from sage.all import *
-def is_full_space(K):
+def iso_space(K):
r"""
- Return whether or not this cone is equal to its ambient vector space.
+ Construct the space `W \times W^{\perp}` isomorphic to the ambient space
+ of ``K`` where `W` is equal to the span of ``K``.
+ """
+ V = K.lattice().vector_space()
- OUTPUT:
+ # Create the space W \times W^{\perp} isomorphic to V.
+ # First we get an orthogonal (but not normal) basis...
+ M = matrix(V.base_field(), K.rays())
+ W_basis,_ = M.gram_schmidt()
- ``True`` if this cone is the entire vector space and ``False``
- otherwise.
+ W = V.subspace_with_basis(W_basis)
+ W_perp = W.complement()
- EXAMPLES:
+ return W.cartesian_product(W_perp)
- A ray in two dimensions is not equal to the entire space::
- sage: K = Cone([(1,0)])
- sage: is_full_space(K)
- False
+def ips_iso(K):
+ r"""
+ Construct the IPS isomorphism and its inverse from our paper.
- Neither is the nonnegative orthant::
+ Given a cone ``K``, the returned isomorphism will split its ambient
+ vector space `V` into a cartesian product `W \times W^{\perp}` where
+ `W` equals the span of ``K``.
+ """
+ V = K.lattice().vector_space()
+ V_iso = iso_space(K)
+ (W, W_perp) = V_iso.cartesian_factors()
- sage: K = Cone([(1,0),(0,1)])
- sage: is_full_space(K)
- False
+ # A space equivalent to V, but using our basis.
+ V_user = V.subspace_with_basis( W.basis() + W_perp.basis() )
- The right half-space contains a vector subspace, but it is still not
- equal to the entire plane::
+ def phi(v):
+ # Write v in terms of our custom basis, where the first dim(W)
+ # coordinates are for the W-part of the basis.
+ cs = V_user.coordinates(v)
- sage: K = Cone([(1,0),(-1,0),(0,1)])
- sage: is_full_space(K)
- False
+ w1 = sum([ V_user.basis()[idx]*cs[idx]
+ for idx in range(0, W.dimension()) ])
+ w2 = sum([ V_user.basis()[idx]*cs[idx]
+ for idx in range(W.dimension(), V.dimension()) ])
- But if we include nonnegative sums from both axes, then the resulting
- cone is the entire two-dimensional space::
+ return V_iso( (w1, w2) )
- sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
- sage: is_full_space(K)
- True
- """
- return K.linear_subspace() == K.lattice().vector_space()
+ def phi_inv( pair ):
+ # Crash if the arguments are in the wrong spaces.
+ V_iso(pair)
+
+ #w = sum([ sub_w[idx]*W.basis()[idx] for idx in range(0,m) ])
+ #w_prime = sum([ sub_w_prime[idx]*W_perp.basis()[idx]
+ # for idx in range(0,n-m) ])
+
+ return sum( pair.cartesian_factors() )
+
+
+ return (phi,phi_inv)
+
+
+
+def unrestrict_span(K, K2=None):
+ if K2 is None:
+ K2 = K
+
+ _,phi_inv = ips_iso(K2)
+ V_iso = iso_space(K2)
+ (W, W_perp) = V_iso.cartesian_factors()
+
+ rays = []
+ for r in K.rays():
+ w = sum([ r[idx]*W.basis()[idx] for idx in range(0,len(r)) ])
+ pair = V_iso( (w, W_perp.zero()) )
+ rays.append( phi_inv(pair) )
+
+ L = ToricLattice(W.dimension() + W_perp.dimension())
+
+ return Cone(rays, lattice=L)
-def random_cone(min_dim=0, max_dim=None, min_rays=0, max_rays=None):
+
+def intersect_span(K1, K2):
r"""
- Generate a random rational convex polyhedral cone.
+ Return a new cone obtained by intersecting ``K1`` with the span of ``K2``.
+ """
+ L = K1.lattice()
- 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.
+ if L.rank() != K2.lattice().rank():
+ raise ValueError('K1 and K2 must belong to lattices of the same rank.')
- 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``.
+ SL_gens = list(K2.rays())
+ span_K2_gens = SL_gens + [ -g for g in SL_gens ]
- .. NOTE:
+ # The lattices have the same rank (see above) so this should work.
+ span_K2 = Cone(span_K2_gens, L)
+ return K1.intersection(span_K2)
- 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.
+
+
+def restrict_span(K, K2=None):
+ r"""
+ Restrict ``K`` into its own span, or the span of another cone.
INPUT:
- - ``min_dim`` (default: zero) -- A nonnegative integer representing the
- minimum dimension of the ambient lattice.
