X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;ds=sidebyside;f=mjo%2Fcone%2Fcone.py;h=6d7d2d9b4d2b379bc6041fe089e5a6ea38b8e48a;hb=342d9147356f7757bed2f9165a600a9e5ec0a5e2;hp=60f9c34ec8bc271d65812859f51ca77636c8cbbc;hpb=874e3ce831e0b1901b3c280a32ffe18e36f54959;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 60f9c34..6d7d2d9 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -7,326 +7,202 @@ addsitedir(abspath('../../')) from sage.all import * -def project_span(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: - - The projected cone should always be solid:: - - sage: K = random_cone() - 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() - sage: K_S = project_span(K) - sage: P = project_span(K_S.dual()).dual() - sage: P.is_proper() - True +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()``. - """ - F = K.lattice().base_field() - Q = K.lattice().quotient(K.sublattice_complement()) - vecs = [ vector(F, reversed(list(Q(r)))) for r in K.rays() ] + INPUT: - L = None - if len(vecs) == 0: - L = ToricLattice(0) + - ``W`` -- The subspace into which this cone will be restricted. - return Cone(vecs, lattice=L) + OUTPUT: + A new cone in a sublattice corresponding to ``W``. -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 + REFERENCES: -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). + M. Orlitzky. The Lyapunov rank of an improper cone. + http://www.optimization-online.org/DB_HTML/2015/10/5135.html EXAMPLES: - The inverse composed with the isomorphism should be the identity:: + Restricting a solid cone to its own span returns a cone linearly + isomorphic to the original:: - sage: K = random_cone(max_dim=10) - sage: (phi, phi_inv) = span_iso(K) - sage: phi_inv(phi(K)) == 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 - The image of ``K`` under the isomorphism should have full dimension:: + A single ray restricted to its own span has the same representation + regardless of the ambient space:: - sage: K = random_cone(max_dim=10) - sage: (phi, phi_inv) = span_iso(K) - sage: phi(K).dim() == phi(K).lattice_dim() + 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 - """ - phi_domain = K.sublattice().vector_space() - phi_codo = VectorSpace(phi_domain.base_field(), phi_domain.dimension()) - - # S goes from the new space to the cone space. - S = linear_transformation(phi_codo, phi_domain, phi_domain.basis()) - - # 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) - - return Cone(newrays, lattice=L) + Restricting to a trivial space gives the trivial 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()) - - - return (phi, phi_inverse) - - - -def discrete_complementarity_set(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. - - 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))] - - If the cone consists of a single ray, the second components of the - discrete complementarity set should generate the orthogonal - complement of that ray:: - - 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))] - - 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) - [] + 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 complementarity set of the dual can be obtained by switching the - components of the complementarity set of the original cone:: + Restricting a cone to its own span results in a solid cone:: - 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: K_S = _restrict_to_space(K, K.span()) + sage: K_S.is_solid() True - """ - V = K.lattice().vector_space() + Restricting a cone to its own span should not affect the number of + rays in the cone:: - # 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] - - -def LL(K): - r""" - Compute the space `\mathbf{LL}` of all Lyapunov-like transformations - on this 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 - OUTPUT: + Restricting a cone to its own span should not affect its dimension:: - A list of matrices forming a basis for the space of all - Lyapunov-like transformations on the given cone. + 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 - EXAMPLES: + Restricting a cone to its own span should not affects its lineality:: - The trivial cone has no Lyapunov-like transformations:: + 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 - sage: L = ToricLattice(0) - sage: K = Cone([], lattice=L) - sage: LL(K) - [] + Restricting a cone to its own span should not affect the number of + facets it has:: - The Lyapunov-like transformations on the nonnegative orthant are - simply diagonal matrices:: + 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 - sage: K = Cone([(1,)]) - sage: LL(K) - [[1]] + Restricting a solid cone to its own span is a linear isomorphism and + should not affect the dimension of its ambient space:: - sage: K = Cone([(1,0),(0,1)]) - sage: LL(K) - [ - [1 0] [0 0] - [0 0], [0 1] - ] + 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 - 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] - ] + Restricting a solid cone to its own span is a linear isomorphism + that establishes a one-to-one correspondence of discrete + complementarity sets:: - Only the identity matrix is Lyapunov-like on the `L^{3}_{1}` and - `L^{3}_{\infty}` cones [Rudolf et al.]