from sage.all import *
-def project_span(K, K2 = None):
+def iso_space(K):
r"""
- Return a "copy" of ``K`` embeded in a lower-dimensional 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()
+
+ # 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()
+
+ W = V.subspace_with_basis(W_basis)
+ W_perp = W.complement()
+
+ return W.cartesian_product(W_perp)
+
+
+def ips_iso(K):
+ r"""
+ Construct the IPS isomorphism and its inverse from our paper.
+
+ 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()
+
+ # A space equivalent to V, but using our basis.
+ V_user = V.subspace_with_basis( W.basis() + W_perp.basis() )
+
+ 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)
- 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.
+ 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()) ])
+
+ return V_iso( (w1, w2) )
+
+
+ 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 intersect_span(K1, K2):
+ r"""
+ Return a new cone obtained by intersecting ``K1`` with the span of ``K2``.
+ """
+ L = K1.lattice()
+
+ if L.rank() != K2.lattice().rank():
+ raise ValueError('K1 and K2 must belong to lattices of the same rank.')
+
+ SL_gens = list(K2.rays())
+ span_K2_gens = SL_gens + [ -g for g in SL_gens ]
+
+ # The lattices have the same rank (see above) so this should work.
+ span_K2 = Cone(span_K2_gens, L)
+ return K1.intersection(span_K2)
+
+
+
+def restrict_span(K, K2=None):
+ r"""
+ Restrict ``K`` into its own span, or the span of another cone.
+
+ INPUT:
+
+ - ``K2`` -- another cone whose lattice has the same rank as this cone.
+
+ OUTPUT:
+
+ A new cone in a sublattice.
EXAMPLES::
- 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: K = Cone([(1,)])
+ sage: restrict_span(K) == K
+ True
- sage: K = Cone([(1,0,0), (0,1,0)])
- sage: K2 = Cone([(0,1)])
- sage: project_span(K, K2).rays()
+ 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
+
+ TESTS:
+
+ The projected cone should always be solid::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 10)
+ sage: K_S = restrict_span(K)
+ sage: K_S.is_solid()
+ True
+
+ And the resulting cone should live in a space having the same
+ dimension as the space we restricted it to::
+
+ 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
+
+ This function has ``unrestrict_span()`` as its inverse::
+
+ 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
+
+ This function should not affect the dimension of a cone::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 10)
+ sage: K.dim() == restrict_span(K).dim()
+ True
+
+ Nor should it affect the lineality of a cone::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 10)
+ sage: lineality(K) == lineality(restrict_span(K))
+ True
+
+ No matter which space we restrict to, the lineality should not
+ increase::
+
+ 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
+
+ If we do this according to our paper, then the result is proper::
+
+ 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
+
+ If ``K`` is strictly convex, then both ``K_W`` and
+ ``K_star_W.dual()`` should equal ``K`` (after we unrestrict)::
+
+ 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: 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^{*}`::
+
+ 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
"""
- # Allow us to use a second cone to generate the subspace into
- # which we're "projecting."
if K2 is None:
K2 = K
- # Use these to generate the new cone.
- cs1 = K.rays().matrix().columns()
+ phi,_ = ips_iso(K2)
+ (W, W_perp) = iso_space(K2).cartesian_factors()
+
+ ray_pairs = [ phi(r) for r in K.rays() ]
+
+ if any([ w2 != W_perp.zero() for (_, w2) in ray_pairs ]):
+ msg = 'Cone has nonzero components in W-perp!'
+ raise ValueError(msg)
+
+ # 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.
- # And use these to figure out which indices to drop.
- cs2 = K2.rays().matrix().columns()
+ 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
- perp_idxs = []
+ If our cone is all of `\mathbb{R}^{2}`, then its codimension is zero::
- for idx in range(0, len(cs2)):
- if cs2[idx].is_zero():
- perp_idxs.append(idx)
+ 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::
- solid_cols = [ cs1[idx] for idx in range(0,len(cs1))
- if not idx in perp_idxs
- and not idx >= len(cs2) ]
+ sage: K = Cone([], lattice=ToricLattice(0))
+ sage: codim(K)
+ 0
- m = matrix(solid_cols)
- L = ToricLattice(len(m.rows()))
- J = Cone(m.transpose(), lattice=L)
- return J
+ 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
"""
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]
+ 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
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 for any cone.
+ not possible (see the first reference).
.. note::
sage: lyapunov_rank(octant)
3
+ 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: 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)])
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 [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()
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: set_random_seed()
+ sage: K = random_cone(max_dim=10)
sage: b = lyapunov_rank(K)
sage: n = K.lattice_dim()
sage: b == n-1
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: set_random_seed()
+ sage: K = random_cone(max_dim=10)
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: 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
+
"""
- return len(LL(K))
+ K_orig = K
+ beta = 0
+
+ m = K.dim()
+ n = K.lattice_dim()
+ l = lineality(K)
+
+ if m < n:
+ # K is not solid, project onto its span.
+ K = restrict_span(K)
+
+ # 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()) ]
+
+ #if Ks != Js:
+ # print [ list(r) for r in K_orig.rays() ]
+
+ # Lemma 3
+ beta += m * l
+
+ beta += len(LL(K))
+ return beta
projects / sage.d.git / blobdiff