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``.
+
+ REFERENCES:
- 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.
+ .. [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_rays = 8)
- sage: K.contains(random_element(K))
+ 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_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)) ]
-
- # 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
+ 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())
+
+ # Since ``S`` is a subspace, its dual is its orthogonal complement
+ # (albeit in the wrong lattice).
+ S_perp = Cone(S.dual(), K.lattice())
+ P = K.intersection(S_perp)
+ return (P,S)
def positive_operator_gens(K):
r"""