_:: + 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 - 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] - ] + 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: 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: 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 - TESTS: + 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:: - 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: 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 - 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 + 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 """ - 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] - - # 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) - - # Turn our matrices into long vectors... - vectors = [ W(m.list()) for m in tensor_products ] - - # Vector space representation of Lyapunov-like matrices - # (i.e. vec(L) where L is Luapunov-like). - LL_vector = W.span(vectors).complement() - - # 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 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) - return matrix_basis + # 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 (or bilinearity) rank of this cone. + Compute the Lyapunov rank of this cone. - The Lyapunov rank of a cone can be thought of in (mainly) two ways: - - 1. The dimension of the Lie algebra of the automorphism group of the - cone. - - 2. The dimension of the linear space of all Lyapunov-like - transformations on the cone. - - INPUT: - - A closed, convex polyhedral 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: - 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). + A nonnegative integer representing the Lyapunov rank of this cone. - .. note:: - - In the references, the cones are always assumed to be proper. We - do not impose this restriction. - - .. seealso:: - - :meth:`is_proper` + 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: @@ -339,21 +215,21 @@ def lyapunov_rank(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. + .. [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. - .. [Orlitzky/Gowda] M. Orlitzky and M. S. Gowda. The Lyapunov Rank of an - Improper Cone. Work in-progress. + M. Orlitzky. The Lyapunov rank of an improper cone. + http://www.optimization-online.org/DB_HTML/2015/10/5135.html - .. [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. + 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 et al.]_:: + [Rudolf]_:: sage: positives = Cone([(1,)]) sage: lyapunov_rank(positives) @@ -365,21 +241,53 @@ def lyapunov_rank(K): 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 et al.]_:: + [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 et al.]_:: + 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 - The Lyapunov rank should be additive on a product of cones - [Rudolf et al.]_:: + 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)]) @@ -387,7 +295,7 @@ def lyapunov_rank(K): sage: lyapunov_rank(K) == lyapunov_rank(L31) + lyapunov_rank(octant) True - Two isomorphic cones should have the same Lyapunov rank [Rudolf et al.]_. + 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}`:: @@ -396,7 +304,7 @@ def lyapunov_rank(K): 3 The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` - itself [Rudolf et al.]_:: + itself [Rudolf]_:: sage: K = Cone([(2,2,4), (-1,9,0), (2,0,6)]) sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) @@ -404,19 +312,34 @@ def lyapunov_rank(K): TESTS: - The Lyapunov rank should be additive on a product of cones - [Rudolf et al.]_:: + The Lyapunov rank should be additive on a product of proper cones + [Rudolf]_:: - sage: K1 = random_cone(max_dim=10, max_rays=10) - sage: K2 = random_cone(max_dim=10, max_rays=10) + 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 et al.]