X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=aeec0c90b5f582c8ba858d4b616fff191fb6d529;hb=c2d15c0884c8b92483f58826747887bd2bcdcdeb;hp=ff7d195d134c15943dbb75c1f26b741bb4a0afba;hpb=10142e85f34c47fa35df002f519d1d58a79a74f4;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index ff7d195..aeec0c9 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -1,615 +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. - - EXAMPLES:: - - sage: K = Cone([(1,)]) - sage: project_span(K) == K - True - - 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 - - TESTS: + Determine whether or not ``L`` is positive on ``K``. - The projected cone should always be solid:: - - sage: K = random_cone(max_dim = 10) - sage: K_S = project_span(K) - sage: K_S.is_solid() - True - - If we do this according to our paper, then the result is proper:: - - sage: K = random_cone(max_dim = 10) - sage: K_S = project_span(K) - sage: P = project_span(K_S.dual()).dual() - sage: P.is_proper() - True - - """ - L = K.lattice() - F = L.base_field() - Q = L.quotient(K.sublattice_complement()) - vecs = [ vector(F, reversed(list(Q(r)))) for r in K.rays() ] - - newL = None - if len(vecs) == 0: - newL = ToricLattice(0) - - return Cone(vecs, lattice=newL) + 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``. + 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. + INPUT: -def lineality(K): - r""" - Compute the lineality of this cone. + - ``L`` -- A matrix over either an exact ring or ``SR``. - The lineality of a cone is the dimension of the largest linear - subspace contained in that cone. + - ``K`` -- A polyhedral closed convex cone. OUTPUT: - A nonnegative integer; the dimension of the largest subspace - contained within this cone. + If the base ring of ``L`` is exact, then ``True`` will be returned if + and only if ``L`` is positive on ``K``. - REFERENCES: + If the base ring of ``L`` is ``SR``, then the situation is more + complicated: - .. [Rockafellar] R.T. Rockafellar. Convex Analysis. Princeton - University Press, Princeton, 1970. + - ``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. - EXAMPLES: + .. SEEALSO:: - The lineality of the nonnegative orthant is zero, since it clearly - contains no lines:: + :func:`is_cross_positive_on`, + :func:`is_Z_operator_on`, + :func:`is_lyapunov_like_on` - sage: K = Cone([(1,0,0), (0,1,0), (0,0,1)]) - sage: lineality(K) - 0 + EXAMPLES: - However, if we add another ray so that the entire `x`-axis belongs - to the cone, then the resulting cone will have lineality one:: + Nonnegative matrices are positive operators on the nonnegative + orthant:: - sage: K = Cone([(1,0,0), (-1,0,0), (0,1,0), (0,0,1)]) - sage: lineality(K) - 1 + 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 - If our cone is all of `\mathbb{R}^{2}`, then its lineality is equal - to the dimension of the ambient space (i.e. two):: + TESTS: - sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)]) - sage: lineality(K) - 2 + The identity operator is always positive:: - Per the definition, the lineality of the trivial cone in a trivial - space is zero:: + 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 - sage: K = Cone([], lattice=ToricLattice(0)) - sage: lineality(K) - 0 + The "zero" operator is always positive:: - TESTS: + 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 - The lineality of a cone should be an integer between zero and the - dimension of the ambient space, inclusive:: + Everything in ``K.positive_operators_gens()`` should be + positive on ``K``:: - sage: K = random_cone(max_dim = 10) - sage: l = lineality(K) - sage: l in ZZ + 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: (0 <= l) and (l <= K.lattice_dim()) + sage: all([ is_positive_on(L.change_ring(SR),K) # long time + ....: for L in K.positive_operators_gens() ]) # long time True - A strictly convex cone should have lineality zero:: + Technically we could test this, but for now only closed convex cones + are supported as our ``K`` argument:: + + 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. - sage: K = random_cone(max_dim = 10, strictly_convex = True) - sage: lineality(K) - 0 + 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 K.linear_subspace().dimension() + + 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 codim(K): +def is_cross_positive_on(L,K): r""" - Compute the codimension of this cone. + Determine whether or not ``L`` is cross-positive on ``K``. - The codimension of a cone is the dimension of the space of all - elements perpendicular to every element of the cone. In other words, - the codimension is the difference between the dimension of the - ambient space and the dimension of the cone itself. + 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 nonnegative integer representing the dimension of the space of all - elements perpendicular to this cone. + INPUT: - .. seealso:: + - ``L`` -- A matrix over either an exact ring or ``SR``. - :meth:`dim`, :meth:`lattice_dim` + - ``K`` -- A polyhedral closed convex cone. - EXAMPLES: + OUTPUT: - The codimension of the nonnegative orthant is zero, since the span of - its generators equals the entire ambient space:: + If the base ring of ``L`` is exact, then ``True`` will be returned if + and only if ``L`` is cross-positive on ``K``. - sage: K = Cone([(1,0,0), (0,1,0), (0,0,1)]) - sage: codim(K) - 0 + If the base ring of ``L`` is ``SR``, then the situation is more + complicated: - However, if we remove a ray so that the entire cone