X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;ds=sidebyside;f=mjo%2Fcone%2Fcone.py;h=c71a24cbee0c9d0857fd3c8d8fe2ecbb0042238c;hb=a4109ab945f5d3ed94207c936e60b5b187ae450b;hp=3b724a7c0fe8d0305351aae778b23978ed79e8f4;hpb=b1f56b9fd21ce64f0d4613d1d07e090e89eba621;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 3b724a7..c71a24c 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -1,413 +1,771 @@ -# 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 * - -def random_cone(min_dim=0, max_dim=None, min_rays=0, max_rays=None): +def is_lyapunov_like(L,K): r""" - Generate a random rational convex polyhedral cone. - - Lower and upper bounds may be provided for both the dimension of the - ambient space and the number of generating rays of the cone. If a - lower bound is left unspecified, it defaults to zero. Unspecified - upper bounds will be chosen randomly. - - The lower bound on the number of rays is limited to twice the - maximum dimension of the ambient vector space. To see why, consider - the space $\mathbb{R}^{2}$, and suppose we have generated four rays, - $\left\{ \pm e_{1}, \pm e_{2} \right\}$. Clearly any other ray in - the space is a nonnegative linear combination of those four, - so it is hopeless to generate more. It is therefore an error - to request more in the form of ``min_rays``. + Determine whether or not ``L`` is Lyapunov-like on ``K``. - .. NOTE: + 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. - If you do not explicitly request more than ``2 * max_dim`` rays, - a larger number may still be randomly generated. In that case, - the returned cone will simply be equal to the entire space. + There are faster ways of checking this property. For example, we + could compute a `lyapunov_like_basis` of the cone, and then test + whether or not the given matrix is contained in the span of that + basis. The value of this function is that it works on symbolic + matrices. INPUT: - - ``min_dim`` (default: zero) -- A nonnegative integer representing the - minimum dimension of the ambient lattice. + - ``L`` -- A linear transformation or matrix. - - ``max_dim`` (default: random) -- A nonnegative integer representing - the maximum dimension of the ambient - lattice. - - - ``min_rays`` (default: zero) -- A nonnegative integer representing the - minimum number of generating rays of the - cone. - - - ``max_rays`` (default: random) -- A nonnegative integer representing the - maximum number of generating rays of - the cone. + - ``K`` -- A polyhedral closed convex cone. OUTPUT: - A new, randomly generated cone. - - A ``ValueError` will be thrown under the following conditions: + ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``, + and ``False`` otherwise. - * Any of ``min_dim``, ``max_dim``, ``min_rays``, or ``max_rays`` - are negative. + .. WARNING:: - * ``max_dim`` is less than ``min_dim``. + 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. - * ``max_rays`` is less than ``min_rays``. + REFERENCES: - * ``min_rays`` is greater than twice ``max_dim``. + M. Orlitzky. The Lyapunov rank of an improper cone. + http://www.optimization-online.org/DB_HTML/2015/10/5135.html EXAMPLES: - If we set the lower/upper bounds to zero, then our result is - predictable:: + The identity is always Lyapunov-like in a nontrivial space:: - sage: random_cone(0,0,0,0) - 0-d cone in 0-d lattice N + sage: set_random_seed() + sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8) + sage: L = identity_matrix(K.lattice_dim()) + sage: is_lyapunov_like(L,K) + True - We can predict the dimension when ``min_dim == max_dim``:: + As is the "zero" transformation:: - sage: random_cone(min_dim=4, max_dim=4, min_rays=0, max_rays=0) - 0-d cone in 4-d lattice N + sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8) + sage: R = K.lattice().vector_space().base_ring() + sage: L = zero_matrix(R, K.lattice_dim()) + sage: is_lyapunov_like(L,K) + True - Likewise for the number of rays when ``min_rays == max_rays``:: + Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like + on ``K``:: - sage: random_cone(min_dim=10, max_dim=10, min_rays=10, max_rays=10) - 10-d cone in 10-d lattice N + sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6) + sage: all([ is_lyapunov_like(L,K) for L in K.lyapunov_like_basis() ]) + True - TESTS: + """ + return all([(L*x).inner_product(s) == 0 + for (x,s) in K.discrete_complementarity_set()]) - It's hard to test the output of a random process, but we can at - least make sure that we get a cone back:: - sage: from sage.geometry.cone import is_Cone # long time - sage: K = random_cone() # long time - sage: is_Cone(K) # long time - True +def positive_operator_gens(K1, K2 = None): + r""" + Compute generators of the cone of positive operators on this cone. A + linear operator on a cone is positive if the image of the cone under + the operator is a subset of the cone. This concept can be extended + to two cones, where the image of the first cone under a positive + operator is a subset of the second cone. - The upper/lower bounds are respected:: + INPUT: - sage: K = random_cone(min_dim=5, max_dim=10, min_rays=3, max_rays=4) - sage: 5 <= K.lattice_dim() and K.lattice_dim() <= 10 - True - sage: 3 <= K.nrays() and K.nrays() <= 4 - True + - ``K2`` -- (default: ``K1``) the codomain cone; the image of this + cone under the returned operators is a subset of ``K2``. - Ensure that an exception is raised when either lower bound is greater - than its respective upper bound:: + OUTPUT: - sage: random_cone(min_dim=5, max_dim=2) - Traceback (most recent call last): - ... - ValueError: max_dim cannot be less than min_dim. + A list of `m`-by-``n`` matrices where ``m == K2.lattice_dim()`` and + ``n == K1.lattice_dim()``. Each matrix ``P`` in the list should have + the property that ``P*x`` is an element of ``K2`` whenever ``x`` is + an element of ``K1``. Moreover, any nonnegative linear combination of + these matrices shares the same property. - sage: random_cone(min_rays=5, max_rays=2) - Traceback (most recent call last): - ... - ValueError: max_rays cannot be less than min_rays. + REFERENCES: - And if we request too many rays:: + .. [Orlitzky-Pi-Z] + M. Orlitzky. + Positive and Z-operators on closed convex cones. - sage: random_cone(min_rays=5, max_dim=1) - Traceback (most recent call last): - ... - ValueError: min_rays cannot be larger than twice max_dim. + .. [Tam] + B.-S. Tam. + Some results of polyhedral cones and simplicial cones. + Linear and Multilinear Algebra, 4:4 (1977) 281--284. - """ + EXAMPLES: - # Catch obvious mistakes so that we can generate clear error - # messages. - - if min_dim < 0: - raise ValueError('min_dim must be nonnegative.') - - if min_rays < 0: - raise ValueError('min_rays must be nonnegative.') - - if max_dim is not None: - if max_dim < 0: - raise ValueError('max_dim must be nonnegative.') - if (max_dim < min_dim): - raise ValueError('max_dim cannot be less than min_dim.') - if min_rays > 2*max_dim: - raise ValueError('min_rays cannot be larger than twice max_dim.') - - if max_rays is not None: - if max_rays < 0: - raise ValueError('max_rays must be nonnegative.') - if (max_rays < min_rays): - raise ValueError('max_rays cannot be less than min_rays.') - - - def random_min_max(l,u): - r""" - We need to handle two cases for the upper bounds, and we need to do - the same thing for max_dim/max_rays. So we consolidate the logic here. - """ - if u is None: - # The upper bound is unspecified; return a random integer - # in [l,infinity). - return l + ZZ.random_element().abs() - else: - # We have an upper bound, and it's greater than or equal - # to our lower bound. So we generate a random integer in - # [0,u-l], and then add it to l to get something in - # [l,u]. To understand the "+1", check the - # ZZ.random_element() docs. - return l + ZZ.random_element(u - l + 1) - - def is_full_space(K): - r""" - Is this cone equivalent to the full ambient vector