from sage.all import *
-def project_span(K, K2 = None):
+def basically_the_same(K1,K2):
r"""
- Return a "copy" of ``K`` embeded in a lower-dimensional space.
+ ``True`` if ``K1`` and ``K2`` are basically the same, and ``False``
+ otherwise. This is intended as a lazy way to check whether or not
+ ``K1`` and ``K2`` are linearly isomorphic (i.e. ``A(K1) == K2`` for
+ some invertible linear transformation ``A``).
+ """
+ if K1.lattice_dim() != K2.lattice_dim():
+ return False
+
+ if K1.nrays() != K2.nrays():
+ return False
+
+ if K1.dim() != K2.dim():
+ return False
+
+ if K1.lineality() != K2.lineality():
+ return False
+
+ if K1.is_solid() != K2.is_solid():
+ return False
+
+ if K1.is_strictly_convex() != K2.is_strictly_convex():
+ return False
+
+ if len(LL(K1)) != len(LL(K2)):
+ return False
+
+ C_of_K1 = discrete_complementarity_set(K1)
+ C_of_K2 = discrete_complementarity_set(K2)
+ if len(C_of_K1) != len(C_of_K2):
+ return False
+
+ if len(K1.facets()) != len(K2.facets()):
+ return False
+
+ return True
+
+
+
+def rho(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.
- 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.
+ 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([(1,0,0), (0,1,0)])
- sage: K2 = Cone([(0,1)])
- sage: project_span(K, K2).rays()
+ sage: K = Cone([(1,)])
+ sage: rho(K) == K
+ True
+
+ sage: K2 = Cone([(1,0)])
+ sage: rho(K2).rays()
N(1)
in 1-d lattice N
+ sage: K3 = Cone([(1,0,0)])
+ sage: rho(K3).rays()
+ N(1)
+ in 1-d lattice N
+ sage: rho(K2) == rho(K3)
+ True
+
+ TESTS:
+
+ The projected cone should always be solid::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8)
+ sage: K_S = rho(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 = 8)
+ sage: K_S = rho(K, K.dual() )
+ sage: K_S.lattice_dim() == K.dual().dim()
+ True
+
+ This function should not affect the dimension of a cone::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8)
+ sage: K.dim() == rho(K).dim()
+ True
+
+ Nor should it affect the lineality of a cone::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8)
+ sage: K.lineality() == rho(K).lineality()
+ True
+
+ No matter which space we restrict to, the lineality should not
+ increase::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8)
+ sage: K.lineality() >= rho(K).lineality()
+ True
+ sage: K.lineality() >= rho(K, K.dual()).lineality()
+ True
+
+ If we do this according to our paper, then the result is proper::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=False)
+ sage: K_S = rho(K)
+ sage: K_SP = rho(K_S.dual()).dual()
+ sage: K_SP.is_proper()
+ True
+ sage: K_SP = rho(K_S, K_S.dual())
+ sage: K_SP.is_proper()
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=False)
+ sage: K_S = rho(K)
+ sage: K_SP = rho(K_S.dual()).dual()
+ sage: K_SP.is_proper()
+ True
+ sage: K_SP = rho(K_S, K_S.dual())
+ sage: K_SP.is_proper()
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=True)
+ sage: K_S = rho(K)
+ sage: K_SP = rho(K_S.dual()).dual()
+ sage: K_SP.is_proper()
+ True
+ sage: K_SP = rho(K_S, K_S.dual())
+ sage: K_SP.is_proper()
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=True)
+ sage: K_S = rho(K)
+ sage: K_SP = rho(K_S.dual()).dual()
+ sage: K_SP.is_proper()
+ True
+ sage: K_SP = rho(K_S, K_S.dual())
+ sage: K_SP.is_proper()
+ True
+
+ 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 rho::
+
+ sage: set_random_seed()
+ sage: J = random_cone(max_dim = 8, solid=False, strictly_convex=False)
+ sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
+ sage: K_W = rho(K, J)
+ sage: K_star_W_star = rho(K.dual(), J).dual()
+ sage: basically_the_same(K_W, K_star_W_star)
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: J = random_cone(max_dim = 8, solid=True, strictly_convex=False)
+ sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
+ sage: K_W = rho(K, J)
+ sage: K_star_W_star = rho(K.dual(), J).dual()
+ sage: basically_the_same(K_W, K_star_W_star)
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: J = random_cone(max_dim = 8, solid=False, strictly_convex=True)
+ sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
+ sage: K_W = rho(K, J)
+ sage: K_star_W_star = rho(K.dual(), J).dual()
+ sage: basically_the_same(K_W, K_star_W_star)
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: J = random_cone(max_dim = 8, solid=True, strictly_convex=True)
+ sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
+ sage: K_W = rho(K, J)
+ sage: K_star_W_star = rho(K.dual(), J).dual()
+ sage: basically_the_same(K_W, K_star_W_star)
+ 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()
+ # First we project K onto the span of K2. This will explode if the
+ # rank of ``K2.lattice()`` doesn't match ours.
