X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=5f058ca9a8c055f972ffffaa216975517e0dc723;hb=9bab3c89cc7d669e7b99295900c9f590e5525079;hp=424a907dc75d1ed7d4a1a59e1e8eee7b9f64d75e;hpb=08ff39a270e90e6263339da12144ddd2bfc6f3f5;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 424a907..5f058ca 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -8,425 +8,651 @@ addsitedir(abspath('../../')) from sage.all import * -def is_full_space(K): +def _restrict_to_space(K, W): r""" - Return whether or not this cone is equal to its ambient vector space. + Restrict this cone a subspace of its ambient space. + + INPUT: + + - ``W`` -- The subspace into which this cone will be restricted. OUTPUT: - ``True`` if this cone is the entire vector space and ``False`` - otherwise. + A new cone in a sublattice corresponding to ``W``. EXAMPLES: - A ray in two dimensions is not equal to the entire space:: + When this cone is solid, restricting it into its own span should do + nothing:: - sage: K = Cone([(1,0)]) - sage: is_full_space(K) - False + sage: K = Cone([(1,)]) + sage: _restrict_to_space(K, K.span()) == K + True - Neither is the nonnegative orthant:: + A single ray restricted into its own span gives the same output + regardless of the ambient space:: + + sage: K2 = Cone([(1,0)]) + sage: K2_S = _restrict_to_space(K2, K2.span()).rays() + sage: K2_S + N(1) + in 1-d lattice N + sage: K3 = Cone([(1,0,0)]) + 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 - sage: K = Cone([(1,0),(0,1)]) - sage: is_full_space(K) - False + TESTS: - The right half-space contains a vector subspace, but it is still not - equal to the entire plane:: + The projected cone should always be solid:: - sage: K = Cone([(1,0),(-1,0),(0,1)]) - sage: is_full_space(K) - False + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim = 8) + sage: _restrict_to_space(K, K.span()).is_solid() + True - But if we include nonnegative sums from both axes, then the resulting - cone is the entire two-dimensional space:: + And the resulting cone should live in a space having the same + dimension as the space we restricted it to:: - sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) - sage: is_full_space(K) + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim = 8) + sage: K_P = _restrict_to_space(K, K.dual().span()) + sage: K_P.lattice_dim() == K.dual().dim() True - """ - return K.linear_subspace() == K.lattice().vector_space() + This function should not affect the dimension of a cone:: + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim = 8) + sage: K.dim() == _restrict_to_space(K,K.span()).dim() + True -def random_cone(min_dim=0, max_dim=None, min_rays=0, max_rays=None): - r""" - Generate a random rational convex polyhedral cone. + Nor should it affect the lineality of a 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. + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim = 8) + sage: K.lineality() == _restrict_to_space(K, K.span()).lineality() + True - 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``. + No matter which space we restrict to, the lineality should not + increase:: - .. NOTE: + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim = 8) + sage: S = K.span(); P = K.dual().span() + sage: K.lineality() >= _restrict_to_space(K,S).lineality() + True + sage: K.lineality() >= _restrict_to_space(K,P).lineality() + True - 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. + If we do this according to our paper, then the result is proper:: - INPUT: + 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.dual(), K_S.dual().span()).dual() + sage: K_SP.is_proper() + True + sage: K_SP = _restrict_to_space(K_S, K_S.dual().span()) + sage: K_SP.is_proper() + True - - ``min_dim`` (default: zero) -- A nonnegative integer representing the - minimum dimension of the ambient lattice. + Test the proposition in our paper concerning the duals and + restrictions. Generate a random cone, then create a subcone of + it. The operation of dual-taking should then commute with + _restrict_to_space:: + + sage: set_random_seed() + sage: J = random_cone(max_ambient_dim = 8) + sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) + sage: K_W_star = _restrict_to_space(K, J.span()).dual() + sage: K_star_W = _restrict_to_space(K.dual(), J.span()) + sage: _basically_the_same(K_W_star, K_star_W) + True + + """ + # First we want to intersect ``K`` with ``W``. The easiest way to + # do this is via cone intersection, so we turn the subspace ``W`` + # into a cone. + W_cone = Cone(W.basis() + [-b for b in W.basis()], lattice=K.lattice()) + K = K.intersection(W_cone) - - ``max_dim`` (default: random) -- A nonnegative integer representing - the maximum dimension of