from sage.all import *
-def is_lyapunov_like(L,K):
- r"""
- Determine whether or not ``L`` is Lyapunov-like on ``K``.
-
- We say that ``L`` is Lyapunov-like on ``K`` if `\left\langle
- L\left\lparenx\right\rparen,s\right\rangle = 0` for all pairs
- `\left\langle x,s \right\rangle` in the complementarity set of
- ``K``. It is known [Orlitzky]_ that this property need only be
- checked for generators of ``K`` and its dual.
-
- INPUT:
-
- - ``L`` -- A linear transformation or matrix.
-
- - ``K`` -- A polyhedral closed convex cone.
-
- OUTPUT:
-
- ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``,
- and ``False`` otherwise.
-
- .. WARNING::
-
- If this function returns ``True``, then ``L`` is Lyapunov-like
- on ``K``. However, if ``False`` is returned, that could mean one
- of two things. The first is that ``L`` is definitely not
- Lyapunov-like on ``K``. The second is more of an "I don't know"
- answer, returned (for example) if we cannot prove that an inner
- product is zero.
-
- REFERENCES:
-
- M. Orlitzky. The Lyapunov rank of an improper cone.
- http://www.optimization-online.org/DB_HTML/2015/10/5135.html
-
- EXAMPLES:
-
- 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: 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: R = K.lattice().vector_space().base_ring()
- sage: L = zero_matrix(R, K.lattice_dim())
- sage: is_lyapunov_like(L,K)
- True
-
- 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: all([ is_lyapunov_like(L,K) for L in K.lyapunov_like_basis() ])
- True
-
- """
- return all([(L*x).inner_product(s) == 0
- 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()
- 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
- """
- # 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)
-
- return (P,S)
-
-
-def positive_operator_gens(K):
- r"""
- Compute generators of the cone of positive operators on this cone.
-
- OUTPUT:
-
- A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``.
- Each matrix ``P`` in the list should have the property that ``P*x``
- is an element of ``K`` whenever ``x`` is an element of
- ``K``. Moreover, any nonnegative linear combination of these
- matrices shares the same property.
-
- 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,)])
- sage: positive_operator_gens(K)
- [[1]]
-
- sage: K = Cone([(1,0),(0,1)])
- sage: positive_operator_gens(K)
- [
- [1 0] [0 1] [0 0] [0 0]
- [0 0], [0 0], [1 0], [0 1]
- ]
-
- Every operator is positive on the ambient vector space::
-
- sage: K = Cone([(1,),(-1,)])
- sage: K.is_full_space()
- True
- sage: positive_operator_gens(K)
- [[1], [-1]]
-
- sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
- sage: K.is_full_space()
- 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]
- ]
-
- TESTS:
-
- 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()) 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().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().list())
- sage: K.contains(P*K.random_element())
- True
-
- The dimension of the cone of positive operators is given by the
- corollary in my paper::
-
- 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: actual = Cone([p.list() for p in pi_of_K], lattice=L).dim()
- sage: expected = n**2 - l*(m - l) - (n - m)*m
- sage: actual == expected
- True
-
- The lineality of the cone of positive operators is given by the
- corollary in my paper::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=5)
- sage: n = K.lattice_dim()
- sage: pi_of_K = positive_operator_gens(K)
- sage: L = ToricLattice(n**2)
- sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).lineality()
- sage: expected = n**2 - K.dim()*K.dual().dim()
- sage: actual == expected
- True
-
- The cone ``K`` is proper if and only if the cone of positive
- operators on ``K`` is proper::
-
- 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([p.list() for p in pi_of_K], lattice=L)
- sage: K.is_proper() == pi_cone.is_proper()
- 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
- # two values to construct the appropriate "long vector" space.
- F = K.lattice().base_field()
- n = K.lattice_dim()
-
- tensor_products = [ s.tensor_product(x) for x in K for s in K.dual() ]
-
- # Convert those tensor products to long vectors.
- W = VectorSpace(F, n**2)
- vectors = [ W(tp.list()) for tp in tensor_products ]
-
- # Create the *dual* cone of the positive operators, expressed as
- # long vectors..
- pi_dual = Cone(vectors, ToricLattice(W.dimension()))
-
- # Now compute the desired cone from its dual...
