-# 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 *
-# 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::
-
- 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
-
- Ensure that an exception is raised when either lower bound is greater
- than its respective upper bound::
-
- sage: random_cone(min_dim=5, max_dim=2)
- Traceback (most recent call last):
- ...
- ValueError: max_dim cannot be less than min_dim.
-
- sage: random_cone(min_rays=5, max_rays=2)
- Traceback (most recent call last):
- ...
- ValueError: max_rays cannot be less than min_rays.
-
- And if we request too many rays::
-
- sage: random_cone(min_rays=5, max_dim=1)
- Traceback (most recent call last):
- ...
- ValueError: min_rays cannot be larger than twice max_dim.
-
- """
-
- # 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)]
-
- # (The lattice parameter is required when no rays are given, so we
- # pass it just in case ``r == 0``).
- K = Cone(rays, lattice=L)
-
- # 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)
-
- return K
-
-
-def discrete_complementarity_set(K):
- r"""
- Compute the discrete complementarity set of this cone.
-
- The complementarity set of this cone is the set of all orthogonal
- pairs `(x,s)` such that `x` is in this cone, and `s` is in its
- dual. The discrete complementarity set restricts `x` and `s` to be
- generators of their respective cones.
-
- OUTPUT:
-
- A list of pairs `(x,s)` such that,
-
- * `x` is in this cone.
- * `x` is a generator of this cone.
- * `s` is in this cone's dual.
- * `s` is a generator of this cone's dual.
- * `x` and `s` are orthogonal.
-
- EXAMPLES:
-
- The discrete complementarity set of the nonnegative orthant consists
- of pairs of standard basis vectors::
-
- sage: K = Cone([(1,0),(0,1)])
- sage: discrete_complementarity_set(K)
- [((1, 0), (0, 1)), ((0, 1), (1, 0))]
-
- If the cone consists of a single ray, the second components of the
- discrete complementarity set should generate the orthogonal
- complement of that ray::
-
- sage: K = Cone([(1,0)])
- sage: discrete_complementarity_set(K)
- [((1, 0), (0, 1)), ((1, 0), (0, -1))]
- sage: K = Cone([(1,0,0)])
- sage: discrete_complementarity_set(K)
- [((1, 0, 0), (0, 1, 0)),
- ((1, 0, 0), (0, -1, 0)),
- ((1, 0, 0), (0, 0, 1)),
- ((1, 0, 0), (0, 0, -1))]
-
- When the cone is the entire space, its dual is the trivial cone, so
- the discrete complementarity set is empty::
-
- sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
- sage: discrete_complementarity_set(K)
- []
-
- TESTS:
-
- The complementarity set of the dual can be obtained by switching the
- components of the complementarity set of the original cone::
-
- sage: K1 = random_cone(max_dim=10, max_rays=10)
- sage: K2 = K1.dual()
- sage: expected = [(x,s) for (s,x) in discrete_complementarity_set(K2)]
- sage: actual = discrete_complementarity_set(K1)
- sage: actual == expected
- True
-
- """
- V = K.lattice().vector_space()
-
- # Convert the rays to vectors so that we can compute inner
- # products.
- xs = [V(x) for x in K.rays()]
- ss = [V(s) for s in K.dual().rays()]
-
- return [(x,s) for x in xs for s in ss if x.inner_product(s) == 0]
-
-
-def lyapunov_rank(K):
- r"""
- Compute the Lyapunov (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:
-
- A closed, convex polyhedral 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 the first reference).
-
- .. note::
-
- In the references, the cones are always assumed to be proper. We
- do not impose this restriction.
-
- .. seealso::
-
- :meth:`is_proper`
-
- ALGORITHM:
-
- The codimension formula from the second reference is used. We find
- all pairs `(x,s)` in the complementarity set of `K` such that `x`
- and `s` are rays of our cone. It is known that these vectors are
- sufficient to apply the codimension formula. Once we have all such
- pairs, we "brute force" the codimension formula by finding all
- linearly-independent `xs^{T}`.
-
- REFERENCES:
-
- 1. 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
- 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`::
-
- sage: positives = Cone([(1,)])
- sage: lyapunov_rank(positives)
- 1
- 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: lyapunov_rank(octant)
- 3
-
- The `L^{3}_{1}` cone is known to have a Lyapunov rank of one::
-
- 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::
-
- 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::
-
- 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: K = L31.cartesian_product(octant)
- 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
- octant in `\mathbb{R}^{3}`::
-
- sage: K = Cone([(1,2,3), (-1,1,0), (1,0,6)])
- sage: lyapunov_rank(K)
- 3
-
- The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
- itself::
-
- sage: K = Cone([(2,2,4), (-1,9,0), (2,0,6)])
- sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
- True
-
- TESTS:
-
- The Lyapunov rank should be additive on a product of cones::
-
- sage: K1 = random_cone(max_dim=10, max_rays=10)
- sage: K2 = random_cone(max_dim=10, max_rays=10)
- sage: K = K1.cartesian_product(K2)
- sage: lyapunov_rank(K) == lyapunov_rank(K1) + lyapunov_rank(K2)
- True
-
- The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
- itself::
-
- 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())
-
- vectors = [phi(m) for m in matrices]
-
- return (W.dimension() - W.span(vectors).rank())
+def LL_cone(K):
+ gens = K.lyapunov_like_basis()
+ L = ToricLattice(K.lattice_dim()**2)
+ return Cone(( g.list() for g in gens ), lattice=L, check=False)
+
+def Sigma_cone(K):
+ gens = K.cross_positive_operators_gens()
+ L = ToricLattice(K.lattice_dim()**2)
+ return Cone(( g.list() for g in gens ), lattice=L, check=False)
+
+def Z_cone(K):
+ gens = K.Z_operators_gens()
+ L = ToricLattice(K.lattice_dim()**2)
+ return Cone(( g.list() for g in gens ), lattice=L, check=False)
+
+def pi_cone(K1, K2=None):
+ if K2 is None:
+ K2 = K1
+ gens = K1.positive_operators_gens(K2)
+ L = ToricLattice(K1.lattice_dim()*K2.lattice_dim())
+ return Cone(( g.list() for g in gens ), lattice=L, check=False)
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