X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=b7456e21abc170d6479f9010a2b7ccdd5ab9438d;hb=1bbade9f41ffbfe366b15d0db657f666bc1f025d;hp=6ade5e628f1035c99294048c7fb55b4b9c1204d9;hpb=7d2f3fba7f494158dbce5f7a3eca1d15ee7f577e;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 6ade5e6..b7456e2 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -1,345 +1,23 @@ -# 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): - 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. - - INPUT: - - - ``min_dim`` (default: zero) -- A nonnegative integer representing the - minimum dimension of the ambient lattice. - - - ``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. - - OUTPUT: - - A new, randomly generated cone. - - EXAMPLES: - - If we set the lower/upper bounds to zero, then our result is - predictable:: - - sage: random_cone(0,0,0,0) - 0-d cone in 0-d lattice N - - In fact, as long as we ask for zero rays, we should be able to predict - the output when ``min_dim == max_dim``:: - - sage: random_cone(min_dim=4, max_dim=4, min_rays=0, max_rays=0) - 0-d cone in 4-d lattice N - - 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:: - - sage: from sage.geometry.cone import is_Cone # long time - sage: K = random_cone() # long time - sage: is_Cone(K) # long time - True - - Ensure that an exception is raised when either lower bound is greater - than its respective upper bound:: - - sage: random_cone(min_dim=5, max_dim=2) - Traceback (most recent call last): - ... - ValueError: max_dim must be greater than or equal to min_dim. - - sage: random_cone(min_rays=5, max_rays=2) - Traceback (most recent call last): - ... - ValueError: max_rays must be greater than or equal to min_rays. - - """ - - # 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 (min_dim > max_dim): - raise ValueError('max_dim must be greater than or equal to min_dim.') - - if max_rays is not None: - if max_rays < 0: - raise ValueError('max_rays must be nonnegative.') - if (min_rays > max_rays): - raise ValueError('max_rays must be greater than or equal to 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) - 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. - return Cone(rays, lattice=L) - - -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) - [] - - TESTS: - - The complementarity set of the dual can be obtained by switching the - components of the complementarity set of the original 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 - True - - """ - V = K.lattice().vector_space() - - # Convert the rays to vectors so that we can compute inner - # products. - xs = [V(x) for x in K.rays()] - ss = [V(s) for s in K.dual().rays()] - - return [(x,s) for x in xs for s in ss if x.inner_product(s) == 0] - - -def lyapunov_rank(K): - r""" - Compute the Lyapunov (or bilinearity) 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. - - 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). - - .. note:: - - In the references, the cones are always assumed to be proper. We - do not impose this restriction. - - .. seealso:: - - :meth:`is_proper` - - ALGORITHM: - - 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}`. - - REFERENCES: - - 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. - - 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. - - EXAMPLES: - - The nonnegative orthant in `\mathbb{R}^{n}` always has rank `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 - - The `L^{3}_{1}` cone is known to have a Lyapunov rank of one:: - - 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:: - - 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:: - - 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. 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:: - - sage: K = Cone([(2,2,4), (-1,9,0), (2,0,6)]) - sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) - True - - TESTS: - - The Lyapunov rank should be additive on a product of cones:: - - 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) - True - - The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` - itself:: - - sage: K = random_cone(max_dim=10, max_rays=10) - sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) - 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. - # - # 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) - - def phi(m): - r""" - Convert a matrix to a vector isomorphically. - """ - return W(m.list()) - - vectors = [phi(m) for m in matrices] - - return (W.dimension() - W.span(vectors).rank()) +def LL_cone(K): + gens = K.lyapunov_like_basis() + L = ToricLattice(K.lattice_dim()**2) + return Cone(( g.list() for g in gens ), lattice=L, check=False) + +def Sigma_cone(K): + gens = K.cross_positive_operators_gens() + L = ToricLattice(K.lattice_dim()**2) + return Cone(( g.list() for g in gens ), lattice=L, check=False) + +def Z_cone(K): + gens = K.Z_operators_gens() + L = ToricLattice(K.lattice_dim()**2) + return Cone(( g.list() for g in gens ), lattice=L, check=False) + +def pi_cone(K1, K2=None): + if K2 is None: + K2 = K1 + gens = K1.positive_operators_gens(K2) + L = ToricLattice(K1.lattice_dim()*K2.lattice_dim()) + return Cone(( g.list() for g in gens ), lattice=L, check=False)