+ - ``K2`` -- another cone whose lattice has the same rank as this cone.
- - ``max_dim`` (default: random) -- A nonnegative integer representing
- the maximum dimension of the ambient
- lattice.
+ OUTPUT:
- - ``min_rays`` (default: zero) -- A nonnegative integer representing the
- minimum number of generating rays of the
- cone.
+ A new cone in a sublattice.
- - ``max_rays`` (default: random) -- A nonnegative integer representing the
- maximum number of generating rays of
- the cone.
+ EXAMPLES::
- OUTPUT:
+ sage: K = Cone([(1,)])
+ sage: restrict_span(K) == K
+ True
- A new, randomly generated cone.
+ sage: K2 = Cone([(1,0)])
+ sage: restrict_span(K2).rays()
+ N(1)
+ in 1-d lattice N
+ sage: K3 = Cone([(1,0,0)])
+ sage: restrict_span(K3).rays()
+ N(1)
+ in 1-d lattice N
+ sage: restrict_span(K2) == restrict_span(K3)
+ True
- A ``ValueError` will be thrown under the following conditions:
+ TESTS:
- * Any of ``min_dim``, ``max_dim``, ``min_rays``, or ``max_rays``
- are negative.
+ The projected cone should always be solid::
- * ``max_dim`` is less than ``min_dim``.
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 10)
+ sage: K_S = restrict_span(K)
+ sage: K_S.is_solid()
+ True
- * ``max_rays`` is less than ``min_rays``.
+ And the resulting cone should live in a space having the same
+ dimension as the space we restricted it to::
- * ``min_rays`` is greater than twice ``max_dim``.
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 10)
+ sage: K_S = restrict_span( intersect_span(K, K.dual()), K.dual() )
+ sage: K_S.lattice_dim() == K.dual().dim()
+ True
- EXAMPLES:
+ This function has ``unrestrict_span()`` as its inverse::
- If we set the lower/upper bounds to zero, then our result is
- predictable::
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 10, solid=True)
+ sage: J = restrict_span(K)
+ sage: K == unrestrict_span(J,K)
+ True
- sage: random_cone(0,0,0,0)
- 0-d cone in 0-d lattice N
+ This function should not affect the dimension of a cone::
- We can predict the dimension when ``min_dim == max_dim``::
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 10)
+ sage: K.dim() == restrict_span(K).dim()
+ True
- sage: random_cone(min_dim=4, max_dim=4, min_rays=0, max_rays=0)
- 0-d cone in 4-d lattice N
+ Nor should it affect the lineality of a cone::
- Likewise for the number of rays when ``min_rays == max_rays``::
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 10)
+ sage: lineality(K) == lineality(restrict_span(K))
+ True
- sage: random_cone(min_dim=10, max_dim=10, min_rays=10, max_rays=10)
- 10-d cone in 10-d lattice N
+ No matter which space we restrict to, the lineality should not
+ increase::
- TESTS:
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 10)
+ sage: J = intersect_span(K, K.dual())
+ sage: lineality(K) >= lineality(restrict_span(J, K.dual()))
+ True
- It's hard to test the output of a random process, but we can at
- least make sure that we get a cone back::
+ If we do this according to our paper, then the result is proper::
- 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: K = random_cone(max_dim = 10)
+ sage: K_S = restrict_span(K)
+ sage: P = restrict_span(K_S.dual()).dual()
+ sage: P.is_proper()
True
- The upper/lower bounds are respected::
+ If ``K`` is strictly convex, then both ``K_W`` and
+ ``K_star_W.dual()`` should equal ``K`` (after we unrestrict)::
- 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: set_random_seed()
+ sage: K = random_cone(max_dim = 10, strictly_convex=True)
+ sage: K_W = restrict_span(intersect_span(K,K.dual()), K.dual())
+ sage: K_star_W_star = restrict_span(K.dual()).dual()
+ sage: j1 = unrestrict_span(K_W, K.dual())
+ sage: j2 = unrestrict_span(K_star_W_star, K.dual())
+ sage: j1 == j2
True
- sage: 3 <= K.nrays() and K.nrays() <= 4
+ sage: j1 == K
True
+ sage: K; [ list(r) for r in K.rays() ]
+
+ Test the proposition in our paper concerning the duals, where the
+ subspace `W` is the span of `K^{*}`::
- Ensure that an exception is raised when either lower bound is greater
- than its respective upper bound::
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 10, solid=False, strictly_convex=False)
+ sage: K_W = restrict_span(intersect_span(K,K.dual()), K.dual())
+ sage: K_star_W_star = restrict_span(K.dual(), K.dual()).dual()
+ sage: K_W.nrays() == K_star_W_star.nrays()
+ True
+ sage: K_W.dim() == K_star_W_star.dim()
+ True
+ sage: lineality(K_W) == lineality(K_star_W_star)
+ True
+ sage: K_W.is_solid() == K_star_W_star.is_solid()
+ True
+ sage: K_W.is_strictly_convex() == K_star_W_star.is_strictly_convex()
+ True
+
+ """
+ if K2 is None:
+ K2 = K
- sage: random_cone(min_dim=5, max_dim=2)
- Traceback (most recent call last):
- ...