the ambient - lattice. + # We've already intersected K with the span of K2, so every + # generator of K should belong to W now. + K_W_rays = [ W.coordinate_vector(r) for r in K.rays() ] - - ``min_rays`` (default: zero) -- A nonnegative integer representing the - minimum number of generating rays of the - cone. + L = ToricLattice(W.dimension()) + return Cone(K_W_rays, lattice=L) + + +def lyapunov_rank(K): + r""" + Compute the Lyapunov rank of this cone. - - ``max_rays`` (default: random) -- A nonnegative integer representing the - maximum number of generating rays of - the cone. + The Lyapunov rank of a cone is the dimension of the space of its + Lyapunov-like transformations -- that is, the length of a + :meth:`lyapunov_like_basis`. Equivalently, the Lyapunov rank is the + dimension of the Lie algebra of the automorphism group of the cone. OUTPUT: - A new, randomly generated cone. + A nonnegative integer representing the Lyapunov rank of this cone. + + If the ambient space is trivial, the Lyapunov rank will be zero. + Otherwise, if the dimension of the ambient vector space is `n`, then + the resulting Lyapunov rank will be between `1` and `n` inclusive. A + Lyapunov rank of `n-1` is not possible [Orlitzky]_. + + ALGORITHM: - A ``ValueError` will be thrown under the following conditions: + 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}`. - * Any of ``min_dim``, ``max_dim``, ``min_rays``, or ``max_rays`` - are negative. + REFERENCES: - * ``max_dim`` is less than ``min_dim``. + .. [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. - * ``max_rays`` is less than ``min_rays``. + M. Orlitzky. The Lyapunov rank of an improper cone. + http://www.optimization-online.org/DB_HTML/2015/10/5135.html - * ``min_rays`` is greater than twice ``max_dim``. + 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: - If we set the lower/upper bounds to zero, then our result is - predictable:: + The nonnegative orthant in `\mathbb{R}^{n}` always has rank `n` + [Rudolf]_:: - sage: random_cone(0,0,0,0) - 0-d cone in 0-d lattice N + 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 - We can predict the dimension when ``min_dim == max_dim``:: + The full space `\mathbb{R}^{n}` has Lyapunov rank `n^{2}` + [Orlitzky]_:: - sage: random_cone(min_dim=4, max_dim=4, min_rays=0, max_rays=0) - 0-d cone in 4-d lattice N + sage: R5 = VectorSpace(QQ, 5) + sage: gs = R5.basis() + [ -r for r in R5.basis() ] + sage: K = Cone(gs) + sage: lyapunov_rank(K) + 25 - Likewise for the number of rays when ``min_rays == max_rays``:: + The `L^{3}_{1}` cone is known to have a Lyapunov rank of one + [Rudolf]_:: - sage: random_cone(min_dim=10, max_dim=10, min_rays=10, max_rays=10) - 10-d cone in 10-d lattice N + sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)]) + sage: lyapunov_rank(L31) + 1 + + Likewise for the `L^{3}_{\infty}` cone [Rudolf]_:: + + sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)]) + sage: lyapunov_rank(L3infty) + 1 + + A single ray in `n` dimensions should have Lyapunov rank `n^{2} - n + + 1` [Orlitzky]_:: + + sage: K = Cone([(1,0,0,0,0)]) + sage: lyapunov_rank(K) + 21 + sage: K.lattice_dim()**2 - K.lattice_dim() + 1 + 21 + + A subspace (of dimension `m`) in `n` dimensions should have a + Lyapunov rank of `n^{2} - m\left(n - m)` [Orlitzky]_:: + + sage: e1 = (1,0,0,0,0) + sage: neg_e1 = (-1,0,0,0,0) + sage: e2 = (0,1,0,0,0) + sage: neg_e2 = (0,-1,0,0,0) + sage: z = (0,0,0,0,0) + sage: K = Cone([e1, neg_e1, e2, neg_e2, z, z, z]) + sage: lyapunov_rank(K) + 19 + sage: K.lattice_dim()**2 - K.dim()*K.codim() + 19 + + The Lyapunov rank should be additive on a product of proper cones + [Rudolf]_:: + + sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)]) + sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)]) + sage: K = L31.cartesian_product(octant) + sage: lyapunov_rank(K) == lyapunov_rank(L31) + lyapunov_rank(octant) + True + + Two isomorphic cones should have the same Lyapunov rank [Rudolf]_. + The cone ``K`` in the following example is isomorphic to the nonnegative + octant in `\mathbb{R}^{3}`:: + + sage: K = Cone([(1,2,3), (-1,1,0), (1,0,6)]) + sage: lyapunov_rank(K) + 3 + + The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` + itself [Rudolf]_:: + + sage: K = Cone([(2,2,4), (-1,9,0), (2,0,6)]) + sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) + True TESTS: - It's hard to test the output of a random process, but we can at - least make sure that we get a cone back:: + The Lyapunov rank should be additive on a product of proper cones + [Rudolf]_:: - sage: from sage.geometry.cone import is_Cone # long time - sage: K = random_cone() # long time - sage: is_Cone(K) # long time + 