-# Sage doesn't load ~/.sage/init.sage during testing (sage -t), so we
-# have to explicitly mangle our sitedir here so that "mjo.cone"
-# resolves.
-from os.path import abspath
-from site import addsitedir
-addsitedir(abspath('../../'))
-
from sage.all import *
-
-def _basically_the_same(K1, K2):
- r"""
- Test whether or not ``K1`` and ``K2`` are "basically the same."
-
- This is a hack to get around the fact that it's difficult to tell
- when two cones are linearly isomorphic. We have a proposition that
- equates two cones, but represented over `\mathbb{Q}`, they are
- merely linearly isomorphic (not equal). So rather than test for
- equality, we test a list of properties that should be preserved
- under an invertible linear transformation.
-
- OUTPUT:
-
- ``True`` if ``K1`` and ``K2`` are basically the same, and ``False``
- otherwise.
-
- EXAMPLES:
-
- Any proper cone with three generators in `\mathbb{R}^{3}` is
- basically the same as the nonnegative orthant::
-
- sage: K1 = Cone([(1,0,0), (0,1,0), (0,0,1)])
- sage: K2 = Cone([(1,2,3), (3, 18, 4), (66, 51, 0)])
- sage: _basically_the_same(K1, K2)
- True
-
- Negating a cone gives you another cone that is basically the same::
-
- sage: K = Cone([(0,2,-5), (-6, 2, 4), (0, 51, 0)])
- sage: _basically_the_same(K, -K)
- True
-
- TESTS:
-
- Any cone is basically the same as itself::
-
- sage: K = random_cone(max_ambient_dim = 8)
- sage: _basically_the_same(K, K)
- True
-
- After applying an invertible matrix to the rows of a cone, the
- result should be basically the same as the cone we started with::
-
- sage: K1 = random_cone(max_ambient_dim = 8)
- sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular')
- sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice())
- sage: _basically_the_same(K1, K2)
- True
-
- """
- if K1.lattice_dim() != K2.lattice_dim():
- return False
-
- if K1.nrays() != K2.nrays():
- return False
-
- if K1.dim() != K2.dim():
- return False
-
- if K1.lineality() != K2.lineality():
- return False
-
- if K1.is_solid() != K2.is_solid():
- return False
-
- if K1.is_strictly_convex() != K2.is_strictly_convex():
- return False
-
- if len(K1.lyapunov_like_basis()) != len(K2.lyapunov_like_basis()):
- return False
-
- C_of_K1 = K1.discrete_complementarity_set()
- C_of_K2 = K2.discrete_complementarity_set()
- if len(C_of_K1) != len(C_of_K2):
- return False
-
- if len(K1.facets()) != len(K2.facets()):
- return False
-
- return True
-
-
-
-def _restrict_to_space(K, W):
- r"""
- Restrict this cone a subspace of its ambient space.
-
- INPUT:
-
- - ``W`` -- The subspace into which this cone will be restricted.
-
- OUTPUT:
-
- A new cone in a sublattice corresponding to ``W``.
