-# 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(LL(K1)) != len(LL(K2)):
- return False
-
- C_of_K1 = discrete_complementarity_set(K1)
- C_of_K2 = discrete_complementarity_set(K2)
- 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 discrete_complementarity_set(K):
- r"""
- Compute a discrete complementarity set of this cone.
-
- A discrete complementarity set of `K` is the set of all orthogonal
- pairs `(x,s)` such that `x \in G_{1}` and `s \in G_{2}` for some
- generating sets `G_{1}` of `K` and `G_{2}` of its dual. Polyhedral
- convex cones are input in terms of their generators, so "the" (this
- particular) discrete complementarity set corresponds to ``G1
- == K.rays()`` and ``G2 == K.dual().rays()``.
-
- OUTPUT:
-
- A list of pairs `(x,s)` such that,
-
- * Both `x` and `s` are vectors (not rays).
- * `x` is one of ``K.rays()``.
- * `s` is one of ``K.dual().rays()``.
- * `x` and `s` are orthogonal.
-
- REFERENCES:
-
- .. [Orlitzky/Gowda] M. Orlitzky and M. S. Gowda. The Lyapunov Rank of an
- Improper Cone. Work in-progress.
-
- EXAMPLES:
-
- The discrete complementarity set of the nonnegative orthant consists
- of pairs of standard basis vectors::
-
- sage: K = Cone([(1,0),(0,1)])
- sage: discrete_complementarity_set(K)
- [((1, 0), (0, 1)), ((0, 1), (1, 0))]
-
- If the cone consists of a single ray, the second components of the
- discrete complementarity set should generate the orthogonal
- complement of that ray::
-
- sage: K = Cone([(1,0)])
- sage: discrete_complementarity_set(K)
- [((1, 0), (0, 1)), ((1, 0), (0, -1))]
- sage: K = Cone([(1,0,0)])
- sage: discrete_complementarity_set(K)
- [((1, 0, 0), (0, 1, 0)),
- ((1, 0, 0), (0, -1, 0)),
- ((1, 0, 0), (0, 0, 1)),
- ((1, 0, 0), (0, 0, -1))]
-
- When the cone is the entire space, its dual is the trivial cone, so
- the discrete complementarity set is empty::
-
- sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
- sage: discrete_complementarity_set(K)
- []
-
- Likewise when this cone is trivial (its dual is the entire space)::
-
- sage: L = ToricLattice(0)
- sage: K = Cone([], ToricLattice(0))
- sage: discrete_complementarity_set(K)
- []
-
- TESTS:
-
- The complementarity set of the dual can be obtained by switching the
- components of the complementarity set of the original cone::
-
- sage: set_random_seed()
- sage: K1 = random_cone(max_ambient_dim=6)
- sage: K2 = K1.dual()
- sage: expected = [(x,s) for (s,x) in discrete_complementarity_set(K2)]
- sage: actual = discrete_complementarity_set(K1)
- sage: sorted(actual) == sorted(expected)
- True
-
- The pairs in the discrete complementarity set are in fact
- complementary::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=6)
- sage: dcs = discrete_complementarity_set(K)
- sage: sum([x.inner_product(s).abs() for (x,s) in dcs])
- 0
-
- """
- V = K.lattice().vector_space()
-
- # Convert rays to vectors so that we can compute inner products.
- xs = [V(x) for x in K.rays()]
-
- # We also convert the generators of the dual cone so that we
- # return pairs of vectors and not (vector, ray) pairs.
- ss = [V(s) for s in K.dual().rays()]
-
- return [(x,s) for x in xs for s in ss if x.inner_product(s) == 0]
-
-
-def LL(K):
- r"""
- Compute the space `\mathbf{LL}` of all Lyapunov-like transformations
- on this cone.
-
- OUTPUT:
-
- A list of matrices forming a basis for the space of all
- Lyapunov-like transformations on the given cone.
