+++ /dev/null
-"""
-Additional tests for the mjo.cone.cone module. These are extra
-properties that we'd like to check, but which are overkill for inclusion
-into Sage.
-"""
-
-# 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 *
-
-# The double-import is needed to get the underscore methods.
-from mjo.cone.cone import *
-
-
-def _restrict_to_subspace(K, W):
- r"""
- Restrict ``K`` (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 "dual of" 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:
-
- - ``K`` -- The cone to restrict.
-
- - ``W`` -- The subspace into which ``K`` 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_subspace(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: K = Cone([(1,0)])
- sage: K_S = _restrict_to_subspace(K, K.span()).rays()
- sage: K_S
- N(1)
- in 1-d lattice N
- sage: K = Cone([(1,1,1)])
- sage: K_S = _restrict_to_subspace(K, K.span()).rays()
- sage: K_S
- N(1)
- in 1-d lattice N
-
- 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_subspace(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_subspace(K, K.span())
- sage: K_S.is_solid()
- True
-
- Restricting a cone to its 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_subspace(K, K.span())
- sage: K.nrays() == K_S.nrays()
- True
-
- Restricting a cone to its span should not affect its dimension::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 8)
- sage: K_S = _restrict_to_subspace(K, K.span())
- sage: K.dim() == K_S.dim()
- True
-
- Restricting a cone to its span should not affects its lineality::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 8)
- sage: K_S = _restrict_to_subspace(K, K.span())
- sage: K.lineality() == K_S.lineality()
- True
-
- Restricting a cone to its 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_subspace(K, K.span())
- sage: len(K.facets()) == len(K_S.facets())
- True
-
- Restricting a solid cone to its 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_subspace(K, K.span())
- sage: K.lattice_dim() == K_S.lattice_dim()
- True
-
- Restricting a solid cone to its 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_subspace(K, K.span())
- sage: dcs1 = K.discrete_complementarity_set()
- sage: dcs2 = K_S.discrete_complementarity_set()
- sage: len(dcs1) == len(dcs2)
- True
-
- Restricting a solid cone to its span is a linear isomorphism
- under which 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_subspace(K, K.span())
- sage: LL1 = K.lyapunov_like_basis()
- sage: LL2 = K_S.lyapunov_like_basis()
- sage: len(LL1) == len(LL2)
- True
-
- If we restrict a cone to a subspace of its span, the resulting
- cone should have the same dimension as the subspace::
-
- 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_subspace(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_subspace(K, K.span())
- sage: P = K_S.dual().span()
- sage: K_SP = _restrict_to_subspace(K_S, P)
- sage: K_SP.is_proper()
- True
- """
- # We want to intersect this cone with ``W``. We can do that via
- # cone intersection, so we first turn the space ``W`` into a cone.
- W_rays = W.basis() + [ -b for b in W.basis() ]
- W_cone = Cone(W_rays, lattice=K.lattice())
- K = K.intersection(W_cone)
-
- # Now every generator of ``K`` should belong to ``W``.
- K_W_rays = [ W.coordinate_vector(r) for r in K.rays() ]
-
- L = ToricLattice(W.dimension())
- return Cone(K_W_rays, lattice=L)
-
-
-
-#
-# Tests for _restrict_to_subspace.
-#
-def _look_isomorphic(K1, K2):
- r"""
- Test whether or not ``K1`` and ``K2`` look linearly isomorphic.
-
- This is a hack to get around the fact that it's difficult to tell
- when two cones are linearly isomorphic. Instead, we check a list of
- properties that should be preserved under linear isomorphism.
-
- OUTPUT:
-
- ``True`` if ``K1`` and ``K2`` look isomorphic, or ``False``
- if we can prove that they are not isomorphic.
-
- EXAMPLES:
-
- Any proper cone with three generators in `\mathbb{R}^{3}` is
- isomorphic to 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: _look_isomorphic(K1, K2)
- True
-
- Negating a cone gives you an isomorphic cone::
-
- sage: K = Cone([(0,2,-5), (-6, 2, 4), (0, 51, 0)])
- sage: _look_isomorphic(K, -K)
- True
-
- TESTS:
-
- Any cone is isomorphic to itself::
-
- sage: K = random_cone(max_ambient_dim = 8)
- sage: _look_isomorphic(K, K)
- True
-
- After applying an invertible matrix to the rows of a cone, the
- result should is isomorphic to 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: _look_isomorphic(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
-
-
-"""
-Apply _restrict_to_subspace according to our paper (to obtain our main
-result). Test all four parameter combinations::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 8,
- ....: strictly_convex=False,
- ....: solid=False)
- sage: K_S = _restrict_to_subspace(K, K.span())
- sage: K_S2 = K.solid_restriction()
- sage: _look_isomorphic(K_S, K_S2)
- True
- sage: K_SP = _restrict_to_subspace(K_S.dual(), K_S.dual().span()).dual()
- sage: K_SP2 = K_S.strict_quotient()
- sage: K_SP.is_proper()
- True
- sage: K_SP2.is_proper()
- True
- sage: _look_isomorphic(K_SP, K_SP2)
- True
- sage: K_SP = _restrict_to_subspace(K_S, K_S.dual().span())
- sage: K_SP.is_proper()
- True
- sage: _look_isomorphic(K_SP, K_SP2)
- True
-
-::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 8,
- ....: strictly_convex=False,
- ....: solid=True)
- sage: K_S = _restrict_to_subspace(K, K.span())
