X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=aeec0c90b5f582c8ba858d4b616fff191fb6d529;hb=c2d15c0884c8b92483f58826747887bd2bcdcdeb;hp=60f9c34ec8bc271d65812859f51ca77636c8cbbc;hpb=874e3ce831e0b1901b3c280a32ffe18e36f54959;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 60f9c34..aeec0c9 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -1,524 +1,455 @@ -# 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 * +from sage.geometry.cone import is_Cone -def project_span(K): +def is_positive_on(L,K): r""" - Project ``K`` into its own span. + Determine whether or not ``L`` is positive on ``K``. - EXAMPLES:: + We say that ``L`` is positive on a closed convex cone ``K`` if + `L\left\lparen x \right\rparen` belongs to ``K`` for all `x` in + ``K``. This property need only be checked for generators of ``K``. - sage: K = Cone([(1,)]) - sage: project_span(K) == K - True + To reliably check whether or not ``L`` is positive, its base ring + must be either exact (for example, the rationals) or ``SR``. An + exact ring is more reliable, but in some cases a matrix whose + entries contain symbolic constants like ``e`` and ``pi`` will work. - sage: K2 = Cone([(1,0)]) - sage: project_span(K2).rays() - N(1) - in 1-d lattice N - sage: K3 = Cone([(1,0,0)]) - sage: project_span(K3).rays() - N(1) - in 1-d lattice N - sage: project_span(K2) == project_span(K3) - True + INPUT: - TESTS: + - ``L`` -- A matrix over either an exact ring or ``SR``. - The projected cone should always be solid:: + - ``K`` -- A polyhedral closed convex cone. - sage: K = random_cone() - sage: K_S = project_span(K) - sage: K_S.is_solid() - True + OUTPUT: - If we do this according to our paper, then the result is proper:: + If the base ring of ``L`` is exact, then ``True`` will be returned if + and only if ``L`` is positive on ``K``. - sage: K = random_cone() - sage: K_S = project_span(K) - sage: P = project_span(K_S.dual()).dual() - sage: P.is_proper() - True + If the base ring of ``L`` is ``SR``, then the situation is more + complicated: - """ - F = K.lattice().base_field() - Q = K.lattice().quotient(K.sublattice_complement()) - vecs = [ vector(F, reversed(list(Q(r)))) for r in K.rays() ] + - ``True`` will be returned if it can be proven that ``L`` + is positive on ``K``. + - ``False`` will be returned if it can be proven that ``L`` + is not positive on ``K``. + - ``False`` will also be returned if we can't decide; specifically + if we arrive at a symbolic inequality that cannot be resolved. - L = None - if len(vecs) == 0: - L = ToricLattice(0) + .. SEEALSO:: - return Cone(vecs, lattice=L) + :func:`is_cross_positive_on`, + :func:`is_Z_operator_on`, + :func:`is_lyapunov_like_on` + EXAMPLES: -def rename_lattice(L,s): - r""" - Change all names of the given lattice to ``s``. - """ - L._name = s - L._dual_name = s - L._latex_name = s - L._latex_dual_name = s + Nonnegative matrices are positive operators on the nonnegative + orthant:: -def span_iso(K): - r""" - Return an isomorphism (and its inverse) that will send ``K`` into a - lower-dimensional space isomorphic to its span (and back). + sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)]) + sage: L = random_matrix(QQ,3).apply_map(abs) + sage: is_positive_on(L,K) + True - EXAMPLES: + TESTS: - The inverse composed with the isomorphism should be the identity:: + The identity operator is always positive:: - sage: K = random_cone(max_dim=10) - sage: (phi, phi_inv) = span_iso(K) - sage: phi_inv(phi(K)) == K + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=8) + sage: L = identity_matrix(K.lattice_dim()) + sage: is_positive_on(L,K) True - The image of ``K`` under the isomorphism should have full dimension:: + The "zero" operator is always positive:: - sage: K = random_cone(max_dim=10) - sage: (phi, phi_inv) = span_iso(K) - sage: phi(K).dim() == phi(K).lattice_dim() + sage: K = random_cone(max_ambient_dim=8) + sage: R = K.lattice().vector_space().base_ring() + sage: L = zero_matrix(R, K.lattice_dim()) + sage: is_positive_on(L,K) True - """ - phi_domain = K.sublattice().vector_space() - phi_codo = VectorSpace(phi_domain.base_field(), phi_domain.dimension()) + Everything in ``K.positive_operators_gens()`` should be + positive on ``K``:: - # S goes from the new space to the cone space. - S = linear_transformation(phi_codo, phi_domain, phi_domain.basis()) + sage: K = random_cone(max_ambient_dim=5) + sage: all([ is_positive_on(L,K) # long time + ....: for L in K.positive_operators_gens() ]) # long time + True + sage: all([ is_positive_on(L.change_ring(SR),K) # long time + ....: for L in K.positive_operators_gens() ]) # long time + True - # phi goes from the cone space to the new space. - def phi(J_orig): - r""" - Takes a cone ``J`` and sends it into the new space. - """ - newrays = map(S.inverse(), J_orig.rays()) - L = None - if len(newrays) == 0: - L = ToricLattice(0) + Technically we could test this, but for now only closed convex cones + are supported as our ``K`` argument:: - return Cone(newrays, lattice=L) + sage: K = [ vector([1,2,3]), vector([5,-1,7]) ] + sage: L = identity_matrix(3) + sage: is_positive_on(L,K) + Traceback (most recent call last): + ... + TypeError: K must be a Cone. - def phi_inverse(J_sub): - r""" - The inverse to phi which goes from the new space to the cone space. - """ - newrays = map(S, J_sub.rays()) - return Cone(newrays, lattice=K.lattice()) + We can't give reliable answers over inexact rings:: + sage: K = Cone([(1,2,3), (4,5,6)]) + sage: L = identity_matrix(RR,3) + sage: is_positive_on(L,K) + Traceback (most recent call last): + ... + ValueError: The base ring of L is neither SR nor exact. - return (phi, phi_inverse) + """ + if not is_Cone(K): + raise TypeError('K must be a Cone.') + if not L.base_ring().is_exact() and not L.base_ring() is SR: + raise ValueError('The base ring of L is neither SR nor exact.') + if L.base_ring().is_exact(): + # This should be way faster than computing the dual and + # checking a bunch of inequalities, but it doesn't work if + # ``L*x`` is symbolic. For example, ``e in Cone([(1,)])`` + # is true, but returns ``False``. + return all([ L*x in K for x in K ]) + else: + # Fall back to inequality-checking when the entries of ``L`` + # might be symbolic. + return all([ s*(L*x) >= 0 for x in K for s in K.dual() ]) -def discrete_complementarity_set(K): + +def is_cross_positive_on(L,K): r""" - Compute the discrete complementarity set of this cone. + Determine whether or not ``L`` is cross-positive on ``K``. - The complementarity set of this cone is the set of all orthogonal - pairs `(x,s)` such that `x` is in this cone, and `s` is in its - dual. The discrete complementarity set restricts `x` and `s` to be - generators of their respective cones. + We say that ``L`` is cross-positive on a closed convex cone``K`` if + `\left\langle L\left\lparenx\right\rparen,s\right\rangle \ge 0` for + all pairs `\left\langle x,s \right\rangle` in the complementarity + set of ``K``. This property need only be checked for generators of + ``K`` and its dual. - OUTPUT: + To reliably check whether or not ``L`` is cross-positive, its base + ring must be either exact (for example, the rationals) or ``SR``. An + exact ring is more reliable, but in some cases a matrix whose + entries contain symbolic constants like ``e`` and ``pi`` will work. - A list of pairs `(x,s)` such that, + INPUT: - * `x` is in this cone. - * `x` is a generator of this cone. - * `s` is in this cone's dual. - * `s` is a generator of this cone's dual. - * `x` and `s` are orthogonal. + - ``L`` -- A matrix over either an exact ring or ``SR``. - EXAMPLES: + - ``K`` -- A polyhedral closed convex cone. - The discrete complementarity set of the nonnegative orthant consists - of pairs of standard basis vectors:: + OUTPUT: - sage: K = Cone([(1,0),(0,1)]) - sage: discrete_complementarity_set(K) - [((1, 0), (0, 1)), ((0, 1), (1, 0))] + If the base ring of ``L`` is exact, then ``True`` will be returned if + and only if ``L`` is cross-positive on ``K``. - If the cone consists of a single ray, the second components of the - discrete complementarity set should generate the orthogonal - complement of that ray:: + If the base ring of ``L`` is ``SR``, then the situation is more + complicated: - sage: K = Cone([(1,0)]) - sage: discrete_complementarity_set(K) - [((1, 0), (0, 1)), ((1, 0), (0, -1))] - sage: K = Cone([(1,0,0)]) - sage: discrete_complementarity_set(K) - [((1, 0, 0), (0, 1, 0)), - ((1, 0, 0), (0, -1, 0)), - ((1, 0, 0), (0, 0, 1)), - ((1, 0, 0), (0, 0, -1))] + - ``True`` will be returned if it can be proven that ``L`` + is cross-positive on ``K``. + - ``False`` will be returned if it can be proven that ``L`` + is not cross-positive on ``K``. + - ``False`` will also be returned if we can't decide; specifically + if we arrive at a symbolic inequality that cannot be resolved. - When the cone is the entire space, its dual is the trivial cone, so - the discrete complementarity set is empty:: + .. SEEALSO:: - sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) - sage: discrete_complementarity_set(K) - [] + :func:`is_positive_on`, + :func:`is_Z_operator_on`, + :func:`is_lyapunov_like_on` - TESTS: + EXAMPLES: - The complementarity set of the dual can be obtained by switching the - components of the complementarity set of the original cone:: + The identity operator is always cross-positive:: - sage: K1 = random_cone(max_dim=10, max_rays=10) - sage: K2 = K1.dual() - sage: expected = [(x,s) for (s,x) in discrete_complementarity_set(K2)] - sage: actual = discrete_complementarity_set(K1) - sage: actual == expected + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=8) + sage: L = identity_matrix(K.lattice_dim()) + sage: is_cross_positive_on(L,K) True - """ - V = K.lattice().vector_space() - - # Convert the rays to vectors so that we can compute inner - # products. - xs = [V(x) for x in K.rays()] - ss = [V(s) for s in K.dual().rays()] - - return [(x,s) for x in xs for s in ss if x.inner_product(s) == 0] - + The "zero" operator is always cross-positive:: -def LL(K): - r""" - Compute the space `\mathbf{LL}` of all Lyapunov-like transformations - on this cone. - - OUTPUT: + sage: K = random_cone(max_ambient_dim=8) + sage: R = K.lattice().vector_space().base_ring() + sage: L = zero_matrix(R, K.lattice_dim()) + sage: is_cross_positive_on(L,K) + True - A list of matrices forming a basis for the space of all - Lyapunov-like transformations on the given cone. + TESTS: - EXAMPLES: + Everything in ``K.cross_positive_operators_gens()`` should be + cross-positive on ``K``:: - The trivial cone has no Lyapunov-like transformations:: + sage: K = random_cone(max_ambient_dim=5) + sage: all([ is_cross_positive_on(L,K) # long time + ....: for L in K.cross_positive_operators_gens() ]) # long time + True + sage: all([ is_cross_positive_on(L.change_ring(SR),K) # long time + ....: for L in K.cross_positive_operators_gens() ]) # long time + True - sage: L = ToricLattice(0) - sage: K = Cone([], lattice=L) - sage: LL(K) - [] + Technically we could test this, but for now only closed convex cones + are supported as our ``K`` argument:: - The Lyapunov-like transformations on the nonnegative orthant are - simply diagonal matrices:: + sage: L = identity_matrix(3) + sage: K = [ vector([8,2,-8]), vector([5,-5,7]) ] + sage: is_cross_positive_on(L,K) + Traceback (most recent call last): + ... + TypeError: K must be a Cone. - sage: K = Cone([(1,)]) - sage: LL(K) - [[1]] + We can't give reliable answers over inexact rings:: - sage: K = Cone([(1,0),(0,1)]) - sage: LL(K) - [ - [1 0] [0 0] - [0 0], [0 1] - ] + sage: K = Cone([(1,2,3), (4,5,6)]) + sage: L = identity_matrix(RR,3) + sage: is_cross_positive_on(L,K) + Traceback (most recent call last): + ... + ValueError: The base ring of L is neither SR nor exact. - sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)]) - sage: LL(K) - [ - [1 0 0] [0 0 0] [0 0 0] - [0 0 0] [0 1 0] [0 0 0] - [0 0 0], [0 0 0], [0 0 1] - ] - - Only the identity matrix is Lyapunov-like on the `L^{3}_{1}` and - `L^{3}_{\infty}` cones [Rudolf et al.]_:: - - sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)]) - sage: LL(L31) - [ - [1 0 0] - [0 1 0] - [0 0 1] - ] - - sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)]) - sage: LL(L3infty) - [ - [1 0 0] - [0 1 0] - [0 0 1] - ] + """ + if not is_Cone(K): + raise TypeError('K must be a Cone.') + if not L.base_ring().is_exact() and not L.base_ring() is SR: + raise ValueError('The base ring of L is neither SR nor exact.') - TESTS: + return all([ s*(L*x) >= 0 + for (x,s) in K.discrete_complementarity_set() ]) - The inner product `\left< L\left(x\right), s \right>` is zero for - every pair `\left( x,s \right)` in the discrete complementarity set - of the cone:: +def is_Z_operator_on(L,K): + r""" + Determine whether or not ``L`` is a Z-operator on ``K``. - sage: K = random_cone(max_dim=8, max_rays=10) - sage: C_of_K = discrete_complementarity_set(K) - sage: l = [ (L*x).inner_product(s) for (x,s) in C_of_K for L in LL(K) ] - sage: sum(map(abs, l)) - 0 + We say that ``L`` is a Z-operator on a closed convex cone``K`` if + `\left\langle L\left\lparenx\right\rparen,s\right\rangle \le 0` for + all pairs `\left\langle x,s \right\rangle` in the complementarity + set of ``K``. It is known that this property need only be checked + for generators of ``K`` and its dual. - """ - V = K.lattice().vector_space() + A matrix is a Z-operator on ``K`` if and only if its negation is a + cross-positive operator on ``K``. - C_of_K = discrete_complementarity_set(K) + To reliably check whether or not ``L`` is a Z operator, its base + ring must be either exact (for example, the rationals) or ``SR``. An + exact ring is more reliable, but in some cases a matrix whose + entries contain symbolic constants like ``e`` and ``pi`` will work. - tensor_products = [s.tensor_product(x) for (x,s) in C_of_K] + INPUT: - # Sage doesn't think matrices are vectors, so we have to convert - # our matrices to vectors explicitly before we can figure out how - # many are linearly-indepenedent. - # - # The space W has the same base ring as V, but dimension - # dim(V)^2. So it has the same dimension as the space of linear - # transformations on V. In other words, it's just the right size - # to create an isomorphism between it and our matrices. - W = VectorSpace(V.base_ring(), V.dimension()**2) + - ``L`` -- A matrix over either an exact ring or ``SR``. - # Turn our matrices into long vectors... - vectors = [ W(m.list()) for m in tensor_products ] + - ``K`` -- A polyhedral closed convex cone. - # Vector space representation of Lyapunov-like matrices - # (i.e. vec(L) where L is Luapunov-like). - LL_vector = W.span(vectors).complement() + OUTPUT: - # Now construct an ambient MatrixSpace in which to stick our - # transformations. - M = MatrixSpace(V.base_ring(), V.dimension()) + If the base ring of ``L`` is exact, then ``True`` will be returned if + and only if ``L`` is a Z-operator on ``K``. - matrix_basis = [ M(v.list()) for v in LL_vector.basis() ] + If the