space? - """ - return K.lines().dim() == K.lattice_dim() - - d = random_min_max(min_dim, max_dim) - r = random_min_max(min_rays, max_rays) - - L = ToricLattice(d) - - # The rays are trickier to generate, since we could generate v and - # 2*v as our "two rays." In that case, the resuting cone would - # have one generating ray. To avoid such a situation, we start by - # generating ``r`` rays where ``r`` is the number we want to end - # up with... - rays = [L.random_element() for i in range(0, r)] - - # (The lattice parameter is required when no rays are given, so we - # pass it just in case ``r == 0``). - K = Cone(rays, lattice=L) - - # Now if we generated two of the "same" rays, we'll have fewer - # generating rays than ``r``. In that case, we keep making up new - # rays and recreating the cone until we get the right number of - # independent generators. We can obviously stop if ``K`` is the - # entire ambient vector space. - while r > K.nrays() and not is_full_space(K): - rays.append(L.random_element()) - K = Cone(rays) - - return K - - -def discrete_complementarity_set(K): - r""" - Compute the discrete complementarity set of this cone. + Positive operators on the nonnegative orthant are nonnegative matrices:: - 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. + sage: K = Cone([(1,)]) + sage: positive_operator_gens(K) + [[1]] - OUTPUT: + sage: K = Cone([(1,0),(0,1)]) + sage: positive_operator_gens(K) + [ + [1 0] [0 1] [0 0] [0 0] + [0 0], [0 0], [1 0], [0 1] + ] + + The trivial cone in a trivial space has no positive operators:: - A list of pairs `(x,s)` such that, + sage: K = Cone([], ToricLattice(0)) + sage: positive_operator_gens(K) + [] - * `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. + Every operator is positive on the trivial cone:: - EXAMPLES: + sage: K = Cone([(0,)]) + sage: positive_operator_gens(K) + [[1], [-1]] - The discrete complementarity set of the nonnegative orthant consists - of pairs of standard basis vectors:: + sage: K = Cone([(0,0)]) + sage: K.is_trivial() + True + sage: positive_operator_gens(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] + ] - 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:: + Every operator is positive on the ambient vector space:: + + sage: K = Cone([(1,),(-1,)]) + sage: K.is_full_space() + True + sage: positive_operator_gens(K) + [[1], [-1]] sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) - sage: discrete_complementarity_set(K) - [] + sage: K.is_full_space() + True + sage: positive_operator_gens(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] + ] + + A non-obvious application is to find the positive operators on the + right half-plane:: + + sage: K = Cone([(1,0),(0,1),(0,-1)]) + sage: positive_operator_gens(K) + [ + [1 0] [0 0] [ 0 0] [0 0] [ 0 0] + [0 0], [1 0], [-1 0], [0 1], [ 0 -1] + ] TESTS: - The complementarity set of the dual can be obtained by switching the - components of the complementarity set of the original cone:: + Each positive operator generator should send the generators of one + cone into the other cone:: + + sage: set_random_seed() + sage: K1 = random_cone(max_ambient_dim=4) + sage: K2 = random_cone(max_ambient_dim=4) + sage: pi_K1_K2 = positive_operator_gens(K1,K2) + sage: all([ K2.contains(P*x) for P in pi_K1_K2 for x in K1 ]) + True + + Each positive operator generator should send a random element of one + cone into the other cone:: + + sage: set_random_seed() + sage: K1 = random_cone(max_ambient_dim=4) + sage: K2 = random_cone(max_ambient_dim=4) + sage: pi_K1_K2 = positive_operator_gens(K1,K2) + sage: all([ K2.contains(P*K1.random_element(QQ)) for P in pi_K1_K2 ]) + True + + A random element of the positive operator cone should send the + generators of one cone into the other cone:: + + sage: set_random_seed() + sage: K1 = random_cone(max_ambient_dim=4) + sage: K2 = random_cone(max_ambient_dim=4) + sage: pi_K1_K2 = positive_operator_gens(K1,K2) + sage: L = ToricLattice(K1.lattice_dim() * K2.lattice_dim()) + sage: pi_cone = Cone([ g.list() for g in pi_K1_K2 ], + ....: lattice=L, + ....: check=False) + sage: P = matrix(K2.lattice_dim(), + ....: K1.lattice_dim(), + ....: pi_cone.random_element(QQ).list()) + sage: all([ K2.contains(P*x) for x in K1 ]) + True + + A random element of the positive operator cone should send a random + element of one cone into the other cone:: + + sage: set_random_seed() + sage: K1 = random_cone(max_ambient_dim=4) + sage: K2 = random_cone(max_ambient_dim=4) + sage: pi_K1_K2 = positive_operator_gens(K1,K2) + sage: L = ToricLattice(K1.lattice_dim() * K2.lattice_dim()) + sage: pi_cone = Cone([ g.list() for g in pi_K1_K2 ], + ....: lattice=L, + ....: check=False) + sage: P = matrix(K2.lattice_dim(), + ....: K1.lattice_dim(), + ....: pi_cone.random_element(QQ).list()) + sage: K2.contains(P*K1.random_element(ring=QQ)) + True - 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) + The lineality space of the dual of the cone of positive operators + can be computed from the lineality spaces of the cone and its dual:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=4) + sage: pi_of_K = positive_operator_gens(K) + sage: L = ToricLattice(K.lattice_dim()**2) + sage: pi_cone = Cone([ g.list() for g in pi_of_K ], + ....: lattice=L, + ....: check=False) + sage: actual = pi_cone.dual().linear_subspace() + sage: U1 = [ vector((s.tensor_product(x)).list()) + ....: for x in K.lines() + ....: for s in K.dual() ] + sage: U2 = [ vector((s.tensor_product(x)).list()) + ....: for x in K + ....: for s in K.dual().lines() ] + sage: expected = pi_cone.lattice().vector_space().span(U1 + U2) sage: actual == expected True - """ - V = K.lattice().vector_space() + The lineality of the dual of the cone of positive operators + is known from its lineality space:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=4) + sage: n = K.lattice_dim() + sage: m = K.dim() + sage: l = K.lineality() + sage: pi_of_K = positive_operator_gens(K) + sage: L = ToricLattice(n**2) + sage: pi_cone = Cone([p.list() for p in pi_of_K], + ....: lattice=L, + ....: check=False) + sage: actual = pi_cone.dual().lineality() + sage: expected = l*(m - l) + m*(n - m) + sage: actual == expected + True - # 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()] + The dimension of the cone of positive operators is given by the + corollary in my paper:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=4) + sage: n = K.lattice_dim() + sage: m = K.dim() + sage: l = K.lineality() + sage: pi_of_K = positive_operator_gens(K) + sage: L = ToricLattice(n**2) + sage: pi_cone = Cone([p.list() for p in pi_of_K], + ....: lattice=L, + ....: check=False) + sage: actual = pi_cone.dim() + sage: expected = n**2 - l*(m - l) - (n - m)*m + sage: actual == expected + True - return [(x,s) for x in xs for s in ss if x.inner_product(s) == 0] + The trivial cone, full space, and half-plane all give rise to the + expected dimensions:: + sage: n = ZZ.random_element().abs() + sage: K = Cone([[0] * n], ToricLattice(n)) + sage: K.is_trivial() + True + sage: L = ToricLattice(n^2) + sage: pi_of_K = positive_operator_gens(K) + sage: pi_cone = Cone([p.list() for p in pi_of_K], + ....: lattice=L, + ....: check=False) + sage: actual = pi_cone.dim() + sage: actual == n^2 + True + sage: K = K.dual() + sage: K.is_full_space() + True + sage: pi_of_K = positive_operator_gens(K) + sage: pi_cone = Cone([p.list() for p in pi_of_K], + ....: lattice=L, + ....: check=False) + sage: actual = pi_cone.dim() + sage: actual == n^2 + True + sage: K = Cone([(1,0),(0,1),(0,-1)]) + sage: pi_of_K = positive_operator_gens(K) + sage: actual = Cone([p.list() for p in pi_of_K], check=False).dim() + sage: actual == 3 + True -def lyapunov_rank(K): - r""" - Compute the Lyapunov (or bilinearity) rank of this cone. + The lineality of the cone of positive operators follows from the + description of its generators:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=4) + sage: n = K.lattice_dim() + sage: pi_of_K = positive_operator_gens(K) + sage: L = ToricLattice(n**2) + sage: pi_cone = Cone([p.list() for p in pi_of_K], + ....: lattice=L, + ....: check=False) + sage: actual = pi_cone.lineality() + sage: expected = n**2 - K.dim()*K.dual().dim() + sage: actual == expected + True - The Lyapunov rank of a cone can be thought of in (mainly) two ways: + The trivial cone, full space, and half-plane all give rise to the + expected linealities:: - 1. The dimension of the Lie algebra of the automorphism group of the - cone. + sage: n = ZZ.random_element().abs() + sage: K = Cone([[0] * n], ToricLattice(n)) + sage: K.is_trivial() + True + sage: L = ToricLattice(n^2) + sage: pi_of_K = positive_operator_gens(K) + sage: pi_cone = Cone([p.list() for p in pi_of_K], + ....: lattice=L, + ....: check=False) + sage: actual = pi_cone.lineality() + sage: actual == n^2 + True + sage: K = K.dual() + sage: K.is_full_space() + True + sage: pi_of_K = positive_operator_gens(K) + sage: pi_cone = Cone([p.list() for p in pi_of_K], lattice=L) + sage: pi_cone.lineality() == n^2 + True + sage: K = Cone([(1,0),(0,1),(0,-1)]) + sage: pi_of_K = positive_operator_gens(K) + sage: pi_cone = Cone([p.list() for p in pi_of_K], check=False) + sage: actual = pi_cone.lineality() + sage: actual == 2 + True - 2. The dimension of the linear space of all Lyapunov-like - transformations on the cone. + A cone is proper if and only if its cone of positive operators + is proper:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=4) + sage: pi_of_K = positive_operator_gens(K) + sage: L = ToricLattice(K.lattice_dim()**2) + sage: pi_cone = Cone([p.list() for p in pi_of_K], + ....: lattice=L, + ....: check=False) + sage: K.is_proper() == pi_cone.is_proper() + True - INPUT: + The positive operators of a permuted cone can be obtained by + conjugation:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=4) + sage: L = ToricLattice(K.lattice_dim()**2) + sage: p = SymmetricGroup(K.lattice_dim()).random_element().matrix() + sage: pK = Cone([ p*k for k in K ], K.lattice(), check=False) + sage: pi_of_pK = positive_operator_gens(pK) + sage: actual = Cone([t.list() for t in pi_of_pK], + ....: lattice=L, + ....: check=False) + sage: pi_of_K = positive_operator_gens(K) + sage: expected = Cone([(p*t*p.inverse()).list() for t in pi_of_K], + ....: lattice=L, + ....: check=False) + sage: actual.is_equivalent(expected) + True - A closed, convex polyhedral cone. + A transformation is positive on a cone if and only if its adjoint is + positive on the dual of that cone:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=4) + sage: F = K.lattice().vector_space().base_field() + sage: n = K.lattice_dim() + sage: L = ToricLattice(n**2) + sage: W = VectorSpace(F, n**2) + sage: pi_of_K = positive_operator_gens(K) + sage: pi_of_K_star = positive_operator_gens(K.dual()) + sage: pi_cone = Cone([p.list() for p in pi_of_K], + ....: lattice=L, + ....: check=False) + sage: pi_star = Cone([p.list() for p in pi_of_K_star], + ....: lattice=L, + ....: check=False) + sage: M = MatrixSpace(F, n) + sage: L = M(pi_cone.random_element(ring=QQ).list()) + sage: pi_star.contains(W(L.transpose().list())) + True - OUTPUT: + sage: L = W.random_element() + sage: L_star = W(M(L.list()).transpose().list()) + sage: pi_cone.contains(L) == pi_star.contains(L_star) + True - 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). + The Lyapunov rank of the positive operator cone is the product of + the Lyapunov ranks of the associated cones if they're all proper:: + + sage: K1 = random_cone(max_ambient_dim=4, + ....: strictly_convex=True, + ....: solid=True) + sage: K2 = random_cone(max_ambient_dim=4, + ....: strictly_convex=True, + ....: solid=True) + sage: pi_K1_K2 = positive_operator_gens(K1,K2) + sage: L = ToricLattice(K1.lattice_dim() * K2.lattice_dim()) + sage: pi_cone = Cone([ g.list() for g in pi_K1_K2 ], + ....: lattice=L, + ....: check=False) + sage: beta1 = K1.lyapunov_rank() + sage: beta2 = K2.lyapunov_rank() + sage: pi_cone.lyapunov_rank() == beta1*beta2 + True - .. note:: + The Lyapunov-like operators on a proper polyhedral positive operator + cone can be computed from the Lyapunov-like operators on the cones + with respect to which the operators are positive:: + + sage: K1 = random_cone(max_ambient_dim=4, + ....: strictly_convex=True, + ....: solid=True) + sage: K2 = random_cone(max_ambient_dim=4, + ....: strictly_convex=True, + ....: solid=True) + sage: pi_K1_K2 = positive_operator_gens(K1,K2) + sage: F = K1.lattice().base_field() + sage: m = K1.lattice_dim() + sage: n = K2.lattice_dim() + sage: L = ToricLattice(m*n) + sage: M1 = MatrixSpace(F, m, m) + sage: M2 = MatrixSpace(F, n, n) + sage: LL_K1 = [ M1(x.list()) for x in K1.dual().lyapunov_like_basis() ] + sage: LL_K2 = [ M2(x.list()) for x in K2.lyapunov_like_basis() ] + sage: tps = [ s.tensor_product(x) for x in LL_K1 for s in LL_K2 ] + sage: W = VectorSpace(F, (m**2)*(n**2)) + sage: expected = span(F, [ W(x.list()) for x in tps ]) + sage: pi_cone = Cone([p.list() for p in pi_K1_K2], + ....: lattice=L, + ....: check=False) + sage: LL_pi = pi_cone.lyapunov_like_basis() + sage: actual = span(F, [ W(x.list()) for x in LL_pi ]) + sage: actual == expected + True - In the references, the cones are always assumed to be proper. We - do not impose this restriction. + """ + if K2 is None: + K2 = K1 - .. seealso:: + # Matrices are not vectors in Sage, so we have to convert them + # to vectors explicitly before we can find a basis. We need these + # two values to construct the appropriate "long vector" space. + F = K1.lattice().base_field() + n = K1.lattice_dim() + m = K2.lattice_dim() - :meth:`is_proper` + tensor_products = [ s.tensor_product(x) for x in K1 for s in K2.dual() ] - ALGORITHM: + # Convert those tensor products to long vectors. + W = VectorSpace(F, n*m) + vectors = [ W(tp.list()) for tp in tensor_products ] - 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}`. + check = True + if K1.is_proper() and K2.is_proper(): + # All of the generators involved are extreme vectors and + # therefore minimal [Tam]_. If this cone is neither solid nor + # strictly convex, then the tensor product of ``s`` and ``x`` + # is the same as that of ``-s`` and ``-x``. However, as a + # /set/, ``tensor_products`` may still be minimal. + check = False - REFERENCES: + # Create the dual cone of the positive operators, expressed as + # long vectors. + pi_dual = Cone(vectors, ToricLattice(W.dimension()), check=check) - 1. M.S. Gowda and J. Tao. On the bilinearity rank of a proper cone - and Lyapunov-like transformations, Mathematical Programming, 147 - (2014) 155-170. + # Now compute the desired cone from its dual... + pi_cone = pi_dual.dual() - 2. 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. + # And finally convert its rays back to matrix representations. + M = MatrixSpace(F, m, n) + return [ M(v.list()) for v in pi_cone ] - EXAMPLES: - The nonnegative orthant in `\mathbb{R}^{n}` always has rank `n`:: +def Z_operator_gens(K): + r""" + Compute generators of the cone of Z-operators on this cone. - 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 + OUTPUT: - The `L^{3}_{1}` cone is known to have a Lyapunov rank of one:: + 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 of + this cone's :meth:`discrete_complementarity_set`. Moreover, any + conic (nonnegative linear) combination of these matrices shares the + same property. - sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)]) - sage: lyapunov_rank(L31) - 1 + REFERENCES: - Likewise for the `L^{3}_{\infty}` cone:: + M. Orlitzky. + Positive and Z-operators on closed convex cones. - sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)]) - sage: lyapunov_rank(L3infty) - 1 + EXAMPLES: - The Lyapunov rank should be additive on a product of cones:: + Z-operators on the nonnegative orthant are just Z-matrices. + That