from sage.all import *
-
-def _restrict_to_space(K, W):
- r"""
- 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:
-
- - ``W`` -- The subspace into which this cone will be restricted.
-
- OUTPUT:
-
- 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:
-
- Restricting a solid cone to its own span returns a cone linearly
- isomorphic to the original::
-
- 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 to its own span has the same representation
- 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([(1,1,1)])
- 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
-
- 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:
-
- 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: K_S = _restrict_to_space(K, K.span())
- sage: K_S.is_solid()
- True
-
- 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_S = _restrict_to_space(K, K.span())
- sage: K.nrays() == K_S.nrays()
- True
-
- 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_S = _restrict_to_space(K, K.span())
- sage: K.dim() == K_S.dim()
- True
-
- 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_S = _restrict_to_space(K, K.span())
- sage: K.lineality() == K_S.lineality()
- True
-
- 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: K_S = _restrict_to_space(K, K.span())
- sage: len(K.facets()) == len(K_S.facets())
- True
-
- 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
-
- 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, solid = True)
- sage: K_S = _restrict_to_space(K, K.span())
- 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
-
- 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
-
- 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: 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
- """
- # 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 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())
- 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 double-import is needed to get the underscore methods.
from mjo.cone.cone import *
-from mjo.cone.cone import _restrict_to_space
#
# Tests for _restrict_to_space.
sage: K = random_cone(max_ambient_dim = 8,
....: strictly_convex=False,
....: solid=False)
- 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_S = K._restrict_to_space(K.span())
+ sage: K_SP = K_S.dual()._restrict_to_space(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 = K_S._restrict_to_space(K_S.dual().span())
sage: K_SP.is_proper()
True
sage: K = random_cone(max_ambient_dim = 8,
....: strictly_convex=True,
....: solid=False)
- 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_S = K._restrict_to_space(K.span())
+ sage: K_SP = K_S.dual()._restrict_to_space(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 = K_S._restrict_to_space(K_S.dual().span())
sage: K_SP.is_proper()
True
sage: K = random_cone(max_ambient_dim = 8,
....: strictly_convex=False,
....: 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_S = K._restrict_to_space(K.span())
+ sage: K_SP = K_S.dual()._restrict_to_space(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 = K_S._restrict_to_space(K_S.dual().span())
sage: K_SP.is_proper()
True
sage: K = random_cone(max_ambient_dim = 8,
....: strictly_convex=True,
....: 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_S = K._restrict_to_space(K.span())
+ sage: K_SP = K_S.dual()._restrict_to_space(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 = K_S._restrict_to_space(K_S.dual().span())
sage: K_SP.is_proper()
True
....: solid=False,
....: strictly_convex=False)
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: K_W_star = K._restrict_to_space(J.span()).dual()
+ sage: K_star_W = K.dual()._restrict_to_space(J.span())
sage: _look_isomorphic(K_W_star, K_star_W)
True
....: solid=True,
....: strictly_convex=False)
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: K_W_star = K._restrict_to_space(J.span()).dual()
+ sage: K_star_W = K.dual()._restrict_to_space(J.span())
sage: _look_isomorphic(K_W_star, K_star_W)
True
....: solid=False,
....: strictly_convex=True)
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: K_W_star = K._restrict_to_space(J.span()).dual()
+ sage: K_star_W = K.dual()._restrict_to_space(J.span())
sage: _look_isomorphic(K_W_star, K_star_W)
True
....: solid=True,
....: strictly_convex=True)
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: K_W_star = K._restrict_to_space(J.span()).dual()
+ sage: K_star_W = K.dual()._restrict_to_space(J.span())
sage: _look_isomorphic(K_W_star, K_star_W)
True
....: solid=True)
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)
+ sage: K1.lyapunov_rank() == K2.lyapunov_rank()
True
::
....: solid=False)
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)
+ sage: K1.lyapunov_rank() == K2.lyapunov_rank()
True
::
....: solid=True)
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)
+ sage: K1.lyapunov_rank() == K2.lyapunov_rank()
True
::
....: solid=False)
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)
+ sage: K1.lyapunov_rank() == K2.lyapunov_rank()
True
The Lyapunov rank of a dual cone should be the same as the original
sage: K = random_cone(max_ambient_dim=8,
....: strictly_convex=False,
....: solid=False)
- sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+ sage: K.lyapunov_rank() == K.dual().lyapunov_rank()
True
::
sage: K = random_cone(max_ambient_dim=8,
....: strictly_convex=False,
....: solid=True)
- sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+ sage: K.lyapunov_rank() == K.dual().lyapunov_rank()
True
::
sage: K = random_cone(max_ambient_dim=8,
....: strictly_convex=True,
....: solid=False)
- sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+ sage: K.lyapunov_rank() == K.dual().lyapunov_rank()
True
::
sage: K = random_cone(max_ambient_dim=8,
....: strictly_convex=True,
....: solid=True)
- sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+ sage: K.lyapunov_rank() == K.dual().lyapunov_rank()
True
The Lyapunov rank of a cone ``K`` is the dimension of
sage: K = random_cone(max_ambient_dim=8,
....: strictly_convex=True,
....: solid=True)
- sage: lyapunov_rank(K) == len(K.lyapunov_like_basis())
+ sage: K.lyapunov_rank() == len(K.lyapunov_like_basis())
True
::
sage: K = random_cone(max_ambient_dim=8,
....: strictly_convex=True,
....: solid=False)
- sage: lyapunov_rank(K) == len(K.lyapunov_like_basis())
+ sage: K.lyapunov_rank() == len(K.lyapunov_like_basis())
True
::
sage: K = random_cone(max_ambient_dim=8,
....: strictly_convex=False,
....: solid=True)
- sage: lyapunov_rank(K) == len(K.lyapunov_like_basis())
+ sage: K.lyapunov_rank() == len(K.lyapunov_like_basis())
True
::
sage: K = random_cone(max_ambient_dim=8,
....: strictly_convex=False,
....: solid=False)
- sage: lyapunov_rank(K) == len(K.lyapunov_like_basis())
+ sage: K.lyapunov_rank() == len(K.lyapunov_like_basis())
True
"""
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