X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=a1ded5270032de21f2647de8453384fde4804c72;hb=090b2c77aa4bd371d66885451f9df44c6b6d818f;hp=baff1a7bbea4943206bc323040ef5d554d40ead4;hpb=cbd0e9a0ed9133de08665a55f146d04cb9536ec6;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index baff1a7..a1ded52 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -1,837 +1,411 @@ -# 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): +def is_lyapunov_like(L,K): r""" - Test whether or not ``K1`` and ``K2`` are "basically the same." + Determine whether or not ``L`` is Lyapunov-like on ``K``. - 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. + 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. - OUTPUT: + INPUT: - ``True`` if ``K1`` and ``K2`` are basically the same, and ``False`` - otherwise. + - ``L`` -- A linear transformation or matrix. - EXAMPLES: + - ``K`` -- A polyhedral closed convex cone. - Any proper cone with three generators in `\mathbb{R}^{3}` is - basically the same as the nonnegative orthant:: + OUTPUT: - 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 + ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``, + and ``False`` otherwise. - Negating a cone gives you another cone that is basically the same:: + .. WARNING:: - sage: K = Cone([(0,2,-5), (-6, 2, 4), (0, 51, 0)]) - sage: _basically_the_same(K, -K) - True + 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. - TESTS: + REFERENCES: - Any cone is basically the same as itself:: + M. Orlitzky. The Lyapunov rank of an improper cone. + http://www.optimization-online.org/DB_HTML/2015/10/5135.html - sage: K = random_cone(max_dim = 8) - sage: _basically_the_same(K, K) - True + EXAMPLES: - After applying an invertible matrix to the rows of a cone, the - result should be basically the same as the cone we started with:: + The identity is always Lyapunov-like in a nontrivial space:: - sage: K1 = random_cone(max_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) + 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 - """ - if K1.lattice_dim() != K2.lattice_dim(): - return False - - if K1.nrays() != K2.nrays(): - return False + As is the "zero" transformation:: - 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(LL(K1)) != len(LL(K2)): - return False - - C_of_K1 = discrete_complementarity_set(K1) - C_of_K2 = discrete_complementarity_set(K2) - if len(C_of_K1) != len(C_of_K2): - return False + 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 - if len(K1.facets()) != len(K2.facets()): - return False + Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like + on ``K``:: - return True + 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 + """ + return all([(L*x).inner_product(s) == 0 + for (x,s) in K.discrete_complementarity_set()]) -def _rho(K, K2=None): +def motzkin_decomposition(K): r""" - Restrict ``K`` into its own span, or the span of another cone. - - INPUT: + Return the pair of components in the motzkin decomposition of this cone. - - ``K2`` -- another cone whose lattice has the same rank as this - cone. + Every convex cone is the direct sum of a strictly convex cone and a + linear subspace. Return a pair ``(P,S)`` of cones such that ``P`` is + strictly convex, ``S`` is a subspace, and ``K`` is the direct sum of + ``P`` and ``S``. OUTPUT: - A new cone in a sublattice. - - EXAMPLES:: - - sage: K = Cone([(1,)]) - sage: _rho(K) == K - True + An ordered pair ``(P,S)`` of closed convex polyhedral cones where + ``P`` is strictly convex, ``S`` is a subspace, and ``K`` is the + direct sum of ``P`` and ``S``. - sage: K2 = Cone([(1,0)]) - sage: _rho(K2).rays() - N(1) - in 1-d lattice N - sage: K3 = Cone([(1,0,0)]) - sage: _rho(K3).rays() - N(1) - in 1-d lattice N - sage: _rho(K2) == _rho(K3) - True - - TESTS: - - The projected cone should always be solid:: - - sage: set_random_seed() - sage: K = random_cone(max_dim = 8) - sage: K_S = _rho(K) - sage: K_S.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_dim = 8) - sage: K_S = _rho(K, K.dual() ) - sage: K_S.