+ span_K2 = Cone(K2.rays() + (-K2).rays(), lattice=K.lattice())
+ K = K.intersection(span_K2)
- # And use these to figure out which indices to drop.
- cs2 = K2.rays().matrix().columns()
+ # Cheat a little to get the subspace span(K2). The paper uses the
+ # rays of K2 as a basis, but everything is invariant under linear
+ # isomorphism (i.e. a change of basis), and this is a little
+ # faster.
+ W = span_K2.linear_subspace()
- perp_idxs = []
+ # We've already intersected K with the span of K2, so every
+ # generator of K should belong to W now.
+ W_rays = [ W.coordinate_vector(r) for r in K.rays() ]
- for idx in range(0, len(cs2)):
- if cs2[idx].is_zero():
- perp_idxs.append(idx)
+ L = ToricLattice(K2.dim())
+ return Cone(W_rays, lattice=L)
- solid_cols = [ cs1[idx] for idx in range(0,len(cs1))
- if not idx in perp_idxs
- and not idx >= len(cs2) ]
-
- m = matrix(solid_cols)
- L = ToricLattice(len(m.rows()))
- J = Cone(m.transpose(), lattice=L)
- return J
def discrete_complementarity_set(K):
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.
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
"""
[0 0 1]
]
+ If our cone is the entire space, then every transformation on it is
+ Lyapunov-like::
+
+ sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)])
+ sage: M = MatrixSpace(QQ,2)
+ sage: M.basis() == LL(K)
+ True
+
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: set_random_seed()
+ sage: K = random_cone(max_dim=8)
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
- Try the formula in my paper::
-
- 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, K_S.dual())
- sage: J_T2 = project_span(K_S.dual()).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
+ 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)
+ 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
"""
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
cone and Lyapunov-like transformations, Mathematical Programming, 147
(2014) 155-170.
+ .. [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.
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()*K.codim()
+ 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=8, strictly_convex=True, solid=True)
+ sage: K2 = random_cone(max_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 dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
itself [Rudolf et al.]_::
- sage: K = random_cone(max_dim=10, max_rays=10)
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim=8)
+ 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=8, strictly_convex=False, solid=False)
+ sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+ True
+
+ Let's check the other permutations as well, just to be sure::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim=8, strictly_convex=False, solid=True)
+ sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim=8, strictly_convex=True, solid=False)
+ sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True)
sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
True
trivial cone in a trivial space as well. However, in zero dimensions,
the Lyapunov rank of the trivial cone will be zero::
- sage: K = random_cone(max_dim=10, strictly_convex=True, solid=True)
+ sage: set_random_seed()
+ sage: K = random_cone(max_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
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=8)
+ 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=8)
+ sage: actual = lyapunov_rank(K)
+ sage: K_S = rho(K)
+ sage: K_SP = rho(K_S.dual()).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 proper cone is just the dimension of ``LL(K)``::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True)
+ sage: lyapunov_rank(K) == len(LL(K))
+ True
+
+ In fact the same can be said of any cone. These additional tests
+ just increase our confidence that the reduction scheme works::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim=8, strictly_convex=True, solid=False)
+ sage: lyapunov_rank(K) == len(LL(K))
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim=8, strictly_convex=False, solid=True)
+ sage: lyapunov_rank(K) == len(LL(K))
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim=8, strictly_convex=False, solid=False)
+ sage: lyapunov_rank(K) == len(LL(K))
+ True
+
+ Test Theorem 3 in [Orlitzky/Gowda]_::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_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
+
"""
- return len(LL(K))
+ beta = 0
+
+ m = K.dim()
+ n = K.lattice_dim()
+ l = K.lineality()
+
+ if m < n:
+ # K is not solid, restrict to its span.
+ K = rho(K)
+
+ # Lemma 2
+ beta += m*(n - m) + (n - m)**2
+
+ 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 = rho(K, K.dual())
+
+ # Lemma 3
+ beta += m * l
+
+ beta += len(LL(K))
+ return beta
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