is contained - within the `x-y`-plane, then the resulting cone will have - codimension one, because the `z`-axis is perpendicular to every - element of the cone:: + - ``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. - sage: K = Cone([(1,0,0), (0,1,0)]) - sage: codim(K) - 1 + .. SEEALSO:: - If our cone is all of `\mathbb{R}^{2}`, then its codimension is zero:: + :func:`is_positive_on`, + :func:`is_Z_operator_on`, + :func:`is_lyapunov_like_on` - sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)]) - sage: codim(K) - 0 + EXAMPLES: - And if the cone is trivial in any space, then its codimension is - equal to the dimension of the ambient space:: + The identity operator is always cross-positive:: - sage: K = Cone([], lattice=ToricLattice(0)) - sage: codim(K) - 0 + 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 - sage: K = Cone([(0,)]) - sage: codim(K) - 1 + The "zero" operator is always cross-positive:: - sage: K = Cone([(0,0)]) - sage: codim(K) - 2 + 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 TESTS: - The codimension of a cone should be an integer between zero and - the dimension of the ambient space, inclusive:: + Everything in ``K.cross_positive_operators_gens()`` should be + cross-positive on ``K``:: - sage: K = random_cone(max_dim = 10) - sage: c = codim(K) - sage: c in ZZ + 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: (0 <= c) and (c <= K.lattice_dim()) + sage: all([ is_cross_positive_on(L.change_ring(SR),K) # long time + ....: for L in K.cross_positive_operators_gens() ]) # long time True - A solid cone should have codimension zero:: + Technically we could test this, but for now only closed convex cones + are supported as our ``K`` argument:: - sage: K = random_cone(max_dim = 10, solid = True) - sage: codim(K) - 0 + 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. - The codimension of a cone is equal to the lineality of its dual:: + We can't give reliable answers over inexact rings:: - sage: K = random_cone(max_dim = 10, solid = True) - sage: codim(K) == lineality(K.dual()) - True + 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. """ - return (K.lattice_dim() - K.dim()) + 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 discrete_complementarity_set(K): +def is_Z_operator_on(L,K): r""" - Compute the discrete complementarity set of this cone. - - 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. - - OUTPUT: - - A list of pairs `(x,s)` such that, - - * `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. + Determine whether or not ``L`` is a Z-operator on ``K``. - EXAMPLES: - - The discrete complementarity set of the nonnegative orthant consists - of pairs of standard basis vectors:: - - sage: K = Cone([(1,0),(0,1)]) - sage: discrete_complementarity_set(K) - [((1, 0), (0, 1)), ((0, 1), (1, 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. - If the cone consists of a single ray, the second components of the - discrete complementarity set should generate the orthogonal - complement of that ray:: + A matrix is a Z-operator on ``K`` if and only if its negation is a + cross-positive operator on ``K``. - 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))] + 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. - When the cone is the entire space, its dual is the trivial cone, so - the discrete complementarity set is empty:: - - sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) - sage: discrete_complementarity_set(K) - [] - - TESTS: + INPUT: - The complementarity set of the dual can be obtained by switching the - components of the complementarity set of the original cone:: + - ``L`` -- A matrix over either an exact ring or ``SR``. - 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 - True + - ``K`` -- A polyhedral closed convex cone. - """ - V = K.lattice().vector_space() + OUTPUT: - # 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()] + If the base ring of ``L`` is exact, then ``True`` will be returned if + and only if ``L`` is a Z-operator on ``K``. - return [(x,s) for x in xs for s in ss if x.inner_product(s) == 0] + If the base ring of ``L`` is ``SR``, then the situation is more + complicated: + - ``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. -def LL(K): - r""" - Compute the space `\mathbf{LL}` of all Lyapunov-like transformations - on this cone. + .. SEEALSO:: - OUTPUT: - - A list of matrices forming a basis for the space of all - Lyapunov-like transformations on the given cone. + :func:`is_positive_on`, + :func:`is_cross_positive_on`, + :func:`is_lyapunov_like_on` EXAMPLES: - The trivial cone has no Lyapunov-like transformations:: - - sage: L = ToricLattice(0) - sage: K = Cone([], lattice=L) - sage: LL(K) - [] - - The Lyapunov-like transformations on the nonnegative orthant are - simply diagonal matrices:: + The identity operator is always a Z-operator:: - sage: K = Cone([(1,)]) - sage: LL(K) - [[1]] + 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 - sage: K = Cone([(1,0),(0,1)]) - sage: LL(K) - [ - [1 0] [0 0] - [0 0], [0 1] - ] + The "zero" operator is always a Z-operator:: - 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] - ] + 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 TESTS: - 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:: - - 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 - - """ - V = K.lattice().vector_space() - - C_of_K = discrete_complementarity_set(K) - - tensor_products = [s.tensor_product(x) for (x,s) in