X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=6d7d2d9b4d2b379bc6041fe089e5a6ea38b8e48a;hb=342d9147356f7757bed2f9165a600a9e5ec0a5e2;hp=7d919e452a103433639507cef5fb5123d59685c8;hpb=5559552f7e3ab5328a42359c2b4c6b238839b060;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 7d919e4..6d7d2d9 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -8,92 +8,24 @@ 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. + Restrict this cone (up to linear isomorphism) to a vector subspace. + + This operation not only restricts the cone to a subspace of its + ambient space, but also represents the rays of the cone in a new + (smaller) lattice corresponding to the subspace. The resulting cone + will be linearly isomorphic **but not equal** to the desired + restriction, since it has likely undergone a change of basis. + + To explain the difficulty, consider the cone ``K = Cone([(1,1,1)])`` + having a single ray. The span of ``K`` is a one-dimensional subspace + containing ``K``, yet we have no way to perform operations like + :meth:`dual` in the subspace. To represent ``K`` in the space + ``K.span()``, we must perform a change of basis and write its sole + ray as ``(1,0,0)``. Now the restricted ``Cone([(1,)])`` is linearly + isomorphic (but of course not equal) to ``K`` interpreted as living + in ``K.span()``. INPUT: @@ -103,16 +35,26 @@ def _restrict_to_space(K, W): A new cone in a sublattice corresponding to ``W``. + REFERENCES: + + M. Orlitzky. The Lyapunov rank of an improper cone. + http://www.optimization-online.org/DB_HTML/2015/10/5135.html + EXAMPLES: - When this cone is solid, restricting it into its own span should do - nothing:: + Restricting a solid cone to its own span returns a cone linearly + isomorphic to the original:: - sage: K = Cone([(1,)]) - sage: _restrict_to_space(K, K.span()) == K + sage: K = Cone([(1,2,3),(-1,1,0),(9,0,-2)]) + sage: K.is_solid() True + sage: _restrict_to_space(K, K.span()).rays() + N(-1, 1, 0), + N( 1, 0, 0), + N( 9, -6, -1) + in 3-d lattice N - A single ray restricted into its own span gives the same output + A single ray restricted to its own span has the same representation regardless of the ambient space:: sage: K2 = Cone([(1,0)]) @@ -120,7 +62,7 @@ def _restrict_to_space(K, W): sage: K2_S N(1) in 1-d lattice N - sage: K3 = Cone([(1,0,0)]) + sage: K3 = Cone([(1,1,1)]) sage: K3_S = _restrict_to_space(K3, K3.span()).rays() sage: K3_S N(1) @@ -128,83 +70,116 @@ def _restrict_to_space(K, W): sage: K2_S == K3_S True + Restricting to a trivial space gives the trivial cone:: + + sage: K = Cone([(8,3,-1,0),(9,2,2,0),(-4,6,7,0)]) + sage: trivial_space = K.lattice().vector_space().span([]) + sage: _restrict_to_space(K, trivial_space) + 0-d cone in 0-d lattice N + TESTS: - The projected cone should always be solid:: + Restricting a cone to its own span results in a solid cone:: sage: set_random_seed() sage: K = random_cone(max_ambient_dim = 8) - sage: _restrict_to_space(K, K.span()).is_solid() + sage: K_S = _restrict_to_space(K, K.span()) + 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:: + Restricting a cone to its own span should not affect the number of + rays in the cone:: 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() + sage: K_S = _restrict_to_space(K, K.span()) + sage: K.nrays() == K_S.nrays() True - This function should not affect the dimension of a cone:: + Restricting a cone to its own span should not affect its dimension:: sage: set_random_seed() sage: K = random_cone(max_ambient_dim = 8) - sage: K.dim() == _restrict_to_space(K,K.span()).dim() + sage: K_S = _restrict_to_space(K, K.span()) + sage: K.dim() == K_S.dim() True - Nor should it affect the lineality of a cone:: + Restricting a cone to its own span should not affects its lineality:: sage: set_random_seed() sage: K = random_cone(max_ambient_dim = 8) - sage: K.lineality() == _restrict_to_space(K, K.span()).lineality() + sage: K_S = _restrict_to_space(K, K.span()) + sage: K.lineality() == K_S.lineality() True - No matter which space we restrict to, the lineality should not - increase:: + Restricting a cone to its own span should not affect the number of + facets it has:: 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() + sage: K_S = _restrict_to_space(K, K.span()) + sage: len(K.facets()) == len(K_S.facets()) True - sage: K.lineality() >= _restrict_to_space(K,P).lineality() + + Restricting