X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=32e13861fdbb0cb2f806f5892cd9c90c1df051f4;hb=3cfc8e228ae337aed975118444a8cbad9a5a7ac3;hp=6d7d2d9b4d2b379bc6041fe089e5a6ea38b8e48a;hpb=342d9147356f7757bed2f9165a600a9e5ec0a5e2;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 6d7d2d9..32e1386 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -7,430 +7,6 @@ addsitedir(abspath('../../')) 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``.