From 342d9147356f7757bed2f9165a600a9e5ec0a5e2 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Mon, 12 Oct 2015 13:03:03 -0400 Subject: [PATCH] Cleanup on _restrict_to_space() tests and documentation. --- mjo/cone/cone.py | 142 +++++++++++++++++++++++++++++++++-------------- 1 file changed, 100 insertions(+), 42 deletions(-) diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 5f058ca..6d7d2d9 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -10,7 +10,22 @@ from sage.all import * 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: @@ -20,16 +35,26 @@ def _restrict_to_space(K, W): 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)]) @@ -37,7 +62,7 @@ def _restrict_to_space(K, W): 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) @@ -45,83 +70,116 @@ def _restrict_to_space(K, W): 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()) -- 2.44.2