a solid cone to its own span is a linear isomorphism and + should not affect the dimension of its ambient space:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim = 8, solid = True) + sage: K_S = _restrict_to_space(K, K.span()) + sage: K.lattice_dim() == K_S.lattice_dim() True - If we do this according to our paper, then the result is proper:: + Restricting a solid cone to its own span is a linear isomorphism + that establishes a one-to-one correspondence of discrete + complementarity sets:: sage: set_random_seed() - sage: K = random_cone(max_ambient_dim = 8) + sage: K = random_cone(max_ambient_dim = 8, solid = True) 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() + sage: dcs_K = K.discrete_complementarity_set() + sage: dcs_K_S = K_S.discrete_complementarity_set() + sage: len(dcs_K) == len(dcs_K_S) True - sage: K_SP = _restrict_to_space(K_S, K_S.dual().span()) - sage: K_SP.is_proper() + + Restricting a solid cone to its own span is a linear isomorphism + under which the Lyapunov rank (the length of a Lyapunov-like basis) + is invariant:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim = 8, solid = True) + sage: K_S = _restrict_to_space(K, K.span()) + sage: len(K.lyapunov_like_basis()) == len(K_S.lyapunov_like_basis()) 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:: + If we restrict a cone to a subspace of its span, the resulting cone + should have the same dimension as the space we restricted it to:: 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) + sage: K = random_cone(max_ambient_dim = 8) + sage: W_basis = random_sublist(K.rays(), 0.5) + sage: W = K.lattice().vector_space().span(W_basis) + sage: K_W = _restrict_to_space(K, W) + sage: K_W.lattice_dim() == W.dimension() True + Through a series of restrictions, any closed convex cone can be + reduced to a cartesian product with a proper factor [Orlitzky]_:: + + 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, K_S.dual().span()) + sage: K_SP.is_proper() + 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. + # We want to intersect ``K`` with ``W``. An easy way to do this is + # via cone intersection, so we turn the space ``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. + # We've already intersected K with W, 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()) @@ -213,26 +188,21 @@ def _restrict_to_space(K, W): 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: + Compute the Lyapunov rank of this cone. - 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. + 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: - 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]_). + 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: @@ -245,21 +215,21 @@ def lyapunov_rank(K): 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. + .. [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. + M. Orlitzky. The Lyapunov rank of an improper cone. + http://www.optimization-online.org/DB_HTML/2015/10/5135.html - .. [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. + 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.]_:: + [Rudolf]_:: sage: positives = Cone([(1,)]) sage: lyapunov_rank(positives) @@ -272,7 +242,7 @@ def lyapunov_rank(K): 3 The full space `\mathbb{R}^{n}` has Lyapunov rank `n^{2}` - [Orlitzky/Gowda]_:: + [Orlitzky]_:: sage: R5 = VectorSpace(QQ, 5) sage: gs = R5.basis() + [ -r for r in R5.basis() ] @@ -281,20 +251,20 @@ def lyapunov_rank(K): 25 The `L^{3}_{1}` cone is known to have a Lyapunov rank of one - [Rudolf et al.]_:: + [Rudolf]_:: 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.]_:: + 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/Gowda]_:: + + 1` [Orlitzky]_:: sage: K = Cone([(1,0,0,0,0)]) sage: lyapunov_rank(K) @@ -303,7 +273,7 @@ def lyapunov_rank(K): 21 A subspace (of dimension `m`) in `n` dimensions should have a - Lyapunov rank of `n^{2} - m\left(n - m)` [Orlitzky/Gowda]_:: + 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) @@ -317,7 +287,7 @@ def lyapunov_rank(K): 19 The Lyapunov rank should be additive on a product of proper cones - [Rudolf et al.]_:: + [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)]) @@ -325,7 +295,7 @@ def lyapunov_rank(K): sage: lyapunov_rank(K) == lyapunov_rank(L31) + lyapunov_rank(octant) True - Two isomorphic cones should have the same Lyapunov rank [Rudolf et al.]