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_dim = 8) - sage: K.dim() == _rho(K).dim() - True - - Nor should it affect the lineality of a cone:: - - sage: set_random_seed() - sage: K = random_cone(max_dim = 8) - sage: K.lineality() == _rho(K).lineality() - True + EXAMPLES: - No matter which space we restrict to, the lineality should not - increase:: + The nonnegative orthant is strictly convex, so it is its own + strictly convex component and its subspace component is trivial:: - sage: set_random_seed() - sage: K = random_cone(max_dim = 8) - sage: K.lineality() >= _rho(K).lineality() + sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)]) + sage: (P,S) = motzkin_decomposition(K) + sage: K.is_equivalent(P) True - sage: K.lineality() >= _rho(K, K.dual()).lineality() + sage: S.is_trivial() True - If we do this according to our paper, then the result is proper:: + Likewise, full spaces are their own subspace components:: - sage: set_random_seed() - sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=False) - sage: K_S = _rho(K) - sage: K_SP = _rho(K_S.dual()).dual() - sage: K_SP.is_proper() - True - sage: K_SP = _rho(K_S, K_S.dual()) - sage: K_SP.is_proper() + sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) + sage: K.is_full_space() True - - :: - - sage: set_random_seed() - sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=False) - sage: K_S = _rho(K) - sage: K_SP = _rho(K_S.dual()).dual() - sage: K_SP.is_proper() + sage: (P,S) = motzkin_decomposition(K) + sage: K.is_equivalent(S) True - sage: K_SP = _rho(K_S, K_S.dual()) - sage: K_SP.is_proper() + sage: P.is_trivial() True - :: - - sage: set_random_seed() - sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=True) - sage: K_S = _rho(K) - sage: K_SP = _rho(K_S.dual()).dual() - sage: K_SP.is_proper() - True - sage: K_SP = _rho(K_S, K_S.dual()) - sage: K_SP.is_proper() - True + TESTS: - :: + A random point in the cone should belong to either the strictly + convex component or the subspace component. If the point is nonzero, + it cannot be in both:: sage: set_random_seed() - sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=True) - sage: K_S = _rho(K) - sage: K_SP = _rho(K_S.dual()).dual() - sage: K_SP.is_proper() - True - sage: K_SP = _rho(K_S, K_S.dual()) - sage: K_SP.is_proper() + sage: K = random_cone(max_ambient_dim=8) + sage: (P,S) = motzkin_decomposition(K) + sage: x = K.random_element() + sage: P.contains(x) or S.contains(x) True - - Test Proposition 7 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 rho:: - - sage: set_random_seed() - sage: J = random_cone(max_dim = 8, solid=False, strictly_convex=False) - sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) - sage: K_W_star = _rho(K, J).dual() - sage: K_star_W = _rho(K.dual(), J) - sage: _basically_the_same(K_W_star, K_star_W) + sage: x.is_zero() or (P.contains(x) != S.contains(x)) True - :: + The strictly convex component should always be strictly convex, and + the subspace component should always be a subspace:: sage: set_random_seed() - sage: J = random_cone(max_dim = 8, solid=True, strictly_convex=False) - sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) - sage: K_W_star = _rho(K, J).dual() - sage: K_star_W = _rho(K.dual(), J) - sage: _basically_the_same(K_W_star, K_star_W) + sage: K = random_cone(max_ambient_dim=8) + sage: (P,S) = motzkin_decomposition(K) + sage: P.is_strictly_convex() True - - :: - - sage: set_random_seed() - sage: J = random_cone(max_dim = 8, solid=False, strictly_convex=True) - sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) - sage: K_W_star = _rho(K, J).dual() - sage: K_star_W = _rho(K.dual(), J) - sage: _basically_the_same(K_W_star, K_star_W) + sage: S.lineality() == S.dim() True - :: + The generators of the strictly convex component are obtained from + the orthogonal projections of the original generators onto the + orthogonal complement of the subspace component:: sage: set_random_seed() - sage: J = random_cone(max_dim = 8, solid=True, strictly_convex=True) - sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) - sage: K_W_star = _rho(K, J).dual() - sage: K_star_W = _rho(K.dual(), J) - sage: _basically_the_same(K_W_star, K_star_W) + sage: K = random_cone(max_ambient_dim=8) + sage: (P,S) = motzkin_decomposition(K) + sage: S_perp = S.linear_subspace().complement() + sage: A = S_perp.matrix().transpose() + sage: proj = A * (A.transpose()*A).inverse() * A.transpose() + sage: expected = Cone([ proj*g for g in K ], K.lattice()) + sage: P.is_equivalent(expected) True - """ - if K2 is None: - K2 = K - - # First we project K onto the span of K2. This will explode if the - # rank of ``K2.lattice()`` doesn't match ours. - span_K2 = Cone(K2.rays() + (-K2).rays(), lattice=K.lattice()) - K = K.intersection(span_K2) + linspace_gens = [ copy(b) for b in K.linear_subspace().basis() ] + linspace_gens += [ -b for b in linspace_gens ] - # Cheat a little to get the subspace span(K2). The paper uses the - # rays of K2 as a basis, but everything is invariant under linear - # isomorphism (i.e. a change of basis), and this is a little - # faster. - W = span_K2.linear_subspace() + S = Cone(linspace_gens, K.lattice()) - # We've already intersected K with the span of K2, so every - # generator of K should belong to W now. - W_rays = [ W.coordinate_vector(r) for r in K.rays() ] + # Since ``S`` is a subspace, its dual is its orthogonal complement + # (albeit in the wrong lattice). + S_perp = Cone(S.dual(), K.lattice()) + P = K.intersection(S_perp) - L = ToricLattice(K2.dim()) - return Cone(W_rays, lattice=L) + return (P,S) - - -def discrete_complementarity_set(K): +def positive_operator_gens(K): r""" - Compute the discrete complementarity set of this cone. - - The complementarity set of a cone is the set of all orthogonal pairs - `(x,s)` such that `x` is in the cone, and `s` is in its dual. The - discrete complementarity set is a subset of the complementarity set - where `x` and `s` are required to be generators of their respective - cones. - - For polyhedral cones, the discrete complementarity set is always - finite. + Compute generators of the cone of positive operators on this cone. OUTPUT: - A list of pairs `(x,s)` such that, - - * Both `x` and `s` are vectors (not rays). - * `x` is a generator of this cone. - * `s` is a generator of this cone's dual. - * `x` and `s` are orthogonal. - - REFERENCES: - - .. [Orlitzky/Gowda] M. Orlitzky and M. S. Gowda. The Lyapunov Rank of an - Improper Cone. Work in-progress. + 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 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) - [] - - Likewise when this cone is trivial (its dual is the entire space):: + The trivial cone in a trivial space has no positive operators:: - sage: L = ToricLattice(0) sage: K = Cone([], ToricLattice(0)) - sage: discrete_complementarity_set(K) + sage: positive_operator_gens(K) [] - TESTS: - - The complementarity set of the dual can be obtained by switching the - components of the complementarity set of the original cone:: - - sage: set_random_seed() - sage: K1 = random_cone(max_dim=6) - sage: K2 = K1.dual() - sage: expected = [(x,s) for (s,x) in discrete_complementarity_set(K2)] - sage: actual = discrete_complementarity_set(K1) - sage: sorted(actual) == sorted(expected) - True - - The pairs in the discrete complementarity set are in fact - complementary:: - - sage: set_random_seed() - sage: K = random_cone(max_dim=6) - sage: dcs = discrete_complementarity_set(K) - sage: sum([x.inner_product(s).abs() for (x,s) in dcs]) - 0 - - """ - V = K.lattice().vector_space() - - # Convert rays to vectors so that we can compute inner products. - xs = [V(x) for x in K.rays()] - - # We also convert the generators of the dual cone so that we - # return pairs of vectors and not (vector, ray) pairs. - ss = [V(s) for s in K.dual().rays()] - - return [(x,s) for x in xs for s in ss if x.inner_product(s) == 0] - - -def LL(K): - r""" - Compute the space `\mathbf{LL}` of all Lyapunov-like transformations - on this cone. - - OUTPUT: - - A list of matrices forming a basis for the space of all - Lyapunov-like transformations on the given cone. - - EXAMPLES: - - The trivial cone has no Lyapunov-like transformations:: - - sage: L = ToricLattice(0) - sage: K = Cone([], lattice=L) - sage: LL(K) - [] - - The Lyapunov-like transformations on the nonnegative orthant are - simply diagonal matrices:: + Positive operators on the nonnegative orthant are nonnegative matrices:: sage: K = Cone([(1,)]) - sage: LL(K) + sage: positive_operator_gens(K) [[1]] sage: K = Cone([(1,0),(0,1)]) - sage: LL(K) + sage: positive_operator_gens(K) [ - [1 0] [0 0] - [0 0], [0 1] + [1 0] [0 1] [0 0] [0 0] + [0 0], [0 0], [1 0], [0 1] ] - sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)]) - sage: LL(K) - [ - [1 0 0] [0 0 0] [0 0 0] - [0 0 0] [0 1 0] [0 0 0] - [0 0 0], [0 0 0], [0 0 1] - ] + Every operator is positive on the ambient vector space:: - Only the identity matrix is Lyapunov-like on the `L^{3}_{1}` and - `L^{3}_{\infty}` cones [Rudolf et al.]_:: - - sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)]) - sage: LL(L31) - [ - [1 0 0] - [0 1 0] - [0 0 1] - ] + sage: K = Cone([(1,),(-1,)]) + sage: K.is_full_space() + True + sage: positive_operator_gens(K) + [[1], [-1]] - sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)]) - sage: LL(L3infty) + sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) + sage: K.is_full_space() + True + sage: positive_operator_gens(K) [ - [1 0 0] - [0 1 0] - [0 0 1] + [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] ] - If our cone is the entire space, then every transformation on it is - Lyapunov-like:: - - sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)]) - sage: M = MatrixSpace(QQ,2) - sage: M.basis() == LL(K) - True - TESTS: - The inner product `\left< L\left(x\right), s \right>` is zero for - every pair `\left( x,s \right)` in the discrete complementarity set - of the cone:: + A positive operator on a cone should send its generators into the cone:: sage: set_random_seed() - sage: K = random_cone(max_dim=8) - sage: C_of_K = discrete_complementarity_set(K) - sage: l = [ (L*x).inner_product(s) for (x,s) in C_of_K for L in LL(K) ] - sage: sum(map(abs, l)) - 0 - - The Lyapunov-like transformations on a cone and its dual are related - by transposition, but we're not guaranteed to compute transposed - elements of `LL\left( K \right)` as our basis for `LL\left( K^{*} - \right)` - - sage: set_random_seed() - sage: K = random_cone(max_dim=8) - sage: LL2 = [ L.transpose() for L in LL(K.dual()) ] - sage: V = VectorSpace( K.lattice().base_field(), K.lattice_dim()^2) - sage: LL1_vecs = [ V(m.list()) for m in LL(K) ] - sage: LL2_vecs = [ V(m.list()) for m in LL2 ] - sage: V.span(LL1_vecs) == V.span(LL2_vecs) - True - - """ - V = K.lattice().vector_space() - - C_of_K = discrete_complementarity_set(K) - - tensor_products = [ s.tensor_product(x) for (x,s) in C_of_K ] - - # Sage doesn't think matrices are vectors, so we have to convert - # our matrices to vectors explicitly before we can figure out how - # many are linearly-indepenedent. - # - # The space W has the same base ring as V, but dimension - # dim(V)^2. So it has the same dimension as the space of linear - # transformations on V. In other words, it's just the right size - # to create an isomorphism between it and our matrices. - W = VectorSpace(V.base_ring(), V.dimension()**2) - - # Turn our matrices into long vectors... - vectors = [ W(m.list()) for m in tensor_products ] - - # Vector space representation of Lyapunov-like matrices - # (i.e. vec(L) where L is Luapunov-like). - LL_vector = W.span(vectors).complement() - - # Now construct an ambient MatrixSpace in which to stick our - # transformations. - M = MatrixSpace(V.base_ring(), V.dimension()) - - matrix_basis = [ M(v.list()) for v in LL_vector.basis() ] - - return matrix_basis - - - -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: - - .. [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()) + sage: K = random_cone(max_ambient_dim=5) + sage: pi_of_K = positive_operator_gens(K) + sage: all([K.contains(p*x) for p in pi_of_K for x in K.rays()]) True - TESTS: - - The Lyapunov rank should be additive on a product of proper cones - [Rudolf et al.]