from sage.all import *
-# TODO: This test fails, maybe due to a bug in the existing cone code.
-# If we request enough generators to span the space, then the returned
-# cone should equal the ambient space::
-#
-# sage: K = random_cone(min_dim=5, max_dim=5, min_rays=10, max_rays=10)
-# sage: K.lines().dimension() == K.lattice_dim()
-# True
-
-def random_cone(min_dim=0, max_dim=None, min_rays=0, max_rays=None):
- r"""
- Generate a random rational convex polyhedral cone.
-
- Lower and upper bounds may be provided for both the dimension of the
- ambient space and the number of generating rays of the cone. If a
- lower bound is left unspecified, it defaults to zero. Unspecified
- upper bounds will be chosen randomly.
-
- The number of generating rays is naturally limited to twice the
- dimension of the ambient space. Take for example $\mathbb{R}^{2}$.
- You could have the generators $\left\{ \pm e_{1}, \pm e_{2}
- \right\}$, with cardinality $4 = 2 \cdot 2$; however any other ray
- in the space is a nonnegative linear combination of those four.
-
- .. NOTE:
-
- If you do not explicitly request more than ``2 * max_dim`` rays,
- a larger number may still be randomly generated. In that case,
- the returned cone will simply be equal to the entire space.
-
- INPUT:
-
- - ``min_dim`` (default: zero) -- A nonnegative integer representing the
- minimum dimension of the ambient lattice.
-
- - ``max_dim`` (default: random) -- A nonnegative integer representing
- the maximum dimension of the ambient
- lattice.
-
- - ``min_rays`` (default: zero) -- A nonnegative integer representing the
- minimum number of generating rays of the
- cone.
-
- - ``max_rays`` (default: random) -- A nonnegative integer representing the
- maximum number of generating rays of
- the cone.
-
- OUTPUT:
-
- A new, randomly generated cone.
-
- A ``ValueError` will be thrown under the following conditions:
-
- * Any of ``min_dim``, ``max_dim``, ``min_rays``, or ``max_rays``
- are negative.
-
- * ``max_dim`` is less than ``min_dim``.
-
- * ``max_rays`` is less than ``min_rays``.
-
- * ``min_rays`` is greater than twice ``max_dim``.
-
- EXAMPLES:
-
- If we set the lower/upper bounds to zero, then our result is
- predictable::
-
- sage: random_cone(0,0,0,0)
- 0-d cone in 0-d lattice N
-
- We can predict the dimension when ``min_dim == max_dim``::
-
- sage: random_cone(min_dim=4, max_dim=4, min_rays=0, max_rays=0)
- 0-d cone in 4-d lattice N
-
- Likewise for the number of rays when ``min_rays == max_rays``::
-
- sage: random_cone(min_dim=10, max_dim=10, min_rays=10, max_rays=10)
- 10-d cone in 10-d lattice N
- TESTS:
-
- It's hard to test the output of a random process, but we can at
- least make sure that we get a cone back::
-
- sage: from sage.geometry.cone import is_Cone # long time
- sage: K = random_cone() # long time
- sage: is_Cone(K) # long time
- True
-
- The upper/lower bounds are respected::
+def project_span(K, K2 = None):
+ r"""
+ Return a "copy" of ``K`` embeded in a lower-dimensional space.
- sage: K = random_cone(min_dim=5, max_dim=10, min_rays=3, max_rays=4)
- sage: 5 <= K.lattice_dim() and K.lattice_dim() <= 10
- True
- sage: 3 <= K.nrays() and K.nrays() <= 4
- True
+ By default, we will project ``K`` into the subspace spanned by its
+ rays. However, if ``K2`` is not ``None``, we will project into the
+ space spanned by the rays of ``K2`` instead.
- Ensure that an exception is raised when either lower bound is greater
- than its respective upper bound::
+ EXAMPLES::
- sage: random_cone(min_dim=5, max_dim=2)
- Traceback (most recent call last):
- ...
- ValueError: max_dim cannot be less than min_dim.
+ sage: K = Cone([(1,0,0), (0,1,0)])
+ sage: project_span(K)
+ 2-d cone in 2-d lattice N
+ sage: project_span(K).rays()
+ N(1, 0),
+ N(0, 1)
+ in 2-d lattice N
- sage: random_cone(min_rays=5, max_rays=2)
- Traceback (most recent call last):
- ...
- ValueError: max_rays cannot be less than min_rays.
+ sage: K = Cone([(1,0,0), (0,1,0)])
+ sage: K2 = Cone([(0,1)])
+ sage: project_span(K, K2).rays()
+ N(1)
+ in 1-d lattice N
- And if we request too many rays::
+ """
+ # Allow us to use a second cone to generate the subspace into
+ # which we're "projecting."
+ if K2 is None:
+ K2 = K
- sage: random_cone(min_rays=5, max_dim=1)
- Traceback (most recent call last):
- ...
- ValueError: min_rays cannot be larger than twice max_dim.
+ # Use these to generate the new cone.
+ cs1 = K.rays().matrix().columns()
- """
+ # And use these to figure out which indices to drop.
+ cs2 = K2.rays().matrix().columns()
- # Catch obvious mistakes so that we can generate clear error
- # messages.
-
- if min_dim < 0:
- raise ValueError('min_dim must be nonnegative.')
-
- if min_rays < 0:
- raise ValueError('min_rays must be nonnegative.')
-
- if max_dim is not None:
- if max_dim < 0:
- raise ValueError('max_dim must be nonnegative.')
- if (max_dim < min_dim):
- raise ValueError('max_dim cannot be less than min_dim.')
