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. 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:
- A random element of the trivial cone is zero::
-
- 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)
-
- 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()
+ 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 pointed_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:
+ The generators of the strictly convex component are obtained from
+ the orthogonal projections of the original generators onto the
+ orthogonal complement of the subspace component::
sage: set_random_seed()
sage: K = random_cone(max_ambient_dim=8)
- sage: (P,S) = pointed_decomposition(K)
- sage: x = random_element(K)
- sage: P.contains(x) or S.contains(x)
- True
- sage: x.is_zero() or (P.contains(x) != S.contains(x))
+ sage: (P,S) = motzkin_decomposition(K)
+ sage: S_perp = S.linear_subspace().complement()
+ sage: A = S_perp.matrix().transpose()
+ sage: proj = A * (A.transpose()*A).inverse() * A.transpose()
+ sage: expected = Cone([ proj*g for g in K ], K.lattice())
+ sage: P.is_equivalent(expected)
True
"""
linspace_gens = [ copy(b) for b in K.linear_subspace().basis() ]
projects / sage.d.git / blobdiff