_. + 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}`:: @@ -334,7 +304,7 @@ def lyapunov_rank(K): 3 The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` - itself [Rudolf et al.]_:: + itself [Rudolf]_:: sage: K = Cone([(2,2,4), (-1,9,0), (2,0,6)]) sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) @@ -343,7 +313,7 @@ def lyapunov_rank(K): TESTS: The Lyapunov rank should be additive on a product of proper cones - [Rudolf et al.]_:: + [Rudolf]_:: sage: set_random_seed() sage: K1 = random_cone(max_ambient_dim=8, @@ -357,7 +327,7 @@ def lyapunov_rank(K): True The Lyapunov rank is invariant under a linear isomorphism - [Orlitzky/Gowda]_:: + [Orlitzky]_:: sage: K1 = random_cone(max_ambient_dim = 8) sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular') @@ -366,7 +336,7 @@ def lyapunov_rank(K): True The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` - itself [Rudolf et al.]_:: + itself [Rudolf]_:: sage: set_random_seed() sage: K = random_cone(max_ambient_dim=8) @@ -390,7 +360,7 @@ def lyapunov_rank(K): sage: b == n-1 False - In fact [Orlitzky/Gowda]_, no closed convex polyhedral cone can have + In fact [Orlitzky]_, no closed convex polyhedral cone can have Lyapunov rank `n-1` in `n` dimensions:: sage: set_random_seed() @@ -401,7 +371,7 @@ def lyapunov_rank(K): False The calculation of the Lyapunov rank of an improper cone can be - reduced to that of a proper cone [Orlitzky/Gowda]_:: + reduced to that of a proper cone [Orlitzky]_:: sage: set_random_seed() sage: K = random_cone(max_ambient_dim=8) @@ -414,15 +384,15 @@ def lyapunov_rank(K): sage: actual == expected True - The Lyapunov rank of any cone is just the dimension of ``K.LL()``:: + 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.LL()) + 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/Gowda]_):: + (a Theorem in [Orlitzky]_):: sage: set_random_seed() sage: K = random_cone(max_ambient_dim=8, @@ -434,7 +404,7 @@ def lyapunov_rank(K): True """ - beta = 0 + beta = 0 # running tally of the Lyapunov rank m = K.dim() n = K.lattice_dim() @@ -456,7 +426,7 @@ def lyapunov_rank(K): # Non-pointed reduction lemma. beta += l * m - beta += len(K.LL()) + beta += len(K.lyapunov_like_basis()) return beta @@ -493,8 +463,8 @@ def is_lyapunov_like(L,K): REFERENCES: - .. [Orlitzky] M. Orlitzky. The Lyapunov rank of an - improper cone (preprint). + M. Orlitzky. The Lyapunov rank of an improper cone. + http://www.optimization-online.org/DB_HTML/2015/10/5135.html EXAMPLES: @@ -514,10 +484,11 @@ def is_lyapunov_like(L,K): sage: is_lyapunov_like(L,K) True - Everything in ``K.LL()`` should be Lyapunov-like on ``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: all([is_lyapunov_like(L,K) for L in K.LL()]) + sage: all([ is_lyapunov_like(L,K) for L in K.lyapunov_like_basis() ]) True """ @@ -686,6 +657,22 @@ def Z_transformations(K): 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 + The trivial cone in a trivial space has no Z-transformations:: sage: K = Cone([], ToricLattice(0)) @@ -697,28 +684,30 @@ def Z_transformations(K): sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) sage: K.is_full_space() True - sage: llvs = span([ vector(l.list()) for l in K.LL() ]) - sage: zvs = span([ vector(z.list()) for z in Z_transformations(K) ]) - sage: zvs == llvs + 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 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: dcs = K.discrete_complementarity_set() - sage: all([z(x).inner_product(s) <= 0 for z in Z_of_K - ....: for (x,s) in dcs]) + 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(min_ambient_dim = 1, max_ambient_dim = 6) - sage: llvs = span([ vector(l.list()) for l in K.LL() ]) + 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() == llvs + sage: z_cone.linear_subspace() == lls True """ @@ -739,15 +728,17 @@ def Z_transformations(K): # 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.. + # Create the *dual* cone of the cross-positive operators, + # expressed as long vectors.. L = ToricLattice(W.dimension()) - Z_dual = Cone(vectors, lattice=L) + Sigma_dual = Cone(vectors, lattice=L) # Now compute the desired cone from its dual... - Z_cone = Z_dual.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(V.base_ring(), V.dimension()) - return [ M(v.list()) for v in Z_cone.rays() ] + return [ -M(v.list()) for v in Sigma_cone.rays() ]