- if min_rays > 2*max_dim:
- raise ValueError('min_rays cannot be larger than twice max_dim.')
-
- if max_rays is not None:
- if max_rays < 0:
- raise ValueError('max_rays must be nonnegative.')
- if (max_rays < min_rays):
- raise ValueError('max_rays cannot be less than min_rays.')
-
-
- def random_min_max(l,u):
- r"""
- We need to handle two cases for the upper bounds, and we need to do
- the same thing for max_dim/max_rays. So we consolidate the logic here.
- """
- if u is None:
- # The upper bound is unspecified; return a random integer
- # in [l,infinity).
- return l + ZZ.random_element().abs()
- else:
- # We have an upper bound, and it's greater than or equal
- # to our lower bound. So we generate a random integer in
- # [0,u-l], and then add it to l to get something in
- # [l,u]. To understand the "+1", check the
- # ZZ.random_element() docs.
- return l + ZZ.random_element(u - l + 1)
-
-
- d = random_min_max(min_dim, max_dim)
- r = random_min_max(min_rays, max_rays)
-
- L = ToricLattice(d)
-
- # The rays are trickier to generate, since we could generate v and
- # 2*v as our "two rays." In that case, the resuting cone would
- # have one generating ray. To avoid such a situation, we start by
- # generating ``r`` rays where ``r`` is the number we want to end
- # up with.
- #
- # However, since we're going to *check* whether or not we actually
- # have ``r``, we need ``r`` rays to be attainable. So we need to
- # limit ``r`` to twice the dimension of the ambient space.
- #
- r = min(r, 2*d)
- rays = [L.random_element() for i in range(0, r)]
+ perp_idxs = []
- # (The lattice parameter is required when no rays are given, so we
- # pass it just in case ``r == 0``).
- K = Cone(rays, lattice=L)
+ for idx in range(0, len(cs2)):
+ if cs2[idx].is_zero():
+ perp_idxs.append(idx)
- # Now if we generated two of the "same" rays, we'll have fewer
- # generating rays than ``r``. In that case, we keep making up new
- # rays and recreating the cone until we get the right number of
- # independent generators.
- while r > K.nrays():
- rays.append(L.random_element())
- K = Cone(rays)
+ solid_cols = [ cs1[idx] for idx in range(0,len(cs1))
+ if not idx in perp_idxs
+ and not idx >= len(cs2) ]
- return K
+ m = matrix(solid_cols)
+ L = ToricLattice(len(m.rows()))
+ J = Cone(m.transpose(), lattice=L)
+ return J
def discrete_complementarity_set(K):
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::
+
+ 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]
+ ]
+
+ 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: K = random_cone(max_dim=8, max_rays=10)
+ 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
+
+ """
+ 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.
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).
+ not possible for any cone.
.. note::
REFERENCES:
- 1. M.S. Gowda and J. Tao. On the bilinearity rank of a proper cone
- and Lyapunov-like transformations, Mathematical Programming, 147
+ .. [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.
- 2. G. Rudolf, N. Noyan, D. Papp, and F. Alizadeh, Bilinear
+ .. [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`::
+ The nonnegative orthant in `\mathbb{R}^{n}` always has rank `n`
+ [Rudolf et al.]_::
sage: positives = Cone([(1,)])
sage: lyapunov_rank(positives)
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: octant = Cone([(1,0,0), (0,1,0), (0,0,1)])
sage: lyapunov_rank(octant)
3
- The `L^{3}_{1}` cone is known to have a Lyapunov rank of one::
+ 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::
+ 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
- The Lyapunov rank should be additive on a product of cones::
+ The Lyapunov rank should be additive on a product of 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: lyapunov_rank(K) == lyapunov_rank(L31) + lyapunov_rank(octant)
True
- Two isomorphic cones should have the same Lyapunov rank. The cone
- ``K`` in the following example is isomorphic to the nonnegative
+ 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)])
3
The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
- itself::
+ itself [Rudolf et al.]_::
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 cones::
+ The Lyapunov rank should be additive on a product of cones
+ [Rudolf et al.]_::
sage: K1 = random_cone(max_dim=10, max_rays=10)
sage: K2 = random_cone(max_dim=10, max_rays=10)
True
The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
- itself::
+ itself [Rudolf et al.]_::
sage: K = random_cone(max_dim=10, max_rays=10)
sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
True
- """
- V = K.lattice().vector_space()
-
- C_of_K = discrete_complementarity_set(K)
-
- matrices = [x.tensor_product(s) 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)
-
- def phi(m):
- r"""
- Convert a matrix to a vector isomorphically.
- """
- return W(m.list())
+ 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::
- vectors = [phi(m) for m in matrices]
+ sage: K = random_cone(max_dim=10, strictly_convex=True, solid=True)
+ sage: b = lyapunov_rank(K)
+ sage: n = K.lattice_dim()
+ sage: (n == 0 or 1 <= b) and b <= n
+ 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: K = random_cone(max_dim=10, max_rays=16)
+ 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: K = random_cone(max_dim=15, max_rays=25)
+ sage: actual = lyapunov_rank(K)
+ sage: K_S = project_span(K)
+ sage: J_T1 = project_span(K_S.dual()).dual()
+ sage: J_T2 = project_span(K, K_S.dual())
+ sage: J_T2 = Cone(J_T2.rays(), lattice=J_T1.lattice())
+ sage: J_T1 == J_T2
+ True
+ sage: J_T = J_T1
+ sage: l = K.linear_subspace().dimension()
+ sage: codim = K.lattice_dim() - K.dim()
+ sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2
+ sage: actual == expected
+ True
- return (W.dimension() - W.span(vectors).rank())
+ """
+ return len(LL(K))