-
- EXAMPLES:
-
- When this cone is solid, restricting it into its own span should do
- nothing::
-
- sage: K = Cone([(1,)])
- sage: _restrict_to_space(K, K.span()) == K
- True
-
- A single ray restricted into its own span gives the same output
- regardless of the ambient space::
-
- sage: K2 = Cone([(1,0)])
- sage: K2_S = _restrict_to_space(K2, K2.span()).rays()
- sage: K2_S
- N(1)
- in 1-d lattice N
- sage: K3 = Cone([(1,0,0)])
- sage: K3_S = _restrict_to_space(K3, K3.span()).rays()
- sage: K3_S
- N(1)
- in 1-d lattice N
- sage: K2_S == K3_S
- True
-
- TESTS:
-
- The projected cone should always be solid::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 8)
- sage: _restrict_to_space(K, K.span()).is_solid()
- True
-
- And the resulting cone should live in a space having the same
- dimension as the space we restricted it to::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 8)
- sage: K_P = _restrict_to_space(K, K.dual().span())
- sage: K_P.lattice_dim() == K.dual().dim()
- True
-
- This function should not affect the dimension of a cone::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 8)
- sage: K.dim() == _restrict_to_space(K,K.span()).dim()
- True
-
- Nor should it affect the lineality of a cone::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 8)
- sage: K.lineality() == _restrict_to_space(K, K.span()).lineality()
- True
-
- No matter which space we restrict to, the lineality should not
- increase::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 8)
- sage: S = K.span(); P = K.dual().span()
- sage: K.lineality() >= _restrict_to_space(K,S).lineality()
- True
- sage: K.lineality() >= _restrict_to_space(K,P).lineality()
- True
-
- If we do this according to our paper, then the result is proper::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 8)
- sage: K_S = _restrict_to_space(K, K.span())
- sage: K_SP = _restrict_to_space(K_S.dual(), K_S.dual().span()).dual()
- sage: K_SP.is_proper()
- True
- sage: K_SP = _restrict_to_space(K_S, K_S.dual().span())
- sage: K_SP.is_proper()
- True
-
- Test the proposition in our paper concerning the duals and
- restrictions. Generate a random cone, then create a subcone of
- it. The operation of dual-taking should then commute with
- _restrict_to_space::
-
- sage: set_random_seed()
- sage: J = random_cone(max_ambient_dim = 8)
- sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
- sage: K_W_star = _restrict_to_space(K, J.span()).dual()
- sage: K_star_W = _restrict_to_space(K.dual(), J.span())
- sage: _basically_the_same(K_W_star, K_star_W)
- True
-
- """
- # First we want to intersect ``K`` with ``W``. The easiest way to
- # do this is via cone intersection, so we turn the subspace ``W``
- # into a cone.
- W_cone = Cone(W.basis() + [-b for b in W.basis()], lattice=K.lattice())
- K = K.intersection(W_cone)
-
- # We've already intersected K with the span of K2, so every
- # generator of K should belong to W now.
- K_W_rays = [ W.coordinate_vector(r) for r in K.rays() ]
-
- L = ToricLattice(W.dimension())
- return Cone(K_W_rays, lattice=L)
-
-
-def lyapunov_rank(K):
- r"""
- Compute the Lyapunov rank of this cone.
-
- The Lyapunov rank of a cone is the dimension of the space of its
- Lyapunov-like transformations -- that is, the length of a
- :meth:`lyapunov_like_basis`. Equivalently, the Lyapunov rank is the
- dimension of the Lie algebra of the automorphism group of the cone.
-
- OUTPUT:
-
- A nonnegative integer representing the Lyapunov rank of this cone.
-
- If the ambient space is trivial, the Lyapunov rank will be zero.
- Otherwise, if the dimension of the ambient vector space is `n`, then
- the resulting Lyapunov rank will be between `1` and `n` inclusive. A
- Lyapunov rank of `n-1` is not possible [Orlitzky]_.
-
- ALGORITHM:
-
- The codimension formula from the second reference is used. We find
- all pairs `(x,s)` in the complementarity set of `K` such that `x`
- and `s` are rays of our cone. It is known that these vectors are
- sufficient to apply the codimension formula. Once we have all such
- pairs, we "brute force" the codimension formula by finding all
- linearly-independent `xs^{T}`.
-
- REFERENCES:
-
- .. [Gowda/Tao] M.S. Gowda and J. Tao. On the bilinearity rank of
- a proper cone and Lyapunov-like transformations. Mathematical
- Programming, 147 (2014) 155-170.
-
- M. Orlitzky. The Lyapunov rank of an improper cone.
- http://www.optimization-online.org/DB_HTML/2015/10/5135.html
-
- G. Rudolf, N. Noyan, D. Papp, and F. Alizadeh, Bilinear
- optimality constraints for the cone of positive polynomials,
- Mathematical Programming, Series B, 129 (2011) 5-31.
