from sage.all import *
-def is_full_space(K):
+def _restrict_to_space(K, W):
r"""
- Return whether or not this cone is equal to its ambient vector space.
+ Restrict this cone a subspace of its ambient space.
+
+ INPUT:
+
+ - ``W`` -- The subspace into which this cone will be restricted.
OUTPUT:
- ``True`` if this cone is the entire vector space and ``False``
- otherwise.
+ A new cone in a sublattice corresponding to ``W``.
EXAMPLES:
- A ray in two dimensions is not equal to the entire space::
+ When this cone is solid, restricting it into its own span should do
+ nothing::
- sage: K = Cone([(1,0)])
- sage: is_full_space(K)
- False
+ sage: K = Cone([(1,)])
+ sage: _restrict_to_space(K, K.span()) == K
+ True
- Neither is the nonnegative orthant::
+ A single ray restricted into its own span gives the same output
+ regardless of the ambient space::
+
+ sage: K2 = Cone([(1,0)])
+ sage: K2_S = _restrict_to_space(K2, K2.span()).rays()
+ sage: K2_S
+ N(1)
+ in 1-d lattice N
+ sage: K3 = Cone([(1,0,0)])
+ sage: K3_S = _restrict_to_space(K3, K3.span()).rays()
+ sage: K3_S
+ N(1)
+ in 1-d lattice N
+ sage: K2_S == K3_S
+ True
- sage: K = Cone([(1,0),(0,1)])
- sage: is_full_space(K)
- False
+ TESTS:
- The right half-space contains a vector subspace, but it is still not
- equal to the entire plane::
+ The projected cone should always be solid::
- sage: K = Cone([(1,0),(-1,0),(0,1)])
- sage: is_full_space(K)
- False
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim = 8)
+ sage: _restrict_to_space(K, K.span()).is_solid()
+ True
- But if we include nonnegative sums from both axes, then the resulting
- cone is the entire two-dimensional space::
+ And the resulting cone should live in a space having the same
+ dimension as the space we restricted it to::
- sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
- sage: is_full_space(K)
+ 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()
True
- """
- return K.linear_subspace() == K.lattice().vector_space()
+ This function should not affect the dimension of a cone::
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim = 8)
+ sage: K.dim() == _restrict_to_space(K,K.span()).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.
+ Nor should it affect the lineality of a 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.
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim = 8)
+ sage: K.lineality() == _restrict_to_space(K, K.span()).lineality()
+ True
- The lower bound on the number of rays is limited to twice the
- maximum dimension of the ambient vector space. To see why, consider
- the space $\mathbb{R}^{2}$, and suppose we have generated four rays,
- $\left\{ \pm e_{1}, \pm e_{2} \right\}$. Clearly any other ray in
- the space is a nonnegative linear combination of those four,
- so it is hopeless to generate more. It is therefore an error
- to request more in the form of ``min_rays``.
+ No matter which space we restrict to, the lineality should not
+ increase::
- .. NOTE:
+ 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()
+ True
+ sage: K.lineality() >= _restrict_to_space(K,P).lineality()
+ True
- 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.
+ If we do this according to our paper, then the result is proper::
- INPUT:
+ 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.dual(), K_S.dual().span()).dual()
+ sage: K_SP.is_proper()
+ True
+ sage: K_SP = _restrict_to_space(K_S, K_S.dual().span())
+ sage: K_SP.is_proper()
+ True
- - ``min_dim`` (default: zero) -- A nonnegative integer representing the
- minimum dimension of the ambient lattice.
+ 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::
+
+ 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)
+ 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.
+ W_cone = Cone(W.basis() + [-b for b in W.basis()], lattice=K.lattice())
+ K = K.intersection(W_cone)
- - ``max_dim`` (default: random) -- A nonnegative integer representing
- the maximum dimension of the ambient
- lattice.
+ # We've already intersected K with the span of K2, so every
+ # generator of K should belong to W now.
+ K_W_rays = [ W.coordinate_vector(r) for r in K.rays() ]
- - ``min_rays`` (default: zero) -- A nonnegative integer representing the
- minimum number of generating rays of the
- cone.