-
- EXAMPLES:
-
- The trivial cone has no Lyapunov-like transformations::
-
- sage: L = ToricLattice(0)
- sage: K = Cone([], lattice=L)
- sage: LL(K)
- []
-
- The Lyapunov-like transformations on the nonnegative orthant are
- simply diagonal matrices::
-
- sage: K = Cone([(1,)])
- sage: LL(K)
- [[1]]
-
- sage: K = Cone([(1,0),(0,1)])
- sage: LL(K)
- [
- [1 0] [0 0]
- [0 0], [0 1]
- ]
-
- sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
- sage: LL(K)
- [
- [1 0 0] [0 0 0] [0 0 0]
- [0 0 0] [0 1 0] [0 0 0]
- [0 0 0], [0 0 0], [0 0 1]
- ]
-
- Only the identity matrix is Lyapunov-like on the `L^{3}_{1}` and
- `L^{3}_{\infty}` cones [Rudolf et al.]_::
-
- sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)])
- sage: LL(L31)
- [
- [1 0 0]
- [0 1 0]
- [0 0 1]
- ]
-
- sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)])
- sage: LL(L3infty)
- [
- [1 0 0]
- [0 1 0]
- [0 0 1]
- ]
-
- If our cone is the entire space, then every transformation on it is
- Lyapunov-like::
-
- sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)])
- sage: M = MatrixSpace(QQ,2)
- sage: M.basis() == LL(K)
- True
-
- TESTS:
-
- The inner product `\left< L\left(x\right), s \right>` is zero for
- every pair `\left( x,s \right)` in the discrete complementarity set
- of the cone::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8)
- sage: C_of_K = discrete_complementarity_set(K)
- sage: l = [ (L*x).inner_product(s) for (x,s) in C_of_K for L in LL(K) ]
- sage: sum(map(abs, l))
- 0
-
- The Lyapunov-like transformations on a cone and its dual are related
- by transposition, but we're not guaranteed to compute transposed
- elements of `LL\left( K \right)` as our basis for `LL\left( K^{*}
- \right)`
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8)
- sage: LL2 = [ L.transpose() for L in LL(K.dual()) ]
- sage: V = VectorSpace( K.lattice().base_field(), K.lattice_dim()^2)
- sage: LL1_vecs = [ V(m.list()) for m in LL(K) ]
- sage: LL2_vecs = [ V(m.list()) for m in LL2 ]
- sage: V.span(LL1_vecs) == V.span(LL2_vecs)
- True
-
- """
- V = K.lattice().vector_space()
-
- C_of_K = discrete_complementarity_set(K)
-
- tensor_products = [ s.tensor_product(x) for (x,s) in C_of_K ]
-
- # 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.
- W = VectorSpace(V.base_ring(), V.dimension()**2)
-
- # Turn our matrices into long vectors...
- vectors = [ W(m.list()) for m in tensor_products ]
-
- # Vector space representation of Lyapunov-like matrices
- # (i.e. vec(L) where L is Luapunov-like).
- LL_vector = W.span(vectors).complement()
-
- # Now construct an ambient MatrixSpace in which to stick our
- # transformations.
- M = MatrixSpace(V.base_ring(), V.dimension())
-
- matrix_basis = [ M(v.list()) for v in LL_vector.basis() ]
-
- return matrix_basis
-
-
-
-def lyapunov_rank(K):
- r"""
- Compute the Lyapunov rank (or bilinearity rank) of this cone.
-
- The Lyapunov rank of a cone can be thought of in (mainly) two ways:
-
- 1. The dimension of the Lie algebra of the automorphism group of the
- cone.
-
- 2. The dimension of the linear space of all Lyapunov-like
- transformations on the cone.
-
- INPUT:
-
- A closed, convex polyhedral cone.
-
- OUTPUT:
-
- An integer representing the Lyapunov rank of the cone. If the
- dimension of the ambient vector space is `n`, then the Lyapunov rank
- will be between `1` and `n` inclusive; however a rank of `n-1` is
- not possible (see [Orlitzky/Gowda]_).
-
- 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.
-
- .. [Orlitzky/Gowda] M. Orlitzky and M. S. Gowda. The Lyapunov Rank of an
- Improper Cone. Work in-progress.
-
- .. [Rudolf et al.] 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 et al.]_::
-
- 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/Gowda]_::
-
- 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 et al.]_::
-
- 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 et al.]_::
-
- 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/Gowda]_::
-
- 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/Gowda]_::
-
- 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 et al.]_::
-
- 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 et al.]_.
- 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 et al.]_::
-
- 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 et al.]_::
-
- 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/Gowda]_::
-
- 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 et al.]_::
-
- 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/Gowda]_, 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/Gowda]_::
-
- 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 any cone is just the dimension of ``LL(K)``::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8)
- sage: lyapunov_rank(K) == len(LL(K))
- True
-
- We can make an imperfect cone perfect by adding a slack variable
- (a Theorem in [Orlitzky/Gowda]_)::
-
- 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
-
- 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(LL(K))
- return beta
-
-
-
-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:
-
- .. [Orlitzky] M. Orlitzky. The Lyapunov rank of an
- improper cone (preprint).
-
- EXAMPLES:
-
- 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: 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: R = K.lattice().vector_space().base_ring()
- sage: L = zero_matrix(R, K.lattice_dim())
- sage: is_lyapunov_like(L,K)
- True
-
- Everything in ``LL(K)`` should be Lyapunov-like on ``K``::
-
- sage: K = random_cone(min_ambient_dim = 1, max_rays = 5)
- sage: all([is_lyapunov_like(L,K) for L in LL(K)])
- True
-
- """
- return all([(L*x).inner_product(s) == 0
- for (x,s) in discrete_complementarity_set(K)])
-
-
-def random_element(K):
- r"""
- Return a random element of ``K`` from its ambient vector space.
-
- ALGORITHM:
-
- 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.
-
- 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.
-
- 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)
-
- TESTS:
-
- Any cone should contain an element of itself::
-
- sage: set_random_seed()
- sage: K = random_cone(max_rays = 8)
- sage: K.contains(random_element(K))
- 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
+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 = 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)