- pi_cone = pi_dual.dual()
-
- # And finally convert its rays back to matrix representations.
- M = MatrixSpace(F, n)
- return [ M(v.list()) for v in pi_cone.rays() ]
-
-
-def Z_transformation_gens(K):
- r"""
- Compute generators of the cone of Z-transformations on this cone.
-
- OUTPUT:
-
- 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.
-
- EXAMPLES:
-
- Z-transformations on the nonnegative orthant are just Z-matrices.
- That is, matrices whose off-diagonal elements are nonnegative::
-
- sage: K = Cone([(1,0),(0,1)])
- sage: Z_transformation_gens(K)
- [
- [ 0 -1] [ 0 0] [-1 0] [1 0] [ 0 0] [0 0]
- [ 0 0], [-1 0], [ 0 0], [0 0], [ 0 -1], [0 1]
- ]
- sage: K = Cone([(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)])
- sage: all([ z[i][j] <= 0 for z in Z_transformation_gens(K)
- ....: for i in range(z.nrows())
- ....: for j in range(z.ncols())
- ....: if i != j ])
- True
-
- The trivial cone in a trivial space has no Z-transformations::
-
- sage: K = Cone([], ToricLattice(0))
- sage: Z_transformation_gens(K)
- []
-
- Z-transformations on a subspace are Lyapunov-like and vice-versa::
-
- sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
- sage: K.is_full_space()
- True
- sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
- sage: zs = span([ vector(z.list()) for z in Z_transformation_gens(K) ])
- sage: zs == lls
- True
-
- TESTS:
-
- The Z-property is possessed by every Z-transformation::
-
- sage: set_random_seed()
- 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
- ....: for (x,s) in dcs])
- True
-
- The lineality space of Z is LL::
-
- sage: set_random_seed()
- 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
- True
-
- And thus, the lineality of Z is the Lyapunov rank::
-
- sage: set_random_seed()
- 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
- # two values to construct the appropriate "long vector" space.
- F = K.lattice().base_field()
- n = K.lattice_dim()
-
- # These tensor products contain generators for the dual cone of
- # the cross-positive transformations.
- tensor_products = [ s.tensor_product(x)
- for (x,s) in K.discrete_complementarity_set() ]
-
- # Turn our matrices into long vectors...
- W = VectorSpace(F, n**2)
- vectors = [ W(m.list()) for m in tensor_products ]
-
- # Create the *dual* cone of the cross-positive operators,
- # expressed as long vectors..
- Sigma_dual = Cone(vectors, lattice=ToricLattice(W.dimension()))
-
- # Now compute the desired cone from its dual...
- Sigma_cone = Sigma_dual.dual()
-
- # And finally convert its rays back to matrix representations.
- # But first, make them negative, so we get Z-transformations and
- # not cross-positive ones.
- M = MatrixSpace(F, n)
- return [ -M(v.list()) for v in Sigma_cone.rays() ]
+def LL_cone(K):
+ gens = K.lyapunov_like_basis()
+ L = ToricLattice(K.lattice_dim()**2)
+ return Cone(( g.list() for g in gens ), lattice=L, check=False)
+def Sigma_cone(K):
+ gens = K.cross_positive_operators_gens()
+ L = ToricLattice(K.lattice_dim()**2)
+ return Cone(( g.list() for g in gens ), lattice=L, check=False)
def Z_cone(K):
- gens = Z_transformation_gens(K)
- L = None
- if len(gens) == 0:
- L = ToricLattice(0)
- return Cone([ g.list() for g in gens ], lattice=L)
-
-def pi_cone(K):
- gens = positive_operator_gens(K)
- L = None
- if len(gens) == 0:
- L = ToricLattice(0)
- return Cone([ g.list() for g in gens ], lattice=L)
+ gens = K.Z_operators_gens()
+ L = ToricLattice(K.lattice_dim()**2)
+ return Cone(( g.list() for g in gens ), lattice=L, check=False)
+
+def pi_cone(K1, K2=None):
+ if K2 is None:
+ K2 = K1
+ gens = K1.positive_operators_gens(K2)
+ L = ToricLattice(K1.lattice_dim()*K2.lattice_dim())
+ return Cone(( g.list() for g in gens ), lattice=L, check=False)
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