The identity is always Lyapunov-like 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)
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)
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: 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() ])
True
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:
+
+ 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``.
- 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.
+ REFERENCES:
+
+ .. [Stoer-Witzgall] J. Stoer and C. Witzgall. Convexity and
+ Optimization in Finite Dimensions I. Springer-Verlag, New
+ York, 1970.
EXAMPLES:
- A random element of the trivial cone is zero::
+ 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([], 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 = Cone([(1,0,0),(0,1,0),(0,0,1)])
+ 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 an 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: (P,S) = motzkin_decomposition(K)
+ sage: x = K.random_element()
+ sage: P.contains(x) or S.contains(x)
+ True
+ sage: x.is_zero() or (P.contains(x) != S.contains(x))
+ True
+
+ 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_rays = 8)
- sage: K.contains(random_element(K))
+ sage: K = random_cone(max_ambient_dim=8)
+ sage: (P,S) = motzkin_decomposition(K)
+ sage: P.is_strictly_convex()
+ True
+ sage: S.lineality() == S.dim()
True
+ 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: 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: 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
"""
- 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)) ]
+ # 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, the rays of its dual generate its
+ # orthogonal complement.
+ S_perp = Cone(S.dual(), K.lattice())
+ P = K.intersection(S_perp)
- # 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
+ return (P,S)
def positive_operator_gens(K):
corollary in my paper::
sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 5)
+ sage: K = random_cone(max_ambient_dim=5)
sage: n = K.lattice_dim()
sage: m = K.dim()
sage: l = K.lineality()
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=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
The lineality space of Z is LL::
sage: set_random_seed()
- sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 6)
+ 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
And thus, the lineality of Z is the Lyapunov rank::
sage: set_random_seed()
- sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 6)
- sage: z_cone = Cone([ z.list() for z in Z_transformation_gens(K) ])
+ sage: K = random_cone(max_ambient_dim=6)
+ 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()
True
+
+ The lineality spaces of pi-star and Z-star are equal:
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=5)
+ 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_star.linear_subspace() == z_star.linear_subspace()
+ 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
projects / sage.d.git / blobdiff