base ring of ``L`` is ``SR``, then the situation is more + complicated: - return matrix_basis + - ``True`` will be returned if it can be proven that ``L`` + is a Z-operator on ``K``. + - ``False`` will be returned if it can be proven that ``L`` + is not a Z-operator on ``K``. + - ``False`` will also be returned if we can't decide; specifically + if we arrive at a symbolic inequality that cannot be resolved. + .. SEEALSO:: + :func:`is_positive_on`, + :func:`is_cross_positive_on`, + :func:`is_lyapunov_like_on` -def lyapunov_rank(K): - r""" - Compute the Lyapunov (or bilinearity) rank of this cone. + EXAMPLES: - The Lyapunov rank of a cone can be thought of in (mainly) two ways: + The identity operator is always a Z-operator:: - 1. The dimension of the Lie algebra of the automorphism group of the - cone. + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=8) + sage: L = identity_matrix(K.lattice_dim()) + sage: is_Z_operator_on(L,K) + True - 2. The dimension of the linear space of all Lyapunov-like - transformations on the cone. + The "zero" operator is always a Z-operator:: - INPUT: + sage: K = random_cone(max_ambient_dim=8) + sage: R = K.lattice().vector_space().base_ring() + sage: L = zero_matrix(R, K.lattice_dim()) + sage: is_Z_operator_on(L,K) + True - A closed, convex polyhedral cone. + TESTS: - OUTPUT: + Everything in ``K.Z_operators_gens()`` should be a Z-operator + on ``K``:: - 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). + sage: K = random_cone(max_ambient_dim=5) + sage: all([ is_Z_operator_on(L,K) # long time + ....: for L in K.Z_operators_gens() ]) # long time + True + sage: all([ is_Z_operator_on(L.change_ring(SR),K) # long time + ....: for L in K.Z_operators_gens() ]) # long time + True - .. note:: + Technically we could test this, but for now only closed convex cones + are supported as our ``K`` argument:: - In the references, the cones are always assumed to be proper. We - do not impose this restriction. + sage: L = identity_matrix(3) + sage: K = [ vector([-4,20,3]), vector([1,-5,2]) ] + sage: is_Z_operator_on(L,K) + Traceback (most recent call last): + ... + TypeError: K must be a Cone. - .. seealso:: - :meth:`is_proper` + We can't give reliable answers over inexact rings:: - ALGORITHM: + sage: K = Cone([(1,2,3), (4,5,6)]) + sage: L = identity_matrix(RR,3) + sage: is_Z_operator_on(L,K) + Traceback (most recent call last): + ... + ValueError: The base ring of L is neither SR nor exact. - 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}`. + """ + return is_cross_positive_on(-L,K) - 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. +def is_lyapunov_like_on(L,K): + r""" + Determine whether or not ``L`` is Lyapunov-like on ``K``. - .. [Orlitzky/Gowda] M. Orlitzky and M. S. Gowda. The Lyapunov Rank of an - Improper Cone. Work in-progress. + We say that ``L`` is Lyapunov-like on a closed convex cone ``K`` if + `\left\langle L\left\lparenx\right\rparen,s\right\rangle = 0` for + all pairs `\left\langle x,s \right\rangle` in the complementarity + set of ``K``. This property need only be checked for generators of + ``K`` and its dual. - .. [Rudolf et al.] 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. + An operator is Lyapunov-like on ``K`` if and only if both the + operator itself and its negation are cross-positive on ``K``. - EXAMPLES: + To reliably check whether or not ``L`` is Lyapunov-like, its base + ring must be either exact (for example, the rationals) or ``SR``. An + exact ring is more reliable, but in some cases a matrix whose + entries contain symbolic constants like ``e`` and ``pi`` will work. - The nonnegative orthant in `\mathbb{R}^{n}` always has rank `n` - [Rudolf et al.]