+ L = ToricLattice(W.dimension())
+ return Cone(K_W_rays, lattice=L)
+
+
+def lyapunov_rank(K):
+ r"""
+ Compute the Lyapunov rank of this cone.
- - ``max_rays`` (default: random) -- A nonnegative integer representing the
- maximum number of generating rays of
- the 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:
- A new, randomly generated cone.
+ 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:
- A ``ValueError` will be thrown under the following conditions:
+ 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}`.
- * Any of ``min_dim``, ``max_dim``, ``min_rays``, or ``max_rays``
- are negative.
+ REFERENCES:
- * ``max_dim`` is less than ``min_dim``.
+ .. [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.
- * ``max_rays`` is less than ``min_rays``.
+ M. Orlitzky. The Lyapunov rank of an improper cone.
+ http://www.optimization-online.org/DB_HTML/2015/10/5135.html
- * ``min_rays`` is greater than twice ``max_dim``.
+ 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:
- If we set the lower/upper bounds to zero, then our result is
- predictable::
+ The nonnegative orthant in `\mathbb{R}^{n}` always has rank `n`
+ [Rudolf]_::
- sage: random_cone(0,0,0,0)
- 0-d cone in 0-d lattice 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
- We can predict the dimension when ``min_dim == max_dim``::
+ The full space `\mathbb{R}^{n}` has Lyapunov rank `n^{2}`
+ [Orlitzky]_::
- sage: random_cone(min_dim=4, max_dim=4, min_rays=0, max_rays=0)
- 0-d cone in 4-d lattice N
+ sage: R5 = VectorSpace(QQ, 5)
+ sage: gs = R5.basis() + [ -r for r in R5.basis() ]
+ sage: K = Cone(gs)
+ sage: lyapunov_rank(K)
+ 25
- Likewise for the number of rays when ``min_rays == max_rays``::
+ The `L^{3}_{1}` cone is known to have a Lyapunov rank of one
+ [Rudolf]_::
- sage: random_cone(min_dim=10, max_dim=10, min_rays=10, max_rays=10)
- 10-d cone in 10-d lattice N
+ 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]_::
+
+ 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]_::
+
+ sage: K = Cone([(1,0,0,0,0)])
+ sage: lyapunov_rank(K)
+ 21
+ sage: K.lattice_dim()**2 - K.lattice_dim() + 1
+ 21
+
+ A subspace (of dimension `m`) in `n` dimensions should have a
+ 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)
+ sage: e2 = (0,1,0,0,0)
+ sage: neg_e2 = (0,-1,0,0,0)
+ sage: z = (0,0,0,0,0)
+ sage: K = Cone([e1, neg_e1, e2, neg_e2, z, z, z])
+ sage: lyapunov_rank(K)
+ 19
+ sage: K.lattice_dim()**2 - K.dim()*K.codim()
+ 19
+
+ The Lyapunov rank should be additive on a product of proper cones
+ [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: 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 [Rudolf]_.
+ 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 [Rudolf]_::
+
+ sage: K = Cone([(2,2,4), (-1,9,0), (2,0,6)])
+ sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+ True
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::
+ The Lyapunov rank should be additive on a product of proper cones
+ [Rudolf]_::
- sage: from sage.geometry.cone import is_Cone # long time
- sage: K = random_cone() # long time
- sage: is_Cone(K) # long time
+ sage: set_random_seed()
+ sage: K1 = random_cone(max_ambient_dim=8,
+ ....: strictly_convex=True,
+ ....: solid=True)
+ sage: K2 = random_cone(max_ambient_dim=8,
+ ....: strictly_convex=True,
+ ....: solid=True)
+ sage: K = K1.cartesian_product(K2)
+ sage: lyapunov_rank(K) == lyapunov_rank(K1) + lyapunov_rank(K2)
True
- The upper/lower bounds are respected::
+ The Lyapunov rank is invariant under a linear isomorphism
+ [Orlitzky]_::
- 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
+ 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: lyapunov_rank(K1) == lyapunov_rank(K2)
True
- sage: 3 <= K.nrays() and K.nrays() <= 4
+
+ The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
+ itself [Rudolf]_::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=8)
+ sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
True
- Ensure that an exception is raised when either lower bound is greater
- than its respective upper bound::
+ 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::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=8,
+ ....: 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
- sage: random_cone(min_dim=5, max_dim=2)
- Traceback (most recent call last):
- ...