-
- EXAMPLES:
-
- The nonnegative orthant in `\mathbb{R}^{n}` always has rank `n`
- [Rudolf]_::
-
- sage: positives = Cone([(1,)])
- sage: lyapunov_rank(positives)
- 1
- sage: quadrant = Cone([(1,0), (0,1)])
- sage: lyapunov_rank(quadrant)
- 2
- sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)])
- sage: lyapunov_rank(octant)
- 3
-
- The full space `\mathbb{R}^{n}` has Lyapunov rank `n^{2}`
- [Orlitzky]_::
-
- sage: R5 = VectorSpace(QQ, 5)
- sage: gs = R5.basis() + [ -r for r in R5.basis() ]
- sage: K = Cone(gs)
- sage: lyapunov_rank(K)
- 25
-
- The `L^{3}_{1}` cone is known to have a Lyapunov rank of one
- [Rudolf]_::
-
- sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)])
- sage: lyapunov_rank(L31)
- 1
-
- Likewise for the `L^{3}_{\infty}` cone [Rudolf]_::
-
- sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)])
- sage: lyapunov_rank(L3infty)
- 1
-
- A single ray in `n` dimensions should have Lyapunov rank `n^{2} - n
- + 1` [Orlitzky]_::
-
- sage: K = Cone([(1,0,0,0,0)])
- sage: lyapunov_rank(K)
- 21
- sage: K.lattice_dim()**2 - K.lattice_dim() + 1
- 21
-
- A subspace (of dimension `m`) in `n` dimensions should have a
- Lyapunov rank of `n^{2} - m\left(n - m)` [Orlitzky]_::
-
- sage: e1 = (1,0,0,0,0)
- sage: neg_e1 = (-1,0,0,0,0)
- sage: e2 = (0,1,0,0,0)
- sage: neg_e2 = (0,-1,0,0,0)
- sage: z = (0,0,0,0,0)
- sage: K = Cone([e1, neg_e1, e2, neg_e2, z, z, z])
- sage: lyapunov_rank(K)
- 19
- sage: K.lattice_dim()**2 - K.dim()*K.codim()
- 19
-
- The Lyapunov rank should be additive on a product of proper cones
- [Rudolf]_::
-
- sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)])
- sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)])
- sage: K = L31.cartesian_product(octant)
- sage: lyapunov_rank(K) == lyapunov_rank(L31) + lyapunov_rank(octant)
- True
-
- Two isomorphic cones should have the same Lyapunov rank [Rudolf]_.
- The cone ``K`` in the following example is isomorphic to the nonnegative
- octant in `\mathbb{R}^{3}`::
-
- sage: K = Cone([(1,2,3), (-1,1,0), (1,0,6)])
- sage: lyapunov_rank(K)
- 3
-
- The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
- itself [Rudolf]_::
-
- sage: K = Cone([(2,2,4), (-1,9,0), (2,0,6)])
- sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
- True
-
- TESTS:
-
- The Lyapunov rank should be additive on a product of proper cones
- [Rudolf]_::
-
- sage: set_random_seed()
- sage: K1 = random_cone(max_ambient_dim=8,
- ....: strictly_convex=True,
- ....: solid=True)
- sage: K2 = random_cone(max_ambient_dim=8,
- ....: strictly_convex=True,
- ....: solid=True)
- sage: K = K1.cartesian_product(K2)
- sage: lyapunov_rank(K) == lyapunov_rank(K1) + lyapunov_rank(K2)
- True
-
- The Lyapunov rank is invariant under a linear isomorphism
- [Orlitzky]_::
-
- sage: K1 = random_cone(max_ambient_dim = 8)
- sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular')
- sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice())
- sage: lyapunov_rank(K1) == lyapunov_rank(K2)
- True
-
- The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
- itself [Rudolf]_::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8)
- sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
- True
-
