From 9d6b29ab0aa0c8c6395834d316d48aa90b3b6c45 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Wed, 6 Jan 2016 09:01:10 -0500 Subject: [PATCH] Remove the cone tests since they all belong to one paper (now in that repo). --- mjo/cone/tests.py | 576 ---------------------------------------------- 1 file changed, 576 deletions(-) delete mode 100644 mjo/cone/tests.py diff --git a/mjo/cone/tests.py b/mjo/cone/tests.py deleted file mode 100644 index 5a910f0..0000000 --- a/mjo/cone/tests.py +++ /dev/null @@ -1,576 +0,0 @@ -""" -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 - -""" -- 2.43.2