- ValueError: max_dim cannot be less than min_dim.
+ In fact [Orlitzky]_, no closed convex polyhedral cone can have
+ Lyapunov rank `n-1` in `n` dimensions::
- sage: random_cone(min_rays=5, max_rays=2)
- Traceback (most recent call last):
- ...
- ValueError: max_rays cannot be less than min_rays.
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=8)
+ sage: b = lyapunov_rank(K)
+ sage: n = K.lattice_dim()
+ sage: b == n-1
+ False
- And if we request too many rays::
+ The calculation of the Lyapunov rank of an improper cone can be
+ reduced to that of a proper cone [Orlitzky]_::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=8)
+ sage: actual = lyapunov_rank(K)
+ sage: K_S = _restrict_to_space(K, K.span())
+ sage: K_SP = _restrict_to_space(K_S.dual(), K_S.dual().span()).dual()
+ sage: l = K.lineality()
+ sage: c = K.codim()
+ sage: expected = lyapunov_rank(K_SP) + K.dim()*(l + c) + c**2
+ sage: actual == expected
+ True
+
+ 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.lyapunov_like_basis())
+ True
+
+ We can make an imperfect cone perfect by adding a slack variable
+ (a Theorem in [Orlitzky]_)::
- sage: random_cone(min_rays=5, max_dim=1)
- Traceback (most recent call last):
- ...
- ValueError: min_rays cannot be larger than twice max_dim.
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=8,
+ ....: strictly_convex=True,
+ ....: solid=True)
+ sage: L = ToricLattice(K.lattice_dim() + 1)
+ sage: K = Cone([ r.list() + [0] for r in K.rays() ], lattice=L)
+ sage: lyapunov_rank(K) >= K.lattice_dim()
+ True
"""
+ beta = 0 # running tally of the Lyapunov rank
- # 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...
- 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. We can obviously stop if ``K`` is the
- # entire ambient vector space.
- while r > K.nrays() and not is_full_space(K):
- rays.append(L.random_element())
- K = Cone(rays)
-
- return K
-
-
-def discrete_complementarity_set(K):
+ m = K.dim()
+ n = K.lattice_dim()
+ l = K.lineality()
+
+ if m < n:
+ # K is not solid, restrict to its span.
+ K = _restrict_to_space(K, K.span())
+
+ # Non-solid reduction lemma.
+ beta += (n - m)*n
+
+ if l > 0:
+ # K is not pointed, restrict to the span of its dual. Uses a
+ # proposition from our paper, i.e. this is equivalent to K =
+ # _rho(K.dual()).dual().
+ K = _restrict_to_space(K, K.dual().span())
+
+ # Non-pointed reduction lemma.
+ beta += l * m
+
+ beta += len(K.lyapunov_like_basis())
+ return beta
+
+
+
+def is_lyapunov_like(L,K):
r"""
- Compute the discrete complementarity set of this cone.
+ Determine whether or not ``L`` is Lyapunov-like on ``K``.
+
+ We say that ``L`` is Lyapunov-like on ``K`` if `\left\langle
+ L\left\lparenx\right\rparen,s\right\rangle = 0` for all pairs
+ `\left\langle x,s \right\rangle` in the complementarity set of
+ ``K``. It is known [Orlitzky]_ that this property need only be
+ checked for generators of ``K`` and its dual.
+
+ INPUT:
+
+ - ``L`` -- A linear transformation or matrix.
- 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.
+ - ``K`` -- A polyhedral closed convex cone.
OUTPUT:
- A list of pairs `(x,s)` such that,
+ ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``,
+ and ``False`` otherwise.