_:: + itself [Rudolf]_:: - sage: K = random_cone(max_dim=10, max_rays=10) + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=8) sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) True @@ -426,7 +349,10 @@ def lyapunov_rank(K): 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: 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 @@ -434,91 +360,385 @@ def lyapunov_rank(K): sage: b == n-1 False - In fact [Orlitzky/Gowda]_, no closed convex polyhedral cone can have + In fact [Orlitzky]_, no closed convex polyhedral cone can have Lyapunov rank `n-1` in `n` dimensions:: - sage: K = random_cone(max_dim=10) + 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/Gowda]_:: + reduced to that of a proper cone [Orlitzky]_:: - sage: K = random_cone(max_dim=15, solid=False, strictly_convex=False) + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=8) 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: 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 - Repeat the previous test with different ``random_cone()`` params:: + The Lyapunov rank of a cone is the size of a :meth:`lyapunov_like_basis`:: - 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: set_random_seed() + sage: K = random_cone(max_ambient_dim=8) + sage: lyapunov_rank(K) == len(K.lyapunov_like_basis()) 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 + 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 - 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 + """ + 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``. + + We say that ``L`` is Lyapunov-like on ``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``. It is known [Orlitzky]_ that this property need only be + checked for generators of ``K`` and its dual. + + INPUT: + + - ``L`` -- A linear transformation or matrix. + + - ``K`` -- A polyhedral closed convex cone. + + OUTPUT: + + ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``, + and ``False`` otherwise. + + .. WARNING:: + + If this function returns ``True``, then ``L`` is Lyapunov-like + on ``K``. However, if ``False`` is returned, that could mean one + of two things. The first is that ``L`` is definitely not + Lyapunov-like on ``K``. The second is more of an "I don't know" + answer, returned (for example) if we cannot prove that an inner + product is zero. + + REFERENCES: + + M. Orlitzky. The Lyapunov rank of an improper cone. + http://www.optimization-online.org/DB_HTML/2015/10/5135.html + + EXAMPLES: + + The identity is always Lyapunov-like in a nontrivial space:: + + sage: set_random_seed() + sage: K = random_cone(min_ambient_dim = 1, max_rays = 8) + sage: L = identity_matrix(K.lattice_dim()) + sage: is_lyapunov_like(L,K) True - 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 + As is the "zero" transformation:: + + sage: K = random_cone(min_ambient_dim = 1, max_rays = 5) + sage: R = K.lattice().vector_space().base_ring() + sage: L = zero_matrix(R, K.lattice_dim()) + sage: is_lyapunov_like(L,K) True - And test with the project_span function:: + Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like + on ``K``:: - 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 + sage: K = random_cone(min_ambient_dim = 1, max_rays = 5) + sage: all([ is_lyapunov_like(L,K) for L in K.lyapunov_like_basis() ]) True """ - return len(LL(K)) + return all([(L*x).inner_product(s) == 0 + for (x,s) in K.discrete_complementarity_set()]) + + +def random_element(K): + r""" + Return a random element of ``K`` from its ambient vector space. + + ALGORITHM: + + The cone ``K`` is specified in terms of its generators, so that + ``K`` is equal to the convex conic combination of those generators. + To choose a random element of ``K``, we assign random nonnegative + coefficients to each generator of ``K`` and construct a new vector + from the scaled rays. + + A vector, rather than a ray, is returned so that the element may + have non-integer coordinates. Thus the element may have an + arbitrarily small norm. + + EXAMPLES: + + A random element of the trivial cone is zero:: + + sage: set_random_seed() + sage: K = Cone([], ToricLattice(0)) + sage: random_element(K) + () + sage: K = Cone([(0,)]) + sage: random_element(K) + (0) + sage: K = Cone([(0,0)]) + sage: random_element(K) + (0, 0) + sage: K = Cone([(0,0,0)]) + sage: random_element(K) + (0, 0, 0) + + TESTS: + + Any cone should contain an element of itself:: + + sage: set_random_seed() + sage: K = random_cone(max_rays = 8) + sage: K.contains(random_element(K)) + True + + """ + V = K.lattice().vector_space() + F = V.base_ring() + coefficients = [ F.random_element().abs() for i in range(K.nrays()) ] + vector_gens = map(V, K.rays()) + scaled_gens = [ coefficients[i]*vector_gens[i] + for i in range(len(vector_gens)) ] + + # Make sure we return a vector. Without the coercion, we might + # return ``0`` when ``K`` has no rays. + v = V(sum(scaled_gens)) + return v + + +def positive_operators(K): + r""" + Compute generators of the cone of positive operators on this cone. + + OUTPUT: + + A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``. + Each matrix ``P`` in the list should have the property that ``P*x`` + is an element of ``K`` whenever ``x`` is an element of + ``K``. Moreover, any nonnegative linear combination of these + matrices shares the same property. + + EXAMPLES: + + The trivial cone in a trivial space has no positive operators:: + + sage: K = Cone([], ToricLattice(0)) + sage: positive_operators(K) + [] + + Positive operators on the nonnegative orthant are nonnegative matrices:: + + sage: K = Cone([(1,)]) + sage: positive_operators(K) + [[1]] + + sage: K = Cone([(1,0),(0,1)]) + sage: positive_operators(K) + [ + [1 0] [0 1] [0 0] [0 0] + [0 0], [0 0], [1 0], [0 1] + ] + + Every operator is positive on the ambient vector space:: + + sage: K = Cone([(1,),(-1,)]) + sage: K.is_full_space() + True + sage: positive_operators(K) + [[1], [-1]] + + sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) + sage: K.is_full_space() + True + sage: positive_operators(K) + [ + [1 0] [-1 0] [0 1] [ 0 -1] [0 0] [ 0 0] [0 0] [ 0 0] + [0 0], [ 0 0], [0 0], [ 0 0], [1 0], [-1 0], [0 1], [ 0 -1] + ] + + TESTS: + + A positive operator on a cone should send its generators into the cone:: + + sage: K = random_cone(max_ambient_dim = 6) + sage: pi_of_K = positive_operators(K) + sage: all([K.contains(p*x) for p in pi_of_K for x in K.rays()]) + True + + """ + # 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. + V = K.lattice().vector_space() + W = VectorSpace(V.base_ring(), V.dimension()**2) + + tensor_products = [ s.tensor_product(x) for x in K for s in K.dual() ] + + # Turn our matrices into long vectors... + vectors = [ W(m.list()) for m in tensor_products ] + + # Create the *dual* cone of the positive operators, expressed as + # long vectors.. + L = ToricLattice(W.dimension()) + pi_dual = Cone(vectors, lattice=L) + + # Now compute the desired cone from its dual... + pi_cone = pi_dual.dual() + + # And finally convert its rays back to matrix representations. + M = MatrixSpace(V.base_ring(), V.dimension()) + + return [ M(v.list()) for v in pi_cone.rays() ] + + +def Z_transformations(K): + r""" + Compute generators of the cone of Z-transformations on this cone. + + OUTPUT: + + A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``. + Each matrix ``L`` in the list should have the property that + ``(L*x).inner_product(s) <= 0`` whenever ``(x,s)`` is an element the + discrete complementarity set of ``K``. Moreover, any nonnegative + linear combination of these matrices shares the same property. + + EXAMPLES: + + Z-transformations on the nonnegative orthant are just Z-matrices. + That is, matrices whose off-diagonal elements are nonnegative:: + + sage: K = Cone([(1,0),(0,1)]) + sage: Z_transformations(K) + [ + [ 0 -1] [ 0 0] [-1 0] [1 0] [ 0 0] [0 0] + [ 0 0], [-1 0], [ 0 0], [0 0], [ 0 -1], [0 1] + ] + sage: K = Cone([(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)]) + sage: all([ z[i][j] <= 0 for z in Z_transformations(K) + ....: for i in range(z.nrows()) + ....: for j in range(z.ncols()) + ....: if i != j ]) + True + + The trivial cone in a trivial space has no Z-transformations:: + + sage: K = Cone([], ToricLattice(0)) + sage: Z_transformations(K) + [] + + Z-transformations on a subspace are Lyapunov-like and vice-versa:: + + sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) + sage: K.is_full_space() + True + sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ]) + sage: zs = span([ vector(z.list()) for z in Z_transformations(K) ]) + sage: zs == lls + True + + TESTS: + + The Z-property is possessed by every Z-transformation:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim = 6) + sage: Z_of_K = Z_transformations(K) + sage: dcs = K.discrete_complementarity_set() + sage: all([(z*x).inner_product(s) <= 0 for z in Z_of_K + ....: for (x,s) in dcs]) + True + + The lineality space of Z is LL:: + + sage: set_random_seed() + sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 6) + sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ]) + sage: z_cone = Cone([ z.list() for z in Z_transformations(K) ]) + sage: z_cone.linear_subspace() == lls + True + + """ + # 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. + V = K.lattice().vector_space() + W = VectorSpace(V.base_ring(), V.dimension()**2) + + C_of_K = K.discrete_complementarity_set() + tensor_products = [ s.tensor_product(x) for (x,s) in C_of_K ] + + # Turn our matrices into long vectors... + vectors = [ W(m.list()) for m in tensor_products ] + + # Create the *dual* cone of the cross-positive operators, + # expressed as long vectors.. + L = ToricLattice(W.dimension()) + Sigma_dual = Cone(vectors, lattice=L) + + # Now compute the desired cone from its dual... + Sigma_cone = Sigma_dual.dual() + + # And finally convert its rays back to matrix representations. + # But first, make them negative, so we get Z-transformations and + # not cross-positive ones. + M = MatrixSpace(V.base_ring(), V.dimension()) + + return [ -M(v.list()) for v in Sigma_cone.rays() ]