X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=5f058ca9a8c055f972ffffaa216975517e0dc723;hb=9bab3c89cc7d669e7b99295900c9f590e5525079;hp=f8b879131908a8d1a15d5069fcc888c5db319b07;hpb=c751dadc58e972026f9bde474909fbe31ff2b0bb;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index f8b8791..5f058ca 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -8,89 +8,6 @@ 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(LL(K1)) != len(LL(K2)): - 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. @@ -211,156 +128,23 @@ def _restrict_to_space(K, W): return Cone(K_W_rays, lattice=L) -def LL(K): - r""" - Compute a basis of 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:: - - sage: K = Cone([(1,)]) - sage: LL(K) - [[1]] - - sage: K = Cone([(1,0),(0,1)]) - sage: LL(K) - [ - [1 0] [0 0] - [0 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] - ] - - 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: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)]) - sage: LL(L3infty) - [ - [1 0 0] - [0 1 0] - [0 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:: - - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=8) - sage: C_of_K = K.discrete_complementarity_set() - 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_ambient_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 = K.discrete_complementarity_set() - - 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 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: + Compute the Lyapunov rank of this cone. - 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: @@ -373,21 +157,21 @@ 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) @@ -400,7 +184,7 @@ 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() ] @@ -409,20 +193,20 @@ 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) @@ -431,7 +215,7 @@ 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) @@ -445,7 +229,7 @@ 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)]) @@ -453,7 +237,7 @@ 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}`:: @@ -462,7 +246,7 @@ 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()) @@ -471,7 +255,7 @@ 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, @@ -485,7 +269,7 @@ 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') @@ -494,7 +278,7 @@ 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) @@ -518,7 +302,7 @@ 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() @@ -529,7 +313,7 @@ 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) @@ -542,15 +326,15 @@ def lyapunov_rank(K): sage: actual == expected True - The Lyapunov rank of any cone is just the dimension of ``LL(K)``:: + 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(LL(K)) + 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, @@ -562,7 +346,7 @@ def lyapunov_rank(K): True """ - beta = 0 + beta = 0 # running tally of the Lyapunov rank m = K.dim() n = K.lattice_dim() @@ -584,7 +368,7 @@ def lyapunov_rank(K): # Non-pointed reduction lemma. beta += l * m - beta += len(LL(K)) + beta += len(K.lyapunov_like_basis()) return beta @@ -621,8 +405,8 @@ 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: @@ -642,10 +426,11 @@ def is_lyapunov_like(L,K): sage: is_lyapunov_like(L,K) True - Everything in ``LL(K)`` 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 LL(K)]) + sage: all([ is_lyapunov_like(L,K) for L in K.lyapunov_like_basis() ]) True """ @@ -765,13 +550,11 @@ 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: all([K.contains(p*x) for p in pi_of_k for x in K.rays()]) + 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 """ - V = K.lattice().vector_space() - # 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. @@ -780,12 +563,10 @@ def positive_operators(K): # 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) - G1 = [ V(x) for x in K.rays() ] - G2 = [ V(s) for s in K.dual().rays() ] - - tensor_products = [ s.tensor_product(x) for x in G1 for s in G2 ] + 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 ] @@ -802,3 +583,104 @@ def positive_operators(K): M = MatrixSpace(V.base_ring(), V.dimension()) return [ M(v.list()) for v in pi_cone.rays() ] + + +def Z_transformations(K): + r""" + Compute generators of the cone of Z-transformations on this cone. + + OUTPUT: + + 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. + + 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)) + sage: Z_transformations(K) + [] + + Z-transformations on a subspace are Lyapunov-like and vice-versa:: + + sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) + 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 == 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]) + 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: 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() == lls + 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 ] + + # Turn our matrices into long vectors... + 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) + + # 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(V.base_ring(), V.dimension()) + + return [ -M(v.list()) for v in Sigma_cone.rays() ]