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,
- ....: strictly_convex=False,
- ....: solid=False)
- sage: K_S = _rho(K)
- sage: K_SP = _rho(K_S.dual()).dual()
- sage: K_SP.is_proper()
- True
- sage: K_SP = _rho(K_S, K_S.dual())
- sage: K_SP.is_proper()
- True
-
- ::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 8,
- ....: strictly_convex=True,
- ....: solid=False)
- sage: K_S = _rho(K)
- sage: K_SP = _rho(K_S.dual()).dual()
- sage: K_SP.is_proper()
- True
- sage: K_SP = _rho(K_S, K_S.dual())
- sage: K_SP.is_proper()
- True
-
- ::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 8,
- ....: strictly_convex=False,
- ....: solid=True)
- sage: K_S = _rho(K)
- sage: K_SP = _rho(K_S.dual()).dual()
- sage: K_SP.is_proper()
- True
- sage: K_SP = _rho(K_S, K_S.dual())
- sage: K_SP.is_proper()
- True
-
- ::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 8,
- ....: strictly_convex=True,
- ....: solid=True)
+ sage: K = random_cone(max_ambient_dim = 8)
sage: K_S = _rho(K)
sage: K_SP = _rho(K_S.dual()).dual()
sage: K_SP.is_proper()
sage: K_SP.is_proper()
True
- Test Proposition 7 in our paper concerning the duals and
+ 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 rho::
sage: set_random_seed()
- sage: J = random_cone(max_ambient_dim = 8,
- ....: solid=False,
- ....: strictly_convex=False)
- sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
- sage: K_W_star = _rho(K, J).dual()
- sage: K_star_W = _rho(K.dual(), J)
- sage: _basically_the_same(K_W_star, K_star_W)
- True
-
- ::
-
- sage: set_random_seed()
- sage: J = random_cone(max_ambient_dim = 8,
- ....: solid=True,
- ....: strictly_convex=False)
- sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
- sage: K_W_star = _rho(K, J).dual()
- sage: K_star_W = _rho(K.dual(), J)
- sage: _basically_the_same(K_W_star, K_star_W)
- True
-
- ::
-
- sage: set_random_seed()
- sage: J = random_cone(max_ambient_dim = 8,
- ....: solid=False,
- ....: strictly_convex=True)
- sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
- sage: K_W_star = _rho(K, J).dual()
- sage: K_star_W = _rho(K.dual(), J)
- sage: _basically_the_same(K_W_star, K_star_W)
- True
-
- ::
-
- sage: set_random_seed()
- sage: J = random_cone(max_ambient_dim = 8,
- ....: solid=True,
- ....: strictly_convex=True)
+ sage: J = random_cone(max_ambient_dim = 8)
sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
sage: K_W_star = _rho(K, J).dual()
sage: K_star_W = _rho(K.dual(), J)
def lyapunov_rank(K):
r"""
- Compute the Lyapunov (or bilinearity) rank of this cone.
+ Compute the Lyapunov rank (or bilinearity rank) of this cone.
The Lyapunov rank of a cone can be thought of in (mainly) two ways:
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 the first reference).
-
- .. note::
-
- In the references, the cones are always assumed to be proper. We
- do not impose this restriction.
-
- .. seealso::
-
- :meth:`is_proper`
+ not possible (see [Orlitzky/Gowda]_).
ALGORITHM:
sage: lyapunov_rank(K1) == lyapunov_rank(K2)
True
- Just to be sure, test a few more::
-
- sage: K1 = random_cone(max_ambient_dim=8,
- ....: strictly_convex=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)
- True
-
- ::
-
- sage: K1 = random_cone(max_ambient_dim=8,
- ....: strictly_convex=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)
- True
-
- ::
-
- sage: K1 = random_cone(max_ambient_dim=8,
- ....: strictly_convex=False,
- ....: 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)
- True
-
- ::
-
- sage: K1 = random_cone(max_ambient_dim=8,
- ....: strictly_convex=False,
- ....: 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)
- True
-
The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
itself [Rudolf et al.]_::
sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
True
- Make sure we exercise the non-strictly-convex/non-solid case::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8,
- ....: strictly_convex=False,
- ....: solid=False)
- sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
- True
-
- Let's check the other permutations as well, just to be sure::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8,
- ....: strictly_convex=False,
- ....: solid=True)
- sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
- True
-
- ::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8,
- ....: strictly_convex=True,
- ....: solid=False)
- sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
- True
-
- ::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8,
- ....: strictly_convex=True,
- ....: solid=True)
- 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
sage: actual == expected
True
- The Lyapunov rank of a proper cone is just the dimension of ``LL(K)``::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8,
- ....: strictly_convex=True,
- ....: solid=True)
- sage: lyapunov_rank(K) == len(LL(K))
- True
-
- In fact the same can be said of any cone. These additional tests
- just increase our confidence that the reduction scheme works::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8,
- ....: strictly_convex=True,
- ....: solid=False)
- sage: lyapunov_rank(K) == len(LL(K))
- 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,
- ....: strictly_convex=False,
- ....: solid=True)
- sage: lyapunov_rank(K) == len(LL(K))
- True
-
- ::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8,
- ....: strictly_convex=False,
- ....: solid=False)
+ sage: K = random_cone(max_ambient_dim=8)
sage: lyapunov_rank(K) == len(LL(K))
True
- Test Theorem 3 in [Orlitzky/Gowda]_::
+ 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,
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