C_of_K] + Everything in ``K.Z_operators_gens()`` should be a Z-operator + on ``K``:: - # 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) + 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 - # Turn our matrices into long vectors... - vectors = [ W(m.list()) for m in tensor_products ] + Technically we could test this, but for now only closed convex cones + are supported as our ``K`` argument:: - # Vector space representation of Lyapunov-like matrices - # (i.e. vec(L) where L is Luapunov-like). - LL_vector = W.span(vectors).complement() + 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. - # Now construct an ambient MatrixSpace in which to stick our - # transformations. - M = MatrixSpace(V.base_ring(), V.dimension()) - matrix_basis = [ M(v.list()) for v in LL_vector.basis() ] + We can't give reliable answers over inexact rings:: - return matrix_basis + 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. + """ + return is_cross_positive_on(-L,K) -def lyapunov_rank(K): +def is_lyapunov_like_on(L,K): r""" - Compute the Lyapunov (or bilinearity) rank of this cone. + Determine whether or not ``L`` is Lyapunov-like on ``K``. - The Lyapunov rank of a cone can be thought of in (mainly) two ways: + 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. - 1. The dimension of the Lie algebra of the automorphism group of the - cone. + An operator is Lyapunov-like on ``K`` if and only if both the + operator itself and its negation are cross-positive on ``K``. - 2. The dimension of the linear space of all Lyapunov-like - transformations on the cone. + 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. INPUT: - A closed, convex polyhedral cone. - - OUTPUT: - - 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). + - ``L`` -- A matrix over either an exact ring or ``SR``. - .. note:: + - ``K`` -- A polyhedral closed convex cone. - In the references, the cones are always assumed to be proper. We - do not impose this restriction. - - .. seealso:: - - :meth:`is_proper` - - ALGORITHM: + OUTPUT: - 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}`. + If the base ring of ``L`` is exact, then ``True`` will be returned if + and only if ``L`` is Lyapunov-like on ``K``. - REFERENCES: + If the base ring of ``L`` is ``SR``, then the situation is more + complicated: - .. [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. + - ``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. - .. [Orlitzky/Gowda] M. Orlitzky and M. S. Gowda. The Lyapunov Rank of an - Improper Cone. Work in-progress. + .. SEEALSO:: - .. [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. + :func:`is_positive_on`, + :func:`is_cross_positive_on`, + :func:`is_Z_operator_on` EXAMPLES: - The nonnegative orthant in `\mathbb{R}^{n}` always has rank `n` - [Rudolf et al.]_:: - - 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/Gowda]_:: - - sage: R5 = VectorSpace(QQ, 5) - sage: gens = R5.basis() + [ -r for r in R5.basis() ] - sage: K = Cone(gens) - sage: lyapunov_rank(K) - 25 - - The `L^{3}_{1}` cone is known to have a Lyapunov rank of one - [Rudolf et al.]_:: - - 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 et al.]_:: - - 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/Gowda]_:: - - 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/Gowda]_:: - - 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: zero = (0,0,0,0,0) - sage: K = Cone([e1, neg_e1, e2, neg_e2, zero, zero, zero]) - sage: lyapunov_rank(K) - 19 - sage: K.lattice_dim()**2 - K.dim()*codim(K) - 19 - - The Lyapunov rank should be additive on a product of proper cones - [Rudolf et al.]_:: - - 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 et al.]_. - 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 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 proper cones - [Rudolf et al.]_:: + The identity operator is always Lyapunov-like:: - sage: K1 = random_cone(max_dim=10, strictly_convex=True, solid=True) - sage: K2 = random_cone(max_dim=10, strictly_convex=True, solid=True) - 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:: + Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like + on ``K``:: - 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 + 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: 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=10) - sage: actual = lyapunov_rank(K) - sage: K_S = project_span(K) - sage: P = project_span(K_S.dual()).dual() - sage: l = lineality(K) - sage: c = codim(K) - sage: expected = lyapunov_rank(P) + K.dim()*(l + c) + c**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 - The Lyapunov rank of a proper cone is just the dimension of ``LL(K)``:: + Technically we could test this, but for now only closed convex cones + are supported as our ``K`` argument:: - sage: K = random_cone(max_dim=10, strictly_convex=True, solid=True) - sage: lyapunov_rank(K) == len(LL(K)) - 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. - """ - beta = 0 + We can't give reliable answers over inexact rings:: - m = K.dim() - n = K.lattice_dim() - l = lineality(K) + 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. - if m < n: - # K is not solid, project onto its span. - K = project_span(K) - - # Lemma 2 - beta += m*(n - m) + (n - m)**2 - - if l > 0: - # K is not pointed, project its dual onto its span. - K = project_span(K.dual()).dual() - - # Lemma 3 - beta += m * l - - beta += len(LL(K)) - return beta + """ + 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)