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.LL()) != len(K2.LL()):
- 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.
+ Restrict this cone (up to linear isomorphism) to a vector subspace.
+
+ This operation not only restricts the cone to a subspace of its
+ ambient space, but also represents the rays of the cone in a new
+ (smaller) lattice corresponding to the subspace. The resulting cone
+ will be linearly isomorphic **but not equal** to the desired
+ restriction, since it has likely undergone a change of basis.
+
+ To explain the difficulty, consider the cone ``K = Cone([(1,1,1)])``
+ having a single ray. The span of ``K`` is a one-dimensional subspace
+ containing ``K``, yet we have no way to perform operations like
+ :meth:`dual` in the subspace. To represent ``K`` in the space
+ ``K.span()``, we must perform a change of basis and write its sole
+ ray as ``(1,0,0)``. Now the restricted ``Cone([(1,)])`` is linearly
+ isomorphic (but of course not equal) to ``K`` interpreted as living
+ in ``K.span()``.
INPUT:
A new cone in a sublattice corresponding to ``W``.
+ REFERENCES:
+
+ M. Orlitzky. The Lyapunov rank of an improper cone.
+ http://www.optimization-online.org/DB_HTML/2015/10/5135.html
+
EXAMPLES:
- When this cone is solid, restricting it into its own span should do
- nothing::
+ Restricting a solid cone to its own span returns a cone linearly
+ isomorphic to the original::
- sage: K = Cone([(1,)])
- sage: _restrict_to_space(K, K.span()) == K
+ sage: K = Cone([(1,2,3),(-1,1,0),(9,0,-2)])
+ sage: K.is_solid()
True
+ sage: _restrict_to_space(K, K.span()).rays()
+ N(-1, 1, 0),
+ N( 1, 0, 0),
+ N( 9, -6, -1)
+ in 3-d lattice N
- A single ray restricted into its own span gives the same output
+ A single ray restricted to its own span has the same representation
regardless of the ambient space::
sage: K2 = Cone([(1,0)])
sage: K2_S
N(1)
in 1-d lattice N
- sage: K3 = Cone([(1,0,0)])
+ sage: K3 = Cone([(1,1,1)])
sage: K3_S = _restrict_to_space(K3, K3.span()).rays()
sage: K3_S
N(1)
sage: K2_S == K3_S
True
+ Restricting to a trivial space gives the trivial cone::
+
+ sage: K = Cone([(8,3,-1,0),(9,2,2,0),(-4,6,7,0)])
+ sage: trivial_space = K.lattice().vector_space().span([])
+ sage: _restrict_to_space(K, trivial_space)
+ 0-d cone in 0-d lattice N
+
TESTS:
- The projected cone should always be solid::
+ Restricting a cone to its own span results in a solid cone::
sage: set_random_seed()
sage: K = random_cone(max_ambient_dim = 8)
- sage: _restrict_to_space(K, K.span()).is_solid()
+ sage: K_S = _restrict_to_space(K, K.span())
+ sage: K_S.is_solid()
True
- And the resulting cone should live in a space having the same
- dimension as the space we restricted it to::
+ Restricting a cone to its own span should not affect the number of
+ rays in the cone::
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()
+ sage: K_S = _restrict_to_space(K, K.span())
+ sage: K.nrays() == K_S.nrays()
True
- This function should not affect the dimension of a cone::
+ Restricting a cone to its own span should not affect its dimension::
sage: set_random_seed()
sage: K = random_cone(max_ambient_dim = 8)
- sage: K.dim() == _restrict_to_space(K,K.span()).dim()
+ sage: K_S = _restrict_to_space(K, K.span())
+ sage: K.dim() == K_S.dim()
True
- Nor should it affect the lineality of a cone::
+ Restricting a cone to its own span should not affects its lineality::
sage: set_random_seed()
sage: K = random_cone(max_ambient_dim = 8)
- sage: K.lineality() == _restrict_to_space(K, K.span()).lineality()
+ sage: K_S = _restrict_to_space(K, K.span())
+ sage: K.lineality() == K_S.lineality()
True
- No matter which space we restrict to, the lineality should not
- increase::
+ Restricting a cone to its own span should not affect the number of
+ facets it has::
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()
+ sage: K_S = _restrict_to_space(K, K.span())
+ sage: len(K.facets()) == len(K_S.facets())
True
- sage: K.lineality() >= _restrict_to_space(K,P).lineality()
+
+ Restricting a solid cone to its own span is a linear isomorphism and
+ should not affect the dimension of its ambient space::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim = 8, solid = True)
+ sage: K_S = _restrict_to_space(K, K.span())
+ sage: K.lattice_dim() == K_S.lattice_dim()
True
- If we do this according to our paper, then the result is proper::
+ Restricting a solid cone to its own span is a linear isomorphism
+ that establishes a one-to-one correspondence of discrete
+ complementarity sets::
sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 8)
+ sage: K = random_cone(max_ambient_dim = 8, solid = True)
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()
+ sage: dcs_K = K.discrete_complementarity_set()
+ sage: dcs_K_S = K_S.discrete_complementarity_set()
+ sage: len(dcs_K) == len(dcs_K_S)
True
- sage: K_SP = _restrict_to_space(K_S, K_S.dual().span())
- sage: K_SP.is_proper()
+
+ Restricting a solid cone to its own span is a linear isomorphism
+ under which the Lyapunov rank (the length of a Lyapunov-like basis)
+ is invariant::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim = 8, solid = True)
+ sage: K_S = _restrict_to_space(K, K.span())
+ sage: len(K.lyapunov_like_basis()) == len(K_S.lyapunov_like_basis())
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::
+ If we restrict a cone to a subspace of its span, the resulting cone
+ should have the same dimension as the space we restricted it to::
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)
+ sage: K = random_cone(max_ambient_dim = 8)
+ sage: W_basis = random_sublist(K.rays(), 0.5)
+ sage: W = K.lattice().vector_space().span(W_basis)
+ sage: K_W = _restrict_to_space(K, W)
+ sage: K_W.lattice_dim() == W.dimension()
True
+ Through a series of restrictions, any closed convex cone can be
+ reduced to a cartesian product with a proper factor [Orlitzky]_::
+
+ 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, K_S.dual().span())
+ sage: K_SP.is_proper()
+ 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.