_:: + The dimension of the cone of positive operators is given by the + corollary in my paper:: sage: set_random_seed() - sage: K1 = random_cone(max_dim=8, strictly_convex=True, solid=True) - sage: K2 = random_cone(max_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_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) + sage: K = random_cone(max_ambient_dim=5) + 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: actual = Cone([p.list() for p in pi_of_K], lattice=L).dim() + sage: expected = n**2 - l*(m - l) - (n - m)*m + sage: actual == expected True - Just to be sure, test a few more:: + The lineality of the cone of positive operators is given by the + corollary in my paper:: - sage: K1 = random_cone(max_dim=8, strictly_convex=True, solid=True) - 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) + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=5) + sage: n = K.lattice_dim() + sage: pi_of_K = positive_operator_gens(K) + sage: L = ToricLattice(n**2) + sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).lineality() + sage: expected = n**2 - K.dim()*K.dual().dim() + sage: actual == expected True - :: + The cone ``K`` is proper if and only if the cone of positive + operators on ``K`` is proper:: - sage: K1 = random_cone(max_dim=8, strictly_convex=True, solid=False) - 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) + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=5) + 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) + sage: K.is_proper() == pi_cone.is_proper() 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() - :: + tensor_products = [ s.tensor_product(x) for x in K for s in K.dual() ] - sage: K1 = random_cone(max_dim=8, strictly_convex=False, solid=True) - 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 + # Convert those tensor products to long vectors. + W = VectorSpace(F, n**2) + vectors = [ W(tp.list()) for tp in tensor_products ] - :: + # Create the *dual* cone of the positive operators, expressed as + # long vectors.. + pi_dual = Cone(vectors, ToricLattice(W.dimension())) - sage: K1 = random_cone(max_dim=8, strictly_convex=False, solid=False) - 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 + # Now compute the desired cone from its dual... + pi_cone = pi_dual.dual() - The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` - itself [Rudolf et al.]_:: + # And finally convert its rays back to matrix representations. + M = MatrixSpace(F, n) + return [ M(v.list()) for v in pi_cone.rays() ] - sage: set_random_seed() - sage: K = random_cone(max_dim=8) - sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) - True - Make sure we exercise the non-strictly-convex/non-solid case:: +def Z_transformation_gens(K): + r""" + Compute generators of the cone of Z-transformations on this cone. - sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=False, solid=False) - sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) - True + OUTPUT: - Let's check the other permutations as well, just to be sure:: + 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: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=False, solid=True) - sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) - True + EXAMPLES: - :: + Z-transformations on the nonnegative orthant are just Z-matrices. + That is, matrices whose off-diagonal elements are nonnegative:: - sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=True, solid=False) - sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) + sage: K = Cone([(1,0),(0,1)]) + sage: Z_transformation_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_transformation_gens(K) + ....: for i in range(z.nrows()) + ....: for j in range(z.ncols()) + ....: if i != j ]) True - :: + The trivial cone in a trivial space has no Z-transformations:: - sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True) - sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) - True + sage: K = Cone([], ToricLattice(0)) + sage: Z_transformation_gens(K) + [] - 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:: + Z-transformations on a subspace are Lyapunov-like and vice-versa:: - sage: set_random_seed() - sage: K = random_cone(max_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 + sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) + sage: K.is_full_space() 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_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_dim=8) - sage: actual = lyapunov_rank(K) - sage: K_S = _rho(K) - sage: K_SP = _rho(K_S.dual()).dual() - sage: l = K.lineality() - sage: c = K.codim() - sage: expected = lyapunov_rank(K_SP) + K.dim()*(l + c) + c**2 - sage: actual == expected + sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ]) + sage: zs = span([ vector(z.list()) for z in Z_transformation_gens(K) ]) + sage: zs == lls True - The Lyapunov rank of a proper cone is just the dimension of ``LL(K)``:: - - sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True) - sage: lyapunov_rank(K) == len(LL(K)) - True + TESTS: - In fact the same can be said of any cone. These additional tests - just increase our confidence that the reduction scheme works:: + The Z-property is possessed by every Z-transformation:: sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=True, solid=False) - sage: lyapunov_rank(K) == len(LL(K)) + sage: K = random_cone(max_ambient_dim=6) + sage: Z_of_K = Z_transformation_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 - :: + The lineality space of Z is LL:: sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=False, solid=True) - sage: lyapunov_rank(K) == len(LL(K)) + 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_transformation_gens(K) ]) + sage: z_cone.linear_subspace() == lls True - :: + And thus, the lineality of Z is the Lyapunov rank:: sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=False, solid=False) - sage: lyapunov_rank(K) == len(LL(K)) + sage: K = random_cone(max_ambient_dim=6) + sage: Z_of_K = Z_transformation_gens(K) + sage: L = ToricLattice(K.lattice_dim()**2) + sage: z_cone = Cone([ z.list() for z in Z_of_K ], lattice=L) + sage: z_cone.lineality() == K.lyapunov_rank() True - Test Theorem 3 in [Orlitzky/Gowda]_:: + The lineality spaces of pi-star and Z-star are equal: sage: set_random_seed() - sage: K = random_cone(max_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() + sage: K = random_cone(max_ambient_dim=5) + sage: pi_of_K = positive_operator_gens(K) + sage: Z_of_K = Z_transformation_gens(K) + sage: L = ToricLattice(K.lattice_dim()**2) + sage: pi_star = Cone([p.list() for p in pi_of_K], lattice=L).dual() + sage: z_star = Cone([ z.list() for z in Z_of_K], lattice=L).dual() + sage: pi_star.linear_subspace() == z_star.linear_subspace() True - """ - beta = 0 - - m = K.dim() + # 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() - l = K.lineality() - - if m < n: - # K is not solid, restrict to its span. - K = _rho(K) - # Lemma 2 - beta += m*(n - m) + (n - m)**2 + # These tensor products contain generators for the dual cone of + # the cross-positive transformations. + tensor_products = [ s.tensor_product(x) + for (x,s) in K.discrete_complementarity_set() ] - 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 = _rho(K, K.dual()) - - # Lemma 3 - beta += m * l + # Turn our matrices into long vectors... + W = VectorSpace(F, n**2) + vectors = [ W(m.list()) for m in tensor_products ] - beta += len(LL(K)) - return beta + # Create the *dual* cone of the cross-positive operators, + # expressed as long vectors.. + Sigma_dual = Cone(vectors, lattice=ToricLattice(W.dimension())) + + # Now compute the desired cone from its dual... + Sigma_cone = Sigma_dual.dual() + + # And finally convert its rays back to matrix representations. + # But first, make them negative, so we get Z-transformations and + # not cross-positive ones. + M = MatrixSpace(F, n) + return [ -M(v.list()) for v in Sigma_cone.rays() ] + + +def Z_cone(K): + gens = Z_transformation_gens(K) + L = None + if len(gens) == 0: + L = ToricLattice(0) + return Cone([ g.list() for g in gens ], lattice=L) + +def pi_cone(K): + gens = positive_operator_gens(K) + L = None + if len(gens) == 0: + L = ToricLattice(0) + return Cone([ g.list() for g in gens ], lattice=L)