- The Lyapunov rank of a proper polyhedral cone in `n` dimensions can
- be any number between `1` and `n` inclusive, excluding `n-1`
- [Gowda/Tao]_. By accident, the `n-1` restriction will hold for the
- trivial cone in a trivial space as well. However, in zero dimensions,
- the Lyapunov rank of the trivial cone will be zero::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8,
- ....: strictly_convex=True,
- ....: solid=True)
- sage: b = lyapunov_rank(K)
- sage: n = K.lattice_dim()
- sage: (n == 0 or 1 <= b) and b <= n
- True
- sage: b == n-1
- False
-
- In fact [Orlitzky]_, no closed convex polyhedral cone can have
- Lyapunov rank `n-1` in `n` dimensions::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8)
- sage: b = lyapunov_rank(K)
- sage: n = K.lattice_dim()
- sage: b == n-1
- False
-
- The calculation of the Lyapunov rank of an improper cone can be
- reduced to that of a proper cone [Orlitzky]_::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8)
- sage: actual = lyapunov_rank(K)
- sage: K_S = _restrict_to_space(K, K.span())
- sage: K_SP = _restrict_to_space(K_S.dual(), K_S.dual().span()).dual()
- sage: l = K.lineality()
- sage: c = K.codim()
- sage: expected = lyapunov_rank(K_SP) + K.dim()*(l + c) + c**2
- sage: actual == expected
- True
-
- The Lyapunov rank of a cone is the size of a :meth:`lyapunov_like_basis`::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8)
- sage: lyapunov_rank(K) == len(K.lyapunov_like_basis())
- True
-
- We can make an imperfect cone perfect by adding a slack variable
- (a Theorem in [Orlitzky]_)::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8,
- ....: strictly_convex=True,
- ....: solid=True)
- sage: L = ToricLattice(K.lattice_dim() + 1)
- sage: K = Cone([ r.list() + [0] for r in K.rays() ], lattice=L)
- sage: lyapunov_rank(K) >= K.lattice_dim()
- True
-
- """
- beta = 0 # running tally of the Lyapunov rank
-
- m = K.dim()
- n = K.lattice_dim()
- l = K.lineality()
-
- if m < n:
- # K is not solid, restrict to its span.
- K = _restrict_to_space(K, K.span())
-
- # Non-solid reduction lemma.
- beta += (n - m)*n
-
- if l > 0:
- # K is not pointed, restrict to the span of its dual. Uses a
- # proposition from our paper, i.e. this is equivalent to K =
- # _rho(K.dual()).dual().
- K = _restrict_to_space(K, K.dual().span())
-
- # Non-pointed reduction lemma.
- beta += l * m
-
- beta += len(K.lyapunov_like_basis())
- return beta
-
-
-
def is_lyapunov_like(L,K):
r"""
Determine whether or not ``L`` is Lyapunov-like on ``K``.
The identity is always Lyapunov-like in a nontrivial space::
sage: set_random_seed()
- sage: K = random_cone(min_ambient_dim = 1, max_rays = 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_rays = 5)
+ 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_rays = 5)
+ 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:
- 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``.
+
+ 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_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)) ]
+ # 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_operators(K):
+def positive_operator_gens(K):
r"""
Compute generators of the cone of positive operators on this cone.