- ValueError: max_dim cannot be less than min_dim.
+ phi,_ = ips_iso(K2)
+ (W, W_perp) = iso_space(K2).cartesian_factors()
- sage: random_cone(min_rays=5, max_rays=2)
- Traceback (most recent call last):
- ...
- ValueError: max_rays cannot be less than min_rays.
+ ray_pairs = [ phi(r) for r in K.rays() ]
- And if we request too many rays::
+ if any([ w2 != W_perp.zero() for (_, w2) in ray_pairs ]):
+ msg = 'Cone has nonzero components in W-perp!'
+ raise ValueError(msg)
- sage: random_cone(min_rays=5, max_dim=1)
- Traceback (most recent call last):
- ...
- ValueError: min_rays cannot be larger than twice max_dim.
+ # Represent the cone in terms of a basis for W, i.e. with smaller
+ # vectors.
+ ws = [ W.coordinate_vector(w1) for (w1, _) in ray_pairs ]
+
+ L = ToricLattice(W.dimension())
+
+ return Cone(ws, lattice=L)
+
+
+
+def lineality(K):
+ r"""
+ Compute the lineality of this cone.
+
+ The lineality of a cone is the dimension of the largest linear
+ subspace contained in that cone.
+
+ OUTPUT:
+
+ A nonnegative integer; the dimension of the largest subspace
+ contained within this cone.
+
+ REFERENCES:
+
+ .. [Rockafellar] R.T. Rockafellar. Convex Analysis. Princeton
+ University Press, Princeton, 1970.
+
+ EXAMPLES:
+
+ The lineality of the nonnegative orthant is zero, since it clearly
+ contains no lines::
+
+ sage: K = Cone([(1,0,0), (0,1,0), (0,0,1)])
+ sage: lineality(K)
+ 0
+
+ However, if we add another ray so that the entire `x`-axis belongs
+ to the cone, then the resulting cone will have lineality one::
+
+ sage: K = Cone([(1,0,0), (-1,0,0), (0,1,0), (0,0,1)])
+ sage: lineality(K)
+ 1
+
+ If our cone is all of `\mathbb{R}^{2}`, then its lineality is equal
+ to the dimension of the ambient space (i.e. two)::
+
+ sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)])
+ sage: lineality(K)
+ 2
+
+ Per the definition, the lineality of the trivial cone in a trivial
+ space is zero::
+
+ sage: K = Cone([], lattice=ToricLattice(0))
+ sage: lineality(K)
+ 0
+
+ TESTS:
+
+ The lineality of a cone should be an integer between zero and the
+ dimension of the ambient space, inclusive::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 10)
+ sage: l = lineality(K)
+ sage: l in ZZ
+ True
+ sage: (0 <= l) and (l <= K.lattice_dim())
+ True
+
+ A strictly convex cone should have lineality zero::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 10, strictly_convex = True)
+ sage: lineality(K)
+ 0
"""
+ return K.linear_subspace().dimension()
+
+
+def codim(K):
+ r"""
+ Compute the codimension of this cone.
+
+ The codimension of a cone is the dimension of the space of all
+ elements perpendicular to every element of the cone. In other words,
+ the codimension is the difference between the dimension of the
+ ambient space and the dimension of the cone itself.
+
+ OUTPUT:
+
+ A nonnegative integer representing the dimension of the space of all
+ elements perpendicular to this cone.