_:: + INPUT: - 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 + - ``L`` -- A matrix over either an exact ring or ``SR``. - The `L^{3}_{1}` cone is known to have a Lyapunov rank of one - [Rudolf et al.]_:: + - ``K`` -- A polyhedral closed convex cone. - sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)]) - sage: lyapunov_rank(L31) - 1 + OUTPUT: - Likewise for the `L^{3}_{\infty}` cone [Rudolf et al.]_:: + If the base ring of ``L`` is exact, then ``True`` will be returned if + and only if ``L`` is Lyapunov-like on ``K``. - sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)]) - sage: lyapunov_rank(L3infty) - 1 + If the base ring of ``L`` is ``SR``, then the situation is more + complicated: - The Lyapunov rank should be additive on a product of cones - [Rudolf et al.]_:: + - ``True`` will be returned if it can be proven that ``L`` + is Lyapunov-like on ``K``. + - ``False`` will be returned if it can be proven that ``L`` + is not Lyapunov-like on ``K``. + - ``False`` will also be returned if we can't decide; specifically + if we arrive at a symbolic inequality that cannot be resolved. - 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 + .. SEEALSO:: - Two isomorphic cones should have the same Lyapunov rank [Rudolf et al.]_. - The cone ``K`` in the following example is isomorphic to the nonnegative - octant in `\mathbb{R}^{3}`:: + :func:`is_positive_on`, + :func:`is_cross_positive_on`, + :func:`is_Z_operator_on` - sage: K = Cone([(1,2,3), (-1,1,0), (1,0,6)]) - sage: lyapunov_rank(K) - 3 + EXAMPLES: - The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` - itself [Rudolf et al.]_:: + Diagonal matrices are Lyapunov-like operators on the nonnegative + orthant:: - sage: K = Cone([(2,2,4), (-1,9,0), (2,0,6)]) - sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) + sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)]) + sage: L = diagonal_matrix(random_vector(QQ,3)) + sage: is_lyapunov_like_on(L,K) True TESTS: - The Lyapunov rank should be additive on a product of cones - [Rudolf et al.]_:: + The identity operator is always Lyapunov-like:: - sage: K1 = random_cone(max_dim=10, max_rays=10) - sage: K2 = random_cone(max_dim=10, max_rays=10) - sage: K = K1.cartesian_product(K2) - sage: lyapunov_rank(K) == lyapunov_rank(K1) + lyapunov_rank(K2) + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=8) + sage: L = identity_matrix(K.lattice_dim()) + sage: is_lyapunov_like_on(L,K) True - The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` - itself [Rudolf et al.]_:: + The "zero" operator is always Lyapunov-like:: - sage: K = random_cone(max_dim=10, max_rays=10) - sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) + sage: K = random_cone(max_ambient_dim=8) + sage: R = K.lattice().vector_space().base_ring() + sage: L = zero_matrix(R, K.lattice_dim()) + sage: is_lyapunov_like_on(L,K) 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: K = random_cone(max_dim=10, 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/Gowda]_, no closed convex polyhedral cone can have - Lyapunov rank `n-1` in `n` dimensions:: - - sage: K = random_cone(max_dim=10) - 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/Gowda]_:: - - sage: K = random_cone(max_dim=15, solid=False, strictly_convex=False) - sage: actual = lyapunov_rank(K) - sage: (phi1, _) = span_iso(K) - sage: K_S = phi1(K) - sage: (phi2, _) = span_iso(K_S.dual()) - sage: J_T = phi2(K_S.dual()).dual() - sage: l = K.linear_subspace().dimension() - sage: codim = K.lattice_dim() - K.dim() - sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2 - sage: actual == expected - True + Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like + on ``K``:: - Repeat the previous test with different ``random_cone()`` params:: - - sage: K = random_cone(max_dim=15, solid=False, strictly_convex=True) - sage: actual = lyapunov_rank(K) - sage: (phi1, _) = span_iso(K) - sage: K_S = phi1(K) - sage: (phi2, _) = span_iso(K_S.dual()) - sage: J_T = phi2(K_S.dual()).dual() - sage: l = K.linear_subspace().dimension() - sage: codim = K.lattice_dim() - K.dim() - sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2 - sage: actual == expected + sage: K = random_cone(max_ambient_dim=5) + sage: all([ is_lyapunov_like_on(L,K) # long time + ....: for L in K.lyapunov_like_basis() ]) # long time True - - sage: K = random_cone(max_dim=15, solid=True, strictly_convex=False) - sage: actual = lyapunov_rank(K) - sage: (phi1, _) = span_iso(K) - sage: K_S = phi1(K) - sage: (phi2, _) = span_iso(K_S.dual()) - sage: J_T = phi2(K_S.dual()).dual() - sage: l = K.linear_subspace().dimension() - sage: codim = K.lattice_dim() - K.dim() - sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2 - sage: actual == expected + sage: all([ is_lyapunov_like_on(L.change_ring(SR),K) # long time + ....: for L in K.lyapunov_like_basis() ]) # long time True - sage: K = random_cone(max_dim=15, solid=True, strictly_convex=True) - sage: actual = lyapunov_rank(K) - sage: (phi1, _) = span_iso(K) - sage: K_S = phi1(K) - sage: (phi2, _) = span_iso(K_S.dual()) - sage: J_T = phi2(K_S.dual()).dual() - sage: l = K.linear_subspace().dimension() - sage: codim = K.lattice_dim() - K.dim() - sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2 - sage: actual == expected - True + Technically we could test this, but for now only closed convex cones + are supported as our ``K`` argument:: - sage: K = random_cone(max_dim=15) - sage: actual = lyapunov_rank(K) - sage: (phi1, _) = span_iso(K) - sage: K_S = phi1(K) - sage: (phi2, _) = span_iso(K_S.dual()) - sage: J_T = phi2(K_S.dual()).dual() - sage: l = K.linear_subspace().dimension() - sage: codim = K.lattice_dim() - K.dim() - sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2 - sage: actual == expected - True + sage: L = identity_matrix(3) + sage: K = [ vector([2,2,-1]), vector([5,4,-3]) ] + sage: is_lyapunov_like_on(L,K) + Traceback (most recent call last): + ... + TypeError: K must be a Cone. - And test with the project_span function:: + We can't give reliable answers over inexact rings:: - sage: K = random_cone(max_dim=15) - sage: actual = lyapunov_rank(K) - sage: K_S = project_span(K) - sage: P = project_span(K_S.dual()).dual() - sage: l = K.linear_subspace().dimension() - sage: codim = K.lattice_dim() - K.dim() - sage: expected = lyapunov_rank(P) + K.dim()*(l + codim) + codim**2 - sage: actual == expected - True + sage: K = Cone([(1,2,3), (4,5,6)]) + sage: L = identity_matrix(RR,3) + sage: is_lyapunov_like_on(L,K) + Traceback (most recent call last): + ... + ValueError: The base ring of L is neither SR nor exact. """ - return len(LL(K)) + if not is_Cone(K): + raise TypeError('K must be a Cone.') + if not L.base_ring().is_exact() and not L.base_ring() is SR: + raise ValueError('The base ring of L is neither SR nor exact.') + + return all([ s*(L*x) == 0 + for (x,s) in K.discrete_complementarity_set() ]) + + +def LL_cone(K): + gens = K.lyapunov_like_basis() + L = ToricLattice(K.lattice_dim()**2) + return Cone([ g.list() for g in gens ], lattice=L, check=False) + +def Sigma_cone(K): + gens = K.cross_positive_operators_gens() + L = ToricLattice(K.lattice_dim()**2) + return Cone([ g.list() for g in gens ], lattice=L, check=False) + +def Z_cone(K): + gens = K.Z_operators_gens() + L = ToricLattice(K.lattice_dim()**2) + return Cone([ g.list() for g in gens ], lattice=L, check=False) + +def pi_cone(K1, K2=None): + if K2 is None: + K2 = K1 + gens = K1.positive_operators_gens(K2) + L = ToricLattice(K1.lattice_dim()*K2.lattice_dim()) + return Cone([ g.list() for g in gens ], lattice=L, check=False)