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.
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:
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)
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() ]
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)
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)
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)])
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}`::
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())
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,
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')
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)
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()
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)
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,
True
"""
- beta = 0
+ beta = 0 # running tally of the Lyapunov rank
m = K.dim()
n = K.lattice_dim()
# Non-pointed reduction lemma.
beta += l * m
- beta += len(LL(K))
+ beta += len(K.lyapunov_like_basis())
return beta
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:
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
"""
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.
# 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 ]
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() ]
projects / sage.d.git / blobdiff