+ # We want to intersect ``K`` with ``W``. An easy way to do this is
+ # via cone intersection, so we turn the space ``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.
+ # We've already intersected K with W, 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())
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.
+ Compute the Lyapunov rank of this cone.
- 2. The dimension of the linear space of all Lyapunov-like
- transformations on the cone.
-
- INPUT:
-
- A closed, convex polyhedral 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:
- 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]_).
+ 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:
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.
+ .. [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.
+ M. Orlitzky. The Lyapunov rank of an improper cone.
+ http://www.optimization-online.org/DB_HTML/2015/10/5135.html
- .. [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.
+ 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.]_::
+ [Rudolf]_::
sage: positives = Cone([(1,)])
sage: lyapunov_rank(positives)
3
The full space `\mathbb{R}^{n}` has Lyapunov rank `n^{2}`
- [Orlitzky/Gowda]_::
+ [Orlitzky]_::
sage: R5 = VectorSpace(QQ, 5)
sage: gs = R5.basis() + [ -r for r in R5.basis() ]
25
The `L^{3}_{1}` cone is known to have a Lyapunov rank of one
- [Rudolf et al.]_::
+ [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 et al.]_::
+ 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/Gowda]_::
+ + 1` [Orlitzky]_::
sage: K = Cone([(1,0,0,0,0)])
sage: lyapunov_rank(K)
21
A subspace (of dimension `m`) in `n` dimensions should have a
- Lyapunov rank of `n^{2} - m\left(n - m)` [Orlitzky/Gowda]_::
+ 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)
19
The Lyapunov rank should be additive on a product of proper cones
- [Rudolf et al.]_::
+ [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: lyapunov_rank(K) == lyapunov_rank(L31) + lyapunov_rank(octant)
True
- Two isomorphic cones should have the same Lyapunov rank [Rudolf et al.]_.
+ 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}`::
3
The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
- itself [Rudolf et al.]_::
+ itself [Rudolf]_::
sage: K = Cone([(2,2,4), (-1,9,0), (2,0,6)])
sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
TESTS:
The Lyapunov rank should be additive on a product of proper cones
- [Rudolf et al.]_::
+ [Rudolf]_::
sage: set_random_seed()
sage: K1 = random_cone(max_ambient_dim=8,
True
The Lyapunov rank is invariant under a linear isomorphism
- [Orlitzky/Gowda]_::
+ [Orlitzky]_::
sage: K1 = random_cone(max_ambient_dim = 8)
sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular')
True
The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
- itself [Rudolf et al.]_::
+ itself [Rudolf]_::
sage: set_random_seed()
sage: K = random_cone(max_ambient_dim=8)
sage: b == n-1
False
- In fact [Orlitzky/Gowda]_, no closed convex polyhedral cone can have
+ In fact [Orlitzky]_, no closed convex polyhedral cone can have
Lyapunov rank `n-1` in `n` dimensions::
sage: set_random_seed()
False
The calculation of the Lyapunov rank of an improper cone can be
- reduced to that of a proper cone [Orlitzky/Gowda]_::
+ reduced to that of a proper cone [Orlitzky]_::
sage: set_random_seed()
sage: K = random_cone(max_ambient_dim=8)
sage: actual == expected
True
- The Lyapunov rank of any cone is just the dimension of ``K.LL()``::
+ 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.LL())
+ 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/Gowda]_)::
+ (a Theorem in [Orlitzky]_)::
sage: set_random_seed()
sage: K = random_cone(max_ambient_dim=8,
True
"""
- beta = 0
+ beta = 0 # running tally of the Lyapunov rank
m = K.dim()
n = K.lattice_dim()
# Non-pointed reduction lemma.
beta += l * m
- beta += len(K.LL())
+ beta += len(K.lyapunov_like_basis())
return beta
REFERENCES:
- .. [Orlitzky] M. Orlitzky. The Lyapunov rank of an
- improper cone (preprint).
+ M. Orlitzky. The Lyapunov rank of an improper cone.
+ http://www.optimization-online.org/DB_HTML/2015/10/5135.html
EXAMPLES:
sage: is_lyapunov_like(L,K)
True
- Everything in ``K.LL()`` should be Lyapunov-like on ``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: all([is_lyapunov_like(L,K) for L in K.LL()])
+ sage: all([ is_lyapunov_like(L,K) for L in K.lyapunov_like_basis() ])
True
"""
sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
sage: K.is_full_space()
True
- sage: llvs = span([ vector(l.list()) for l in K.LL() ])
- sage: zvs = span([ vector(z.list()) for z in Z_transformations(K) ])
- sage: zvs == llvs
+ 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 == lls
True
TESTS:
sage: set_random_seed()
sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 6)
- sage: llvs = span([ vector(l.list()) for l in K.LL() ])
+ 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.linear_subspace() == llvs
+ sage: z_cone.linear_subspace() == lls
True
"""
projects / sage.d.git / blobdiff