- sage: K_S2 = K.solid_restriction()
- sage: _look_isomorphic(K_S, K_S2)
- True
- sage: K_SP = _restrict_to_subspace(K_S.dual(), K_S.dual().span()).dual()
- sage: K_SP2 = K_S.strict_quotient()
- sage: K_SP.is_proper()
- True
- sage: K_SP2.is_proper()
- True
- sage: _look_isomorphic(K_SP, K_SP2)
- True
- sage: K_SP = _restrict_to_subspace(K_S, K_S.dual().span())
- sage: K_SP.is_proper()
- True
- sage: _look_isomorphic(K_SP, K_SP2)
- True
-
-::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 8,
- ....: strictly_convex=True,
- ....: solid=False)
- sage: K_S = _restrict_to_subspace(K, K.span())
- sage: K_S2 = K.solid_restriction()
- sage: _look_isomorphic(K_S, K_S2)
- True
- sage: K_SP = _restrict_to_subspace(K_S.dual(), K_S.dual().span()).dual()
- sage: K_SP2 = K_S.strict_quotient()
- sage: K_SP.is_proper()
- True
- sage: K_SP2.is_proper()
- True
- sage: _look_isomorphic(K_SP, K_SP2)
- True
- sage: K_SP = _restrict_to_subspace(K_S, K_S.dual().span())
- sage: K_SP.is_proper()
- True
- sage: _look_isomorphic(K_SP, K_SP2)
- True
-
-::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 8,
- ....: strictly_convex=True,
- ....: solid=True)
- sage: K_S = _restrict_to_subspace(K, K.span())
- sage: K_S2 = K.solid_restriction()
- sage: _look_isomorphic(K_S, K_S2)
- True
- sage: K_SP = _restrict_to_subspace(K_S.dual(), K_S.dual().span()).dual()
- sage: K_SP2 = K_S.strict_quotient()
- sage: K_SP.is_proper()
- True
- sage: K_SP2.is_proper()
- True
- sage: _look_isomorphic(K_SP, K_SP2)
- True
- sage: K_SP = _restrict_to_subspace(K_S, K_S.dual().span())
- sage: K_SP.is_proper()
- True
- sage: _look_isomorphic(K_SP, K_SP2)
- 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 rho. Test
-all parameter combinations::
-
- 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 = _restrict_to_subspace(K, J.span()).dual()
- sage: K_star_W = _restrict_to_subspace(K.dual(), J.span())
- sage: _look_isomorphic(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 = _restrict_to_subspace(K, J.span()).dual()
- sage: K_star_W = _restrict_to_subspace(K.dual(), J.span())
- sage: _look_isomorphic(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 = _restrict_to_subspace(K, J.span()).dual()
- sage: K_star_W = _restrict_to_subspace(K.dual(), J.span())
- sage: _look_isomorphic(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: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
- sage: K_W_star = _restrict_to_subspace(K, J.span()).dual()
- sage: K_star_W = _restrict_to_subspace(K.dual(), J.span())
- sage: _look_isomorphic(K_W_star, K_star_W)
- True
-
-Ensure that ``__restrict_to_subspace(K, K.span())`` and
-``K.solid_restriction()`` are actually equivalent::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8)
- sage: K1 = _restrict_to_subspace(K, K.span())
- sage: K2 = K.solid_restriction()
- sage: _look_isomorphic(K1,K2)
- True
-
-Ensure that ``K.__restrict_to_subspace(K,K.dual().span())`` and
-``strict_quotient`` are actually equivalent::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=6)
- sage: K1 = _restrict_to_subspace(K, K.dual().span())
- sage: K2 = K.strict_quotient()
- sage: _look_isomorphic(K1,K2)
- True
-"""
-
-
-#
-# Lyapunov rank tests
-#
-"""
-
-The Lyapunov rank is invariant under a linear isomorphism. Check all
-combinations of parameters::
-
- 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: K1.lyapunov_rank() == K2.lyapunov_rank()
- 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: K1.lyapunov_rank() == K2.lyapunov_rank()
- 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: K1.lyapunov_rank() == K2.lyapunov_rank()
- 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: K1.lyapunov_rank() == K2.lyapunov_rank()
- True
-
-The Lyapunov rank of a dual cone should be the same as the original
-cone. Check all combinations of parameters::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8,
- ....: strictly_convex=False,
- ....: solid=False)
- sage: K.lyapunov_rank() == K.dual().lyapunov_rank()
- True
-
-::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8,
- ....: strictly_convex=False,
- ....: solid=True)
- sage: K.lyapunov_rank() == K.dual().lyapunov_rank()
- True
-
-::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8,
- ....: strictly_convex=True,
- ....: solid=False)
- sage: K.lyapunov_rank() == K.dual().lyapunov_rank()
- True
-
-::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8,
- ....: strictly_convex=True,
- ....: solid=True)
- sage: K.lyapunov_rank() == K.dual().lyapunov_rank()
- True
-
-The Lyapunov rank of a cone ``K`` is the dimension of
-``K.lyapunov_like_basis()``. Check all combinations of parameters::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8,
- ....: strictly_convex=True,
- ....: solid=True)
- sage: K.lyapunov_rank() == len(K.lyapunov_like_basis())
- True
-
-::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8,
- ....: strictly_convex=True,
- ....: solid=False)
- sage: K.lyapunov_rank() == len(K.lyapunov_like_basis())
- True
-
-::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8,
- ....: strictly_convex=False,
- ....: solid=True)
- sage: K.lyapunov_rank() == len(K.lyapunov_like_basis())
- True
-
-::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8,
- ....: strictly_convex=False,
- ....: solid=False)
- sage: K.lyapunov_rank() == len(K.lyapunov_like_basis())
- True
-
-"""
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