X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=b7456e21abc170d6479f9010a2b7ccdd5ab9438d;hb=1bbade9f41ffbfe366b15d0db657f666bc1f025d;hp=3f915ed66fb4ea33629bc1c30ff4e021f59f8607;hpb=d3dd6210c93ee4ea0fbeb8e643649a0ab958c796;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 3f915ed..b7456e2 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -1,672 +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 _basically_the_same(K1, K2): - r""" - Test whether or not ``K1`` and ``K2`` are "basically the same." - - This is a hack to get around the fact that it's difficult to tell - when two cones are linearly isomorphic. We have a proposition that - equates two cones, but represented over `\mathbb{Q}`, they are - merely linearly isomorphic (not equal). So rather than test for - equality, we test a list of properties that should be preserved - under an invertible linear transformation. - - OUTPUT: - - ``True`` if ``K1`` and ``K2`` are basically the same, and ``False`` - otherwise. - - EXAMPLES: - - Any proper cone with three generators in `\mathbb{R}^{3}` is - basically the same as the nonnegative orthant:: - - sage: K1 = Cone([(1,0,0), (0,1,0), (0,0,1)]) - sage: K2 = Cone([(1,2,3), (3, 18, 4), (66, 51, 0)]) - sage: _basically_the_same(K1, K2) - True - - Negating a cone gives you another cone that is basically the same:: - - sage: K = Cone([(0,2,-5), (-6, 2, 4), (0, 51, 0)]) - sage: _basically_the_same(K, -K) - True - - TESTS: - - Any cone is basically the same as itself:: - - sage: K = random_cone(max_ambient_dim = 8) - sage: _basically_the_same(K, K) - True - - After applying an invertible matrix to the rows of a cone, the - result should be basically the same as the cone we started with:: - - 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: _basically_the_same(K1, K2) - True - - """ - if K1.lattice_dim() != K2.lattice_dim(): - return False - - if K1.nrays() != K2.nrays(): - return False - - if K1.dim() != K2.dim(): - return False - - if K1.lineality() != K2.lineality(): - return False - - if K1.is_solid() != K2.is_solid(): - return False - - if K1.is_strictly_convex() != K2.is_strictly_convex(): - return False - - if len(K1.LL()) != len(K2.LL()): - return False - - C_of_K1 = K1.discrete_complementarity_set() - C_of_K2 = K2.discrete_complementarity_set() - if len(C_of_K1) != len(C_of_K2): - return False - - if len(K1.facets()) != len(K2.facets()): - return False - - return True - - - -def _restrict_to_space(K, W): - r""" - Restrict this cone a subspace of its ambient space. - - INPUT: - - - ``W`` -- The subspace into which this cone will be restricted. - - OUTPUT: - - A new cone in a sublattice corresponding to ``W``. - - EXAMPLES: - - When this cone is solid, restricting it into its own span should do - nothing:: - - sage: K = Cone([(1,)]) - sage: _restrict_to_space(K, K.span()) == K - True - - 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 - - TESTS: - - The projected cone should always be solid:: - - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim = 8) - sage: _restrict_to_space(K, K.span()).is_solid() - True - - And the resulting cone should live in a space having the same - dimension as the space we restricted it to:: - - 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 - - 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 - - Nor should it affect the lineality of a cone:: - - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim = 8) - sage: K.lineality() == _restrict_to_space(K, K.span()).lineality() - True - - No matter which space we restrict to, the lineality should not - increase:: - - 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 we do this according to our paper, then the result is proper:: - - 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 - - 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) - - # 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() ] - - L = ToricLattice(W.dimension()) - return Cone(K_W_rays, lattice=L) - - -def lyapunov_rank(K): - r""" - Compute the Lyapunov rank (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 [Orlitzky/Gowda]_). - - 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: - - .. [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. - - .. [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. - - EXAMPLES: - - The nonnegative orthant in `\mathbb{R}^{n}` always has rank `n` - [Rudolf et al.]_:: - - sage: positives = Cone([(1,)]) - sage: lyapunov_rank(positives) - 1 - sage: quadrant = Cone([(1,0), (0,1)]) - sage: lyapunov_rank(quadrant) - 2 - sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)]) - sage: lyapunov_rank(octant) - 3 - - The full space `\mathbb{R}^{n}` has Lyapunov rank `n^{2}` - [Orlitzky/Gowda]_:: - - sage: R5 = VectorSpace(QQ, 5) - sage: 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.]_:: - - sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)]) - sage: lyapunov_rank(L31) - 1 - - Likewise for the `L^{3}_{\infty}` cone [Rudolf et al.]_:: - - sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)]) - sage: lyapunov_rank(L3infty) - 1 - - A single ray in `n` dimensions should have Lyapunov rank `n^{2} - n - + 1` [Orlitzky/Gowda]_:: - - sage: K = Cone([(1,0,0,0,0)]) - sage: lyapunov_rank(K) - 21 - sage: K.lattice_dim()**2 - K.lattice_dim() + 1 - 21 - - A subspace (of dimension `m`) in `n` dimensions should have a - Lyapunov rank of `n^{2} - m\left(n - m)` [Orlitzky/Gowda]_:: - - sage: e1 = (1,0,0,0,0) - sage: neg_e1 = (-1,0,0,0,0) - sage: e2 = (0,1,0,0,0) - sage: neg_e2 = (0,-1,0,0,0) - sage: 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 et al.]_:: - - sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)]) - sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)]) - sage: K = L31.cartesian_product(octant) - sage: lyapunov_rank(K) == lyapunov_rank(L31) + lyapunov_rank(octant) - True - - Two isomorphic cones should have the same Lyapunov rank [Rudolf et al.]_. - The cone ``K`` in the following example is isomorphic to the nonnegative - octant in `\mathbb{R}^{3}`:: - - sage: K = Cone([(1,2,3), (-1,1,0), (1,0,6)]) - sage: lyapunov_rank(K) - 3 - - The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` - itself [Rudolf et al.]_:: - - 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 proper cones - [Rudolf et al.]_:: - - 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/Gowda]_:: - - 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.]_:: - - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=8) - sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) - True - - The Lyapunov rank of a proper polyhedral cone in `n` dimensions can - be any number between `1` and `n` inclusive, excluding `n-1` - [Gowda/Tao]_. By accident, the `n-1` restriction will hold for the - trivial cone in a trivial space as well. However, in zero dimensions, - the Lyapunov rank of the trivial cone will be zero:: - - 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 - - In fact [Orlitzky/Gowda]_, no closed convex polyhedral cone can have - Lyapunov rank `n-1` in `n` dimensions:: - - 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]_:: - - 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 any cone is just the dimension of ``K.LL()``:: - - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=8) - sage: lyapunov_rank(K) == len(K.LL()) - True - - We can make an imperfect cone perfect by adding a slack variable - (a Theorem in [Orlitzky/Gowda]_):: - - 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 - - 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.LL()) - 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: - - .. [Orlitzky] M. Orlitzky. The Lyapunov rank of an - improper cone (preprint). - - 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 - - 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.LL()`` 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.LL()]) - 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: - - 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 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)