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 upper/lower bounds are respected:: + The Lyapunov rank is invariant under a linear isomorphism + [Orlitzky]_:: - 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 + 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 - sage: 3 <= K.nrays() and K.nrays() <= 4 + + The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` + itself [Rudolf]_:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=8) + sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) True - Ensure that an exception is raised when either lower bound is greater - than its respective upper bound:: + The Lyapunov rank of a proper polyhedral cone in `n` dimensions can + be any number between `1` and `n` inclusive, excluding `n-1` + [Gowda/Tao]_. By accident, the `n-1` restriction will hold for the + trivial cone in a trivial space as well. However, in zero dimensions, + the Lyapunov rank of the trivial cone will be zero:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=8, + ....: strictly_convex=True, + ....: solid=True) + sage: b = lyapunov_rank(K) + sage: n = K.lattice_dim() + sage: (n == 0 or 1 <= b) and b <= n + True + sage: b == n-1 + False - sage: random_cone(min_dim=5, max_dim=2) - Traceback (most recent call last): - ... - ValueError: max_dim cannot be less than min_dim. + In fact [Orlitzky]_, no closed convex polyhedral cone can have + Lyapunov rank `n-1` in `n` dimensions:: - sage: random_cone(min_rays=5, max_rays=2) - Traceback (most recent call last): - ... - ValueError: max_rays cannot be less than min_rays. + 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 - And if we request too many rays:: + The calculation of the Lyapunov rank of an improper cone can be + reduced to that of a proper cone [Orlitzky]_:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=8) + sage: actual = lyapunov_rank(K) + sage: K_S = _restrict_to_space(K, K.span()) + sage: K_SP = _restrict_to_space(K_S.dual(), K_S.dual().span()).dual() + sage: l = K.lineality() + sage: c = K.codim() + sage: expected = lyapunov_rank(K_SP) + K.dim()*(l + c) + c**2 + sage: actual == expected + True + + The Lyapunov rank of a cone is the size of a :meth:`lyapunov_like_basis`:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=8) + sage: lyapunov_rank(K) == len(K.lyapunov_like_basis()) + True + + We can make an imperfect cone perfect by adding a slack variable + (a Theorem in [Orlitzky]_):: - sage: random_cone(min_rays=5, max_dim=1) - Traceback (most recent call last): - ... - ValueError: min_rays cannot be larger than twice max_dim. + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=8, + ....: strictly_convex=True, + ....: solid=True) + sage: L = ToricLattice(K.lattice_dim() + 1) + sage: K = Cone([ r.list() + [0] for r in K.rays() ], lattice=L) + sage: lyapunov_rank(K) >= K.lattice_dim() + True """ + beta = 0 # running tally of the Lyapunov rank - # 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) - - 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): + 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""" - Compute the discrete complementarity set of this cone. + 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. - 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. + - ``K`` -- A polyhedral closed convex cone. OUTPUT: - A list of pairs `(x,s)` such that, + ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``, + and ``False`` otherwise. + + .. WARNING:: - * `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. + 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 discrete complementarity set of the nonnegative orthant consists - of pairs of standard basis vectors:: + The identity is always Lyapunov-like in a nontrivial space:: - 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: 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 = Cone([(1,0),(-1,0),(0,1),(0,-1)]) - sage: discrete_complementarity_set(K) - [] + 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 + + Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like + on ``K``:: + + 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 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: - The complementarity set of the dual can be obtained by switching the - components of the complementarity set of the original cone:: + Any cone should contain an element of itself:: - 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_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)) ] - # 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] + # 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 lyapunov_rank(K): +def positive_operators(K): r""" - Compute the Lyapunov (or bilinearity) rank of this cone. + Compute generators of the cone of