+
+ .. WARNING::
- * `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.
+ If this function returns ``True``, then ``L`` is Lyapunov-like
+ on ``K``. However, if ``False`` is returned, that could mean one
+ of two things. The first is that ``L`` is definitely not
+ Lyapunov-like on ``K``. The second is more of an "I don't know"
+ answer, returned (for example) if we cannot prove that an inner
+ product is zero.
+
+ REFERENCES:
+
+ M. Orlitzky. The Lyapunov rank of an improper cone.
+ http://www.optimization-online.org/DB_HTML/2015/10/5135.html
EXAMPLES:
- The discrete complementarity set of the nonnegative orthant consists
- of pairs of standard basis vectors::
+ The identity is always Lyapunov-like in a nontrivial space::
- 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: set_random_seed()
+ sage: K = random_cone(min_ambient_dim = 1, max_rays = 8)
+ sage: L = identity_matrix(K.lattice_dim())
+ sage: is_lyapunov_like(L,K)
+ True
- sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
- sage: discrete_complementarity_set(K)
- []
+ As is the "zero" transformation::
+
+ sage: K = random_cone(min_ambient_dim = 1, max_rays = 5)
+ sage: R = K.lattice().vector_space().base_ring()
+ sage: L = zero_matrix(R, K.lattice_dim())
+ sage: is_lyapunov_like(L,K)
+ True
+
+ 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.lyapunov_like_basis() ])
+ True
+
+ """
+ return all([(L*x).inner_product(s) == 0
+ for (x,s) in K.discrete_complementarity_set()])
+
+
+def random_element(K):
+ r"""
+ Return a random element of ``K`` from its ambient vector space.
+
+ ALGORITHM:
+
+ The cone ``K`` is specified in terms of its generators, so that
+ ``K`` is equal to the convex conic combination of those generators.
+ To choose a random element of ``K``, we assign random nonnegative
+ coefficients to each generator of ``K`` and construct a new vector
+ from the scaled rays.
+
+ A vector, rather than a ray, is returned so that the element may
+ have non-integer coordinates. Thus the element may have an
+ arbitrarily small norm.
+
+ EXAMPLES:
+
+ A random element of the trivial cone is zero::
+
+ sage: set_random_seed()
+ sage: K = Cone([], ToricLattice(0))
+ sage: random_element(K)
+ ()
+ sage: K = Cone([(0,)])
+ sage: random_element(K)
+ (0)
+ sage: K = Cone([(0,0)])
+ sage: random_element(K)
+ (0, 0)
+ sage: K = Cone([(0,0,0)])
+ sage: random_element(K)
+ (0, 0, 0)
TESTS:
- The complementarity set of the dual can be obtained by switching the
- components of the complementarity set of the original cone::
+ Any cone should contain an element of itself::
- 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
+ sage: set_random_seed()
+ sage: K = random_cone(max_rays = 8)
+ sage: K.contains(random_element(K))
True
"""
V = K.lattice().vector_space()
+ F = V.base_ring()
+ coefficients = [ F.random_element().abs() for i in range(K.nrays()) ]
+ vector_gens = map(V, K.rays())
+ scaled_gens = [ coefficients[i]*vector_gens[i]
+ for i in range(len(vector_gens)) ]
- # 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]
+ # Make sure we return a vector. Without the coercion, we might
+ # return ``0`` when ``K`` has no rays.
+ v = V(sum(scaled_gens))
+ return v
-def lyapunov_rank(K):
+def positive_operators(K):
r"""
- Compute the Lyapunov (or bilinearity) rank of this cone.
+ Compute generators of the cone of positive operators on this cone.
- The Lyapunov rank of a cone can be thought of in (mainly) two ways:
+ OUTPUT:
- 1. The dimension of the Lie algebra of the automorphism group of the
- cone.
+ A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``.
+ Each matrix ``P`` in the list should have the property that ``P*x``
+ is an element of ``K`` whenever ``x`` is an element of
+ ``K``. Moreover, any nonnegative linear combination of these
+ matrices shares the same property.