+
+ .. seealso::
+
+ :meth:`dim`, :meth:`lattice_dim`
+
+ EXAMPLES:
+
+ The codimension of the nonnegative orthant is zero, since the span of
+ its generators equals the entire ambient space::
+
+ sage: K = Cone([(1,0,0), (0,1,0), (0,0,1)])
+ sage: codim(K)
+ 0
+
+ However, if we remove a ray so that the entire cone is contained
+ within the `x-y`-plane, then the resulting cone will have
+ codimension one, because the `z`-axis is perpendicular to every
+ element of the cone::
+
+ sage: K = Cone([(1,0,0), (0,1,0)])
+ sage: codim(K)
+ 1
+
+ If our cone is all of `\mathbb{R}^{2}`, then its codimension is zero::
- # 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
+ sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)])
+ sage: codim(K)
+ 0
+
+ And if the cone is trivial in any space, then its codimension is
+ equal to the dimension of the ambient space::
+
+ sage: K = Cone([], lattice=ToricLattice(0))
+ sage: codim(K)
+ 0
+
+ sage: K = Cone([(0,)])
+ sage: codim(K)
+ 1
+
+ sage: K = Cone([(0,0)])
+ sage: codim(K)
+ 2
+
+ TESTS:
+
+ The codimension of a cone should be an integer between zero and
+ the dimension of the ambient space, inclusive::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 10)
+ sage: c = codim(K)
+ sage: c in ZZ
+ True
+ sage: (0 <= c) and (c <= K.lattice_dim())
+ True
+
+ A solid cone should have codimension zero::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 10, solid = True)
+ sage: codim(K)
+ 0
+
+ The codimension of a cone is equal to the lineality of its dual::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 10, solid = True)
+ sage: codim(K) == lineality(K.dual())
+ True
+
+ """
+ return (K.lattice_dim() - K.dim())
def discrete_complementarity_set(K):
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: set_random_seed()
+ sage: K1 = random_cone(max_dim=6)
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: sorted(actual) == sorted(expected)
True
"""
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: set_random_seed()
+ 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
+
+ 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_dim=8, max_rays=10)
+ 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 = 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.
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 full space `\mathbb{R}^{n}` has Lyapunov rank `n^{2}`
+ [Orlitzky/Gowda]_::
+
+ sage: R5 = VectorSpace(QQ, 5)
+ sage: gs = R5.basis() + [ -r for r in R5.basis() ]
+ sage: K = Cone(gs)
+ sage: lyapunov_rank(K)
+ 25
+
+ 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::
+ A single ray in `n` dimensions should have Lyapunov rank `n^{2} - n
+ + 1` [Orlitzky/Gowda]_::
+
+ 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/Gowda]_::
+
+ 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()*codim(K)
+ 19
+
+ The Lyapunov rank should be additive on a product of proper 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 proper cones
+ [Rudolf et al.]_::
- sage: K1 = random_cone(max_dim=10, max_rays=10)
- sage: K2 = random_cone(max_dim=10, max_rays=10)
+ sage: set_random_seed()
+ sage: K1 = random_cone(max_dim=10, strictly_convex=True, solid=True)
+ sage: K2 = random_cone(max_dim=10, strictly_convex=True, solid=True)
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::
+ itself [Rudolf et al.]_::
+ sage: set_random_seed()
sage: K = random_cone(max_dim=10, max_rays=10)
sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
True
+ Make sure we exercise the non-strictly-convex/non-solid case::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim=10, strictly_convex=False, solid=False)
+ sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+ True
+
+ 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_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: set_random_seed()
+ sage: K = random_cone(max_dim=10)
+ 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: set_random_seed()
+ sage: K = random_cone(max_dim=10)
+ sage: actual = lyapunov_rank(K)
+ sage: K_S = restrict_span(K)
+ sage: P = restrict_span(K_S.dual()).dual()
+ sage: l = lineality(K)
+ sage: c = codim(K)
+ sage: expected = lyapunov_rank(P) + K.dim()*(l + c) + c**2
+ sage: actual == expected
+ True
+
+ The Lyapunov rank of a proper cone is just the dimension of ``LL(K)``::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim=10, strictly_convex=True, solid=True)
+ sage: lyapunov_rank(K) == len(LL(K))
+ True
+
"""
- V = K.lattice().vector_space()
+ K_orig = K
+ beta = 0
- C_of_K = discrete_complementarity_set(K)
+ m = K.dim()
+ n = K.lattice_dim()
+ l = lineality(K)
- matrices = [x.tensor_product(s) for (x,s) in C_of_K]
+ if m < n:
+ # K is not solid, project onto its span.
+ K = restrict_span(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)
+ # Lemma 2
+ beta += m*(n - m) + (n - m)**2
+
+ if l > 0:
+ # K is not pointed, project its dual onto its span.
+ # Uses a proposition from our paper, i.e. this is
+ # equivalent to K = restrict_span(K.dual()).dual()
+ K = restrict_span(intersect_span(K,K.dual()), K.dual())
+ #K = restrict_span(K.dual()).dual()
+
+ #Ks = [ list(r) for r in sorted(K.rays()) ]
+ #Js = [ list(r) for r in sorted(J.rays()) ]
- def phi(m):
- r"""
- Convert a matrix to a vector isomorphically.
- """
- return W(m.list())
+ #if Ks != Js:
+ # print [ list(r) for r in K_orig.rays() ]
- vectors = [phi(m) for m in matrices]
+ # Lemma 3
+ beta += m * l
- return (W.dimension() - W.span(vectors).rank())
+ beta += len(LL(K))
+ return beta
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