positive operators on this cone. - The Lyapunov rank of a cone can be thought of in (mainly) two ways: + OUTPUT: - 1. The dimension of the Lie algebra of the automorphism group of the - cone. + 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. - 2. The dimension of the linear space of all Lyapunov-like - transformations on the cone. + EXAMPLES: - INPUT: + The trivial cone in a trivial space has no positive operators:: - A closed, convex polyhedral cone. + sage: K = Cone([], ToricLattice(0)) + sage: positive_operators(K) + [] - OUTPUT: + Positive operators on the nonnegative orthant are nonnegative matrices:: - An integer representing the Lyapunov rank of the cone. If the - dimension of the ambient vector space is `n`, then the Lyapunov rank - will be between `1` and `n` inclusive; however a rank of `n-1` is - not possible (see the first reference). + sage: K = Cone([(1,)]) + sage: positive_operators(K) + [[1]] - .. note:: + 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] + ] - In the references, the cones are always assumed to be proper. We - do not impose this restriction. + Every operator is positive on the ambient vector space:: - .. seealso:: + sage: K = Cone([(1,),(-1,)]) + sage: K.is_full_space() + True + sage: positive_operators(K) + [[1], [-1]] - :meth:`is_proper` + 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] + ] - ALGORITHM: + TESTS: - 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}`. + A positive operator on a cone should send its generators into the cone:: - REFERENCES: + 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) - 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. + tensor_products = [ s.tensor_product(x) for x in K for s in K.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. + # Turn our matrices into long vectors... + vectors = [ W(m.list()) for m in tensor_products ] - EXAMPLES: + # Create the *dual* cone of the positive operators, expressed as + # long vectors.. + L = ToricLattice(W.dimension()) + pi_dual = Cone(vectors, lattice=L) - The nonnegative orthant in `\mathbb{R}^{n}` always has rank `n`:: + # Now compute the desired cone from its dual... + pi_cone = pi_dual.dual() - 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 + # And finally convert its rays back to matrix representations. + M = MatrixSpace(V.base_ring(), V.dimension()) - The `L^{3}_{1}` cone is known to have a Lyapunov rank of one:: + return [ M(v.list()) for v in pi_cone.rays() ] - 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:: +def Z_transformations(K): + r""" + Compute generators of the cone of Z-transformations on this cone. - sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)]) - sage: lyapunov_rank(L3infty) - 1 + OUTPUT: - The Lyapunov rank should be additive on a product of cones:: + 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. - 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) + 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 - 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-transformations:: - sage: K = Cone([(1,2,3), (-1,1,0), (1,0,6)]) - sage: lyapunov_rank(K) - 3 + sage: K = Cone([], ToricLattice(0)) + sage: Z_transformations(K) + [] - The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` - itself:: + Z-transformations on a subspace are Lyapunov-like and vice-versa:: - 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),(-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 Lyapunov rank should be additive on a product of cones:: + The Z-property is possessed by every Z-transformation:: - sage: K1 = random_cone(max_dim=10, max_rays=10) - sage: K2 = random_cone(max_dim=10, max_rays=10) - sage: K = K1.cartesian_product(K2) - sage: lyapunov_rank(K) == lyapunov_rank(K1) + lyapunov_rank(K2) + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim = 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 dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` - itself:: + The lineality space of Z is LL:: - 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(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 """ - V = K.lattice().vector_space() - - C_of_K = discrete_complementarity_set(K) - - matrices = [x.tensor_product(s) 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. @@ -435,14 +661,26 @@ def lyapunov_rank(K): # 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) - def phi(m): - r""" - Convert a matrix to a vector isomorphically. - """ - return W(m.list()) + 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() - vectors = [phi(m) for m in matrices] + # 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 (W.dimension() - W.span(vectors).rank()) + return [ -M(v.list()) for v in Sigma_cone.rays() ]