EXAMPLES:
- The trivial cone in a trivial space has no positive operators::
-
- sage: K = Cone([], ToricLattice(0))
- sage: positive_operators(K)
- []
-
Positive operators on the nonnegative orthant are nonnegative matrices::
sage: K = Cone([(1,)])
- sage: positive_operators(K)
+ sage: positive_operator_gens(K)
[[1]]
sage: K = Cone([(1,0),(0,1)])
- sage: positive_operators(K)
+ sage: positive_operator_gens(K)
[
[1 0] [0 1] [0 0] [0 0]
[0 0], [0 0], [1 0], [0 1]
]
+ The trivial cone in a trivial space has no positive operators::
+
+ sage: K = Cone([], ToricLattice(0))
+ sage: positive_operator_gens(K)
+ []
+
+ Every operator is positive on the trivial cone::
+
+ sage: K = Cone([(0,)])
+ sage: positive_operator_gens(K)
+ [[1], [-1]]
+
+ sage: K = Cone([(0,0)])
+ sage: K.is_trivial()
+ 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]
+ ]
+
Every operator is positive on the ambient vector space::
sage: K = Cone([(1,),(-1,)])
sage: K.is_full_space()
True
- sage: positive_operators(K)
+ 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_operators(K)
+ 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]
]
+ A non-obvious application is to find the positive operators on the
+ right half-plane::
+
+ sage: K = Cone([(1,0),(0,1),(0,-1)])
+ sage: positive_operator_gens(K)
+ [
+ [1 0] [0 0] [ 0 0] [0 0] [ 0 0]
+ [0 0], [1 0], [-1 0], [0 1], [ 0 -1]
+ ]
+
TESTS:
- A positive operator on a cone should send its generators into the cone::
+ Each positive operator generator should send the generators of the
+ cone into the cone::
- sage: K = random_cone(max_ambient_dim = 6)
- sage: pi_of_K = positive_operators(K)
- sage: all([K.contains(p*x) for p in pi_of_K for x in K.rays()])
+ 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
"""
- # Sage doesn't think matrices are vectors, so we have to convert
- # our matrices to vectors explicitly before we can figure out how
- # many are linearly-indepenedent.
- #
- # The space W has the same base ring as V, but dimension
- # dim(V)^2. So it has the same dimension as the space of linear
- # transformations on V. In other words, it's just the right size
- # to create an isomorphism between it and our matrices.
- V = K.lattice().vector_space()
- W = VectorSpace(V.base_ring(), V.dimension()**2)
+ # 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() ]
- # Turn our matrices into long vectors...
- vectors = [ W(m.list()) for m in tensor_products ]
+ # 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..
- L = ToricLattice(W.dimension())
- pi_dual = Cone(vectors, lattice=L)
+ 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(V.base_ring(), V.dimension())
-
+ M = MatrixSpace(F, n)
return [ M(v.list()) for v in pi_cone.rays() ]
-def Z_transformations(K):
+def Z_transformation_gens(K):
r"""
Compute generators of the cone of Z-transformations on this cone.
That is, matrices whose off-diagonal elements are nonnegative::
sage: K = Cone([(1,0),(0,1)])
- sage: Z_transformations(K)
+ 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_transformations(K)
+ 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 ])
The trivial cone in a trivial space has no Z-transformations::
sage: K = Cone([], ToricLattice(0))
- sage: Z_transformations(K)
+ sage: Z_transformation_gens(K)
[]
Z-transformations on a subspace are Lyapunov-like and vice-versa::
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_transformations(K) ])
+ sage: zs = span([ vector(z.list()) for z in Z_transformation_gens(K) ])
sage: zs == lls
True
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_transformations(K)
+ 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])
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_transformations(K) ])
+ 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
"""
- # Sage doesn't think matrices are vectors, so we have to convert
- # our matrices to vectors explicitly before we can figure out how
- # many are linearly-indepenedent.
- #
- # The space W has the same base ring as V, but dimension
- # dim(V)^2. So it has the same dimension as the space of linear
- # transformations on V. In other words, it's just the right size
- # to create an isomorphism between it and our matrices.
- V = K.lattice().vector_space()
- W = VectorSpace(V.base_ring(), V.dimension()**2)
-
- C_of_K = K.discrete_complementarity_set()
- tensor_products = [ s.tensor_product(x) for (x,s) in C_of_K ]
+ # 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..
- L = ToricLattice(W.dimension())
- Sigma_dual = Cone(vectors, lattice=L)
+ 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(V.base_ring(), V.dimension())
-
+ M = MatrixSpace(F, n)
return [ -M(v.list()) for v in Sigma_cone.rays() ]
+
+
+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)
projects / sage.d.git / blobdiff