is, matrices whose off-diagonal elements are nonnegative:: - 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) + sage: K = Cone([(1,0),(0,1)]) + sage: Z_operator_gens(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_operator_gens(K) + ....: for i in range(z.nrows()) + ....: for j in range(z.ncols()) + ....: if i != j ]) True - Two isomorphic cones should have the same Lyapunov rank. The cone - ``K`` in the following example is isomorphic to the nonnegative - octant in `\mathbb{R}^{3}`:: + The trivial cone in a trivial space has no Z-operators:: - sage: K = Cone([(1,2,3), (-1,1,0), (1,0,6)]) - sage: lyapunov_rank(K) - 3 + sage: K = Cone([], ToricLattice(0)) + sage: Z_operator_gens(K) + [] - The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` - itself:: + Every operator is a Z-operator on the ambient vector space:: - sage: K = Cone([(2,2,4), (-1,9,0), (2,0,6)]) - sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) + sage: K = Cone([(1,),(-1,)]) + sage: K.is_full_space() True + sage: Z_operator_gens(K) + [[-1], [1]] - TESTS: + sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) + sage: K.is_full_space() + True + sage: Z_operator_gens(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] + ] - The Lyapunov rank should be additive on a product of cones:: + A non-obvious application is to find the Z-operators on the + right half-plane:: - 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: K = Cone([(1,0),(0,1),(0,-1)]) + sage: Z_operator_gens(K) + [ + [-1 0] [1 0] [ 0 0] [0 0] [ 0 0] [0 0] + [ 0 0], [0 0], [-1 0], [1 0], [ 0 -1], [0 1] + ] + + Z-operators 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_operator_gens(K) ]) + sage: zs == lls + True + + TESTS: - The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` - itself:: + The Z-property is possessed by every Z-operator:: - sage: K = random_cone(max_dim=10, max_rays=10) - sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=4) + sage: Z_of_K = Z_operator_gens(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 - """ - V = K.lattice().vector_space() + The lineality space of the cone of Z-operators is the space of + Lyapunov-like operators:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=4) + sage: L = ToricLattice(K.lattice_dim()**2) + sage: Z_cone = Cone([ z.list() for z in Z_operator_gens(K) ], + ....: lattice=L, + ....: check=False) + sage: ll_basis = [ vector(l.list()) for l in K.lyapunov_like_basis() ] + sage: lls = L.vector_space().span(ll_basis) + sage: Z_cone.linear_subspace() == lls + True - C_of_K = discrete_complementarity_set(K) + The lineality of the Z-operators on a cone is the Lyapunov + rank of that cone:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=4) + sage: Z_of_K = Z_operator_gens(K) + sage: L = ToricLattice(K.lattice_dim()**2) + sage: Z_cone = Cone([ z.list() for z in Z_of_K ], + ....: lattice=L, + ....: check=False) + sage: Z_cone.lineality() == K.lyapunov_rank() + True - matrices = [x.tensor_product(s) for (x,s) in C_of_K] + The lineality spaces of the duals of the positive and Z-operator + cones are equal. From this it follows that the dimensions of the + Z-operator cone and positive operator cone are equal:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=4) + sage: pi_of_K = positive_operator_gens(K) + sage: Z_of_K = Z_operator_gens(K) + sage: L = ToricLattice(K.lattice_dim()**2) + sage: pi_cone = Cone([p.list() for p in pi_of_K], + ....: lattice=L, + ....: check=False) + sage: Z_cone = Cone([ z.list() for z in Z_of_K], + ....: lattice=L, + ....: check=False) + sage: pi_cone.dim() == Z_cone.dim() + True + sage: pi_star = pi_cone.dual() + sage: z_star = Z_cone.dual() + sage: pi_star.linear_subspace() == z_star.linear_subspace() + True + + The trivial cone, full space, and half-plane all give rise to the + expected dimensions:: - # 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: n = ZZ.random_element().abs() + sage: K = Cone([[0] * n], ToricLattice(n)) + sage: K.is_trivial() + True + sage: L = ToricLattice(n^2) + sage: Z_of_K = Z_operator_gens(K) + sage: Z_cone = Cone([z.list() for z in Z_of_K], + ....: lattice=L, + ....: check=False) + sage: actual = Z_cone.dim() + sage: actual == n^2 + True + sage: K = K.dual() + sage: K.is_full_space() + True + sage: Z_of_K = Z_operator_gens(K) + sage: Z_cone = Cone([z.list() for z in Z_of_K], + ....: lattice=L, + ....: check=False) + sage: actual = Z_cone.dim() + sage: actual == n^2 + True + sage: K = Cone([(1,0),(0,1),(0,-1)]) + sage: Z_of_K = Z_operator_gens(K) + sage: Z_cone = Cone([z.list() for z in Z_of_K], check=False) + sage: Z_cone.dim() == 3 + True - def phi(m): - r""" - Convert a matrix to a vector isomorphically. - """ - return W(m.list()) + The Z-operators of a permuted cone can be obtained by conjugation:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=4) + sage: L = ToricLattice(K.lattice_dim()**2) + sage: p = SymmetricGroup(K.lattice_dim()).random_element().matrix() + sage: pK = Cone([ p*k for k in K ], K.lattice(), check=False) + sage: Z_of_pK = Z_operator_gens(pK) + sage: actual = Cone([t.list() for t in Z_of_pK], + ....: lattice=L, + ....: check=False) + sage: Z_of_K = Z_operator_gens(K) + sage: expected = Cone([(p*t*p.inverse()).list() for t in Z_of_K], + ....: lattice=L, + ....: check=False) + sage: actual.is_equivalent(expected) + True - vectors = [phi(m) for m in matrices] + An operator is a Z-operator on a cone if and only if its + adjoint is a Z-operator on the dual of that cone:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=4) + sage: F = K.lattice().vector_space().base_field() + sage: n = K.lattice_dim() + sage: L = ToricLattice(n**2) + sage: W = VectorSpace(F, n**2) + sage: Z_of_K = Z_operator_gens(K) + sage: Z_of_K_star = Z_operator_gens(K.dual()) + sage: Z_cone = Cone([p.list() for p in Z_of_K], + ....: lattice=L, + ....: check=False) + sage: Z_star = Cone([p.list() for p in Z_of_K_star], + ....: lattice=L, + ....: check=False) + sage: M = MatrixSpace(F, n) + sage: L = M(Z_cone.random_element(ring=QQ).list()) + sage: Z_star.contains(W(L.transpose().list())) + True - return (W.dimension() - W.span(vectors).rank()) + sage: L = W.random_element() + sage: L_star = W(M(L.list()).transpose().list()) + sage: Z_cone.contains(L) == Z_star.contains(L_star) + True + """ + # Matrices are not vectors in Sage, so we have to convert them + # to vectors explicitly before we can find a basis. We need these + # two values to construct the appropriate "long vector" space. + F = K.lattice().base_field() + n = K.lattice_dim() + + # These tensor products contain generators for the dual cone of + # the cross-positive operators. + tensor_products = [ s.tensor_product(x) + for (x,s) in K.discrete_complementarity_set() ] + + # Turn our matrices into long vectors... + W = VectorSpace(F, n**2) + vectors = [ W(m.list()) for m in tensor_products ] + + check = True + if K.is_proper(): + # All of the generators involved are extreme vectors and + # therefore minimal. If this cone is neither solid nor + # strictly convex, then the tensor product of ``s`` and ``x`` + # is the same as that of ``-s`` and ``-x``. However, as a + # /set/, ``tensor_products`` may still be minimal. + check = False + + # Create the dual cone of the cross-positive operators, + # expressed as long vectors. + Sigma_dual = Cone(vectors, lattice=ToricLattice(W.dimension()), check=check) + + # 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-operators and + # not cross-positive ones. + M = MatrixSpace(F, n) + return [ -M(v.list()) for v in Sigma_cone ] + + +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 Z_cone(K): + gens = Z_operator_gens(K) + L = ToricLattice(K.lattice_dim()**2) + return Cone([ g.list() for g in gens ], lattice=L, check=False) + +def pi_cone(K): + gens = positive_operator_gens(K) + L = ToricLattice(K.lattice_dim()**2) + return Cone([ g.list() for g in gens ], lattice=L, check=False)