X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=a1ded5270032de21f2647de8453384fde4804c72;hb=090b2c77aa4bd371d66885451f9df44c6b6d818f;hp=32e13861fdbb0cb2f806f5892cd9c90c1df051f4;hpb=3cfc8e228ae337aed975118444a8cbad9a5a7ac3;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 32e1386..a1ded52 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -1,10 +1,3 @@ -# 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 is_lyapunov_like(L,K): @@ -47,14 +40,14 @@ def is_lyapunov_like(L,K): 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: 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 As is the "zero" transformation:: - sage: K = random_cone(min_ambient_dim = 1, max_rays = 5) + 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) @@ -63,7 +56,7 @@ def is_lyapunov_like(L,K): Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like on ``K``:: - sage: K = random_cone(min_ambient_dim = 1, max_rays = 5) + 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 @@ -72,64 +65,97 @@ def is_lyapunov_like(L,K): for (x,s) in K.discrete_complementarity_set()]) -def random_element(K): +def motzkin_decomposition(K): r""" - Return a random element of ``K`` from its ambient vector space. + Return the pair of components in the motzkin decomposition of this cone. - ALGORITHM: + 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``. - 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. + OUTPUT: - 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. + 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``. EXAMPLES: - A random element of the trivial cone is zero:: + 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 = 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) + 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: S.is_trivial() + True + + Likewise, full spaces are their own subspace components:: + + sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) + sage: K.is_full_space() + True + sage: (P,S) = motzkin_decomposition(K) + sage: K.is_equivalent(S) + True + sage: P.is_trivial() + True TESTS: - Any cone should contain an element of itself:: + 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_rays = 8) - sage: K.contains(random_element(K)) + 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 + 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: K = random_cone(max_ambient_dim=8) + sage: (P,S) = motzkin_decomposition(K) + sage: P.is_strictly_convex() + True + 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: 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 """ - 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)) ] + linspace_gens = [ copy(b) for b in K.linear_subspace().basis() ] + linspace_gens += [ -b for b in linspace_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 + S = Cone(linspace_gens, K.lattice()) + # 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) -def positive_operators(K): + return (P,S) + +def positive_operator_gens(K): r""" Compute generators of the cone of positive operators on this cone. @@ -146,17 +172,17 @@ def positive_operators(K): The trivial cone in a trivial space has no positive operators:: sage: K = Cone([], ToricLattice(0)) - sage: positive_operators(K) + sage: positive_operator_gens(K) [] Positive operators on the nonnegative orthant are nonnegative matrices:: sage: K = Cone([(1,)]) - sage: positive_operators(K) + sage: positive_operator_gens(K) [[1]] sage: K = Cone([(1,0),(0,1)]) - sage: positive_operators(K) + sage: positive_operator_gens(K) [ [1 0] [0 1] [0 0] [0 0] [0 0], [0 0], [1 0], [0 1] @@ -167,13 +193,13 @@ def positive_operators(K): sage: K = Cone([(1,),(-1,)]) sage: K.is_full_space() True - sage: positive_operators(K) + sage: positive_operator_gens(K) [[1], [-1]] sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) sage: K.is_full_space() True - sage: positive_operators(K) + sage: positive_operator_gens(K) [ [1 0] [-1 0] [0 1] [ 0 -1] [0 0] [ 0 0] [0 0] [ 0 0] [0 0], [ 0 0], [0 0], [ 0 0], [1 0], [-1 0], [0 1], [ 0 -1] @@ -183,43 +209,76 @@ def positive_operators(K): 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: set_random_seed() + 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 + The dimension of the cone of positive operators is given by the + corollary in my paper:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=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 + + The lineality of the cone of positive operators is given by the + corollary in my paper:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=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: 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 """ - # 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) + # 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() ] - # Turn our matrices into long vectors... - vectors = [ W(m.list()) for m in tensor_products ] + # 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.. - L = ToricLattice(W.dimension()) - pi_dual = Cone(vectors, lattice=L) + pi_dual = Cone(vectors, ToricLattice(W.dimension())) # 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()) - + M = MatrixSpace(F, n) return [ M(v.list()) for v in pi_cone.rays() ] -def Z_transformations(K): +def Z_transformation_gens(K): r""" Compute generators of the cone of Z-transformations on this cone. @@ -237,13 +296,13 @@ def Z_transformations(K): That is, matrices whose off-diagonal elements are nonnegative:: sage: K = Cone([(1,0),(0,1)]) - sage: Z_transformations(K) + 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_transformations(K) + 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 ]) @@ -252,7 +311,7 @@ def Z_transformations(K): The trivial cone in a trivial space has no Z-transformations:: sage: K = Cone([], ToricLattice(0)) - sage: Z_transformations(K) + sage: Z_transformation_gens(K) [] Z-transformations on a subspace are Lyapunov-like and vice-versa:: @@ -261,7 +320,7 @@ def Z_transformations(K): 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 = span([ vector(z.list()) for z in Z_transformation_gens(K) ]) sage: zs == lls True @@ -270,8 +329,8 @@ def Z_transformations(K): The Z-property is possessed by every Z-transformation:: sage: set_random_seed() - sage: K = random_cone(max_ambient_dim = 6) - sage: Z_of_K = Z_transformations(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]) @@ -280,34 +339,52 @@ def Z_transformations(K): The lineality space of Z is LL:: sage: set_random_seed() - sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 6) + 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 = 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_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 + + The lineality spaces of pi-star and Z-star are equal: + + sage: set_random_seed() + 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 """ - # 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) - - C_of_K = K.discrete_complementarity_set() - tensor_products = [ s.tensor_product(x) for (x,s) in C_of_K ] + # Matrices are not vectors in Sage, so we have to convert them + # to vectors explicitly before we can find a basis. We need these + # two values to construct the appropriate "long vector" space. + F = K.lattice().base_field() + n = K.lattice_dim() + + # These tensor products contain generators for the dual cone of + # the cross-positive transformations. + tensor_products = [ s.tensor_product(x) + for (x,s) in K.discrete_complementarity_set() ] # Turn our matrices into long vectors... + W = VectorSpace(F, n**2) vectors = [ W(m.list()) for m in tensor_products ] # Create the *dual* cone of the cross-positive operators, # expressed as long vectors.. - L = ToricLattice(W.dimension()) - Sigma_dual = Cone(vectors, lattice=L) + Sigma_dual = Cone(vectors, lattice=ToricLattice(W.dimension())) # Now compute the desired cone from its dual... Sigma_cone = Sigma_dual.dual() @@ -315,6 +392,20 @@ def Z_transformations(K): # 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()) - + 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)