = positive_operator_gens(K)
+ 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=5)
+ 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=5)
+ 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)
+ 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=5)
+ 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)
+ 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=5)
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: L = ToricLattice(K.lattice_dim()**2)
+ sage: pi_cone = Cone([ g.list() for g in pi_of_K ], lattice=L)
+ 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=5)
+ 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)
+ 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
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: actual = Cone([p.list() for p in pi_of_K], lattice=L).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: actual = Cone([p.list() for p in pi_of_K], lattice=L).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]).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: 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: actual = Cone([p.list() for p in pi_of_K], lattice=L).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: actual = Cone([p.list() for p in pi_of_K], lattice=L).lineality()
+ 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]).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)
vectors = [ W(tp.list()) for tp in tensor_products ]
# Create the *dual* cone of the positive operators, expressed as
- # long vectors..
+ # long vectors. WARNING: check=True is necessary even though it
+ # makes Cone() take forever. For an example take
+ # K = Cone([(1,0,0),(0,0,1),(0,0,-1)]).
pi_dual = Cone(vectors, ToricLattice(W.dimension()))
# Now compute the desired cone from its dual...
vectors = [ W(m.list()) for m in tensor_products ]
# Create the *dual* cone of the cross-positive operators,
- # expressed as long vectors..
+ # expressed as long vectors. WARNING: check=True is necessary
+ # even though it makes Cone() take forever. For an example take
+ # K = Cone([(1,0,0),(0,0,1),(0,0,-1)]).
Sigma_dual = Cone(vectors, lattice=ToricLattice(W.dimension()))
# Now compute the desired cone from its dual...
projects / sage.d.git / blobdiff