- 2. The dimension of the linear space of all Lyapunov-like
- transformations on the cone.
+ EXAMPLES:
- INPUT:
+ The trivial cone in a trivial space has no positive operators::
- A closed, convex polyhedral cone.
+ sage: K = Cone([], ToricLattice(0))
+ sage: positive_operators(K)
+ []
- OUTPUT:
+ Positive operators on the nonnegative orthant are nonnegative matrices::
- 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).
+ sage: K = Cone([(1,)])
+ sage: positive_operators(K)
+ [[1]]
- .. note::
+ sage: K = Cone([(1,0),(0,1)])
+ sage: positive_operators(K)
+ [
+ [1 0] [0 1] [0 0] [0 0]
+ [0 0], [0 0], [1 0], [0 1]
+ ]
- In the references, the cones are always assumed to be proper. We
- do not impose this restriction.
+ Every operator is positive on the ambient vector space::
- .. seealso::
+ sage: K = Cone([(1,),(-1,)])
+ sage: K.is_full_space()
+ True
+ sage: positive_operators(K)
+ [[1], [-1]]
- :meth:`is_proper`
+ sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
+ sage: K.is_full_space()
+ True
+ sage: positive_operators(K)
+ [
+ [1 0] [-1 0] [0 1] [ 0 -1] [0 0] [ 0 0] [0 0] [ 0 0]
+ [0 0], [ 0 0], [0 0], [ 0 0], [1 0], [-1 0], [0 1], [ 0 -1]
+ ]
- ALGORITHM:
+ TESTS:
- 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}`.
+ A positive operator on a cone should send its generators into the cone::
- REFERENCES:
+ 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()])
+ 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)
- 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.
+ tensor_products = [ s.tensor_product(x) for x in K for s in K.dual() ]
- 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.
+ # Turn our matrices into long vectors...
+ vectors = [ W(m.list()) for m in tensor_products ]
- EXAMPLES:
+ # Create the *dual* cone of the positive operators, expressed as
+ # long vectors..
+ L = ToricLattice(W.dimension())
+ pi_dual = Cone(vectors, lattice=L)
- The nonnegative orthant in `\mathbb{R}^{n}` always has rank `n`::
+ # Now compute the desired cone from its dual...
+ pi_cone = pi_dual.dual()
- 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
+ # And finally convert its rays back to matrix representations.
+ M = MatrixSpace(V.base_ring(), V.dimension())
- The `L^{3}_{1}` cone is known to have a Lyapunov rank of one::
+ return [ M(v.list()) for v in pi_cone.rays() ]
- 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::
+def Z_transformations(K):
+ r"""
+ Compute generators of the cone of Z-transformations on this cone.
- sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)])
- sage: lyapunov_rank(L3infty)
- 1
+ OUTPUT:
- The Lyapunov rank should be additive on a product of cones::
+ 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.
- 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)
+ 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
- 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}`::
+ The trivial cone in a trivial space has no Z-transformations::
- sage: K = Cone([(1,2,3), (-1,1,0), (1,0,6)])
- sage: lyapunov_rank(K)
- 3
+ sage: K = Cone([], ToricLattice(0))
+ sage: Z_transformations(K)
+ []
- The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
- itself::
+ Z-transformations on a subspace are Lyapunov-like and vice-versa::
- sage: K = Cone([(2,2,4), (-1,9,0), (2,0,6)])
- sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+ 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 Lyapunov rank should be additive on a product of cones::
+ The Z-property is possessed by every Z-transformation::
- 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)
+ 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 dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
- itself::
+ The lineality space of Z is LL::
- sage: K = random_cone(max_dim=10, max_rays=10)
- sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+ 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
"""
- 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.
# 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)
- def phi(m):
- r"""
- Convert a matrix to a vector isomorphically.
- """
- return W(m.list())
+ 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()
- vectors = [phi(m) for m in matrices]
+ # 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 (W.dimension() - W.span(vectors).rank())
+ return [ -M(v.list()) for v in Sigma_cone.rays() ]
projects / sage.d.git / blobdiff