for (x,s) in K.discrete_complementarity_set()])
-def motzkin_decomposition(K):
- r"""
- Return the pair of components in the Motzkin decomposition of this cone.
-
- 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``.
-
- 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:
-
- .. [Stoer-Witzgall] J. Stoer and C. Witzgall. Convexity and
- Optimization in Finite Dimensions I. Springer-Verlag, New
- York, 1970.
-
- EXAMPLES:
-
- The nonnegative orthant is strictly convex, so it is its own
- strictly convex component and its subspace component is trivial::
-
- 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:
-
- 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(ring=QQ)
- 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
-
- 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: 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
- """
- # 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.
- S_perp = Cone(S.dual(), K.lattice(), check=False)
- 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.
``K``. Moreover, any nonnegative linear combination of these
matrices shares the same property.
+ REFERENCES:
+
+ .. [Orlitzky-Pi-Z]
+ M. Orlitzky.
+ Positive operators and Z-transformations on closed convex cones.
+
+ .. [Tam]
+ B.-S. Tam.
+ Some results of polyhedral cones and simplicial cones.
+ Linear and Multilinear Algebra, 4:4 (1977) 281--284.
+
EXAMPLES:
Positive operators on the nonnegative orthant are nonnegative matrices::
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
+ 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)
....: 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
+
+ A transformation is positive on a cone if and only if its adjoint is
+ positive on the dual of that cone::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: F = K.lattice().vector_space().base_field()
+ sage: n = K.lattice_dim()
+ sage: L = ToricLattice(n**2)
+ sage: W = VectorSpace(F, n**2)
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: pi_of_K_star = positive_operator_gens(K.dual())
+ sage: pi_cone = Cone([p.list() for p in pi_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: pi_star = Cone([p.list() for p in pi_of_K_star],
+ ....: lattice=L,
+ ....: check=False)
+ sage: M = MatrixSpace(F, n)
+ sage: L = M(pi_cone.random_element(ring=QQ).list())
+ sage: pi_star.contains(W(L.transpose().list()))
+ True
+
+ sage: L = W.random_element()
+ sage: L_star = W(M(L.list()).transpose().list())
+ sage: pi_cone.contains(L) == pi_star.contains(L_star)
+ 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
vectors = [ W(tp.list()) for tp in tensor_products ]
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).
+ if K.is_proper():
+ # All of the generators involved are extreme vectors and
+ # therefore minimal [Tam]_. If this cone is neither solid nor
+ # strictly convex, then the tensor product of ``s`` and ``x``
+ # is the same as that of ``-s`` and ``-x``. However, as a
+ # /set/, ``tensor_products`` may still be minimal.
check = False
# Create the dual cone of the positive operators, expressed as
A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``.
Each matrix ``L`` in the list should have the property that
- ``(L*x).inner_product(s) <= 0`` whenever ``(x,s)`` is an element the
- discrete complementarity set of ``K``. Moreover, any nonnegative
- linear combination of these matrices shares the same property.
+ ``(L*x).inner_product(s) <= 0`` whenever ``(x,s)`` is an element of
+ this cone's :meth:`discrete_complementarity_set`. Moreover, any
+ conic (nonnegative linear) combination of these matrices shares the
+ same property.
+
+ REFERENCES:
+
+ M. Orlitzky.
+ Positive operators and Z-transformations on closed convex cones.
EXAMPLES:
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)])
....: 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(max_ambient_dim=4)
sage: L = ToricLattice(K.lattice_dim()**2)
- sage: z_cone = Cone([ z.list() for z in Z_transformation_gens(K) ],
+ 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
+ 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=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 ],
+ sage: Z_cone = Cone([ z.list() for z in Z_of_K ],
....: lattice=L,
....: check=False)
- sage: z_cone.lineality() == K.lyapunov_rank()
+ 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=4)
sage: pi_cone = Cone([p.list() for p in pi_of_K],
....: lattice=L,
....: check=False)
- sage: pi_star = pi_cone.dual()
- sage: z_cone = Cone([ z.list() for z in Z_of_K],
+ sage: Z_cone = Cone([ z.list() for z in Z_of_K],
....: lattice=L,
....: check=False)
- sage: z_star = z_cone.dual()
+ 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
+
+ A transformation is a Z-transformation on a cone if and only if its
+ adjoint is a Z-transformation on the dual of that cone::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: F = K.lattice().vector_space().base_field()
+ sage: n = K.lattice_dim()
+ sage: L = ToricLattice(n**2)
+ sage: W = VectorSpace(F, n**2)
+ sage: Z_of_K = Z_transformation_gens(K)
+ sage: Z_of_K_star = Z_transformation_gens(K.dual())
+ sage: Z_cone = Cone([p.list() for p in Z_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: Z_star = Cone([p.list() for p in Z_of_K_star],
+ ....: lattice=L,
+ ....: check=False)
+ sage: M = MatrixSpace(F, n)
+ sage: L = M(Z_cone.random_element(ring=QQ).list())
+ sage: Z_star.contains(W(L.transpose().list()))
+ True
+
+ sage: L = W.random_element()
+ sage: L_star = W(M(L.list()).transpose().list())
+ sage: Z_cone.contains(L) == Z_star.contains(L_star)
+ 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
vectors = [ W(m.list()) for m in tensor_products ]
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).
+ if K.is_proper():
+ # All of the generators involved are extreme vectors and
+ # therefore minimal. If this cone is neither solid nor
+ # strictly convex, then the tensor product of ``s`` and ``x``
+ # is the same as that of ``-s`` and ``-x``. However, as a
+ # /set/, ``tensor_products`` may still be minimal.
check = False
# Create the dual cone of the cross-positive operators,
projects / sage.d.git / blobdiff