-# Sage doesn't load ~/.sage/init.sage during testing (sage -t), so we
-# have to explicitly mangle our sitedir here so that "mjo.cone"
-# resolves.
-from os.path import abspath
-from site import addsitedir
-addsitedir(abspath('../../'))
-
from sage.all import *
-
-def drop_dependent(vs):
- r"""
- Return the largest linearly-independent subset of ``vs``.
- """
- if len(vs) == 0:
- # ...for lazy enough definitions of linearly-independent
- return vs
-
- result = []
- old_V = VectorSpace(vs[0].parent().base_field(), 0)
-
- for v in vs:
- new_V = span(result + [v])
- if new_V.dimension() > old_V.dimension():
- result.append(v)
- old_V = new_V
-
- return result
-
-
-def iso_space(K):
- r"""
- 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.
- W_basis = drop_dependent(K.rays())
- W = V.subspace_with_basis(W_basis)
- W_perp = W.complement()
-
- return W.cartesian_product(W_perp)
-
-
-def ips_iso(K):
+def is_lyapunov_like(L,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)
-
- 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)
-
-
+ Determine whether or not ``L`` is Lyapunov-like on ``K``.
-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.
+ 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:
- - ``K2`` -- another cone whose lattice has the same rank as this cone.
-
- OUTPUT:
-
- A new cone in a sublattice.
-
- EXAMPLES::
-
- sage: K = Cone([(1,)])
- sage: restrict_span(K) == K
- True
-
- 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::
+ - ``L`` -- A linear transformation or matrix.
- 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
+ - ``K`` -- A polyhedral closed convex cone.
- 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
+ OUTPUT:
- This function should not affect the dimension of a cone::
+ ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``,
+ and ``False`` otherwise.
- sage: set_random_seed()
- sage: K = random_cone(max_dim = 10)
- sage: K.dim() == restrict_span(K).dim()
- True
+ .. WARNING::
- Nor should it affect the lineality of a cone::
+ 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.
- sage: set_random_seed()
- sage: K = random_cone(max_dim = 10)
- sage: lineality(K) == lineality(restrict_span(K))
- True
+ REFERENCES:
- No matter which space we restrict to, the lineality should not
- increase::
+ M. Orlitzky. The Lyapunov rank of an improper cone.
+ http://www.optimization-online.org/DB_HTML/2015/10/5135.html
- 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
+ EXAMPLES:
- If we do this according to our paper, then the result is proper::
+ The identity is always Lyapunov-like in a nontrivial space::
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()
+ sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
+ sage: L = identity_matrix(K.lattice_dim())
+ sage: is_lyapunov_like(L,K)
True
- If ``K`` is strictly convex, then both ``K_W`` and
- ``K_star_W.dual()`` should equal ``K`` (after we unrestrict)::
+ As is the "zero" transformation::
- 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
+ sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
+ sage: R = K.lattice().vector_space().base_ring()
+ sage: L = zero_matrix(R, K.lattice_dim())
+ sage: is_lyapunov_like(L,K)
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^{*}`::
+ Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like
+ on ``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()
+ sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6)
+ sage: all([ is_lyapunov_like(L,K) for L in K.lyapunov_like_basis() ])
True
"""
- if K2 is None:
- K2 = K
-
- 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 ]
+ return all([(L*x).inner_product(s) == 0
+ for (x,s) in K.discrete_complementarity_set()])
- L = ToricLattice(W.dimension())
- return Cone(ws, lattice=L)
-
-
-
-def lineality(K):
+def motzkin_decomposition(K):
r"""
- Compute the lineality of this cone.
+ Return the pair of components in the motzkin decomposition of this cone.
- The lineality of a cone is the dimension of the largest linear
- subspace contained in that cone.
+ Every convex cone is the direct sum of a strictly convex cone and a
+ linear subspace. Return a pair ``(P,S)`` of cones such that ``P`` is
+ strictly convex, ``S`` is a subspace, and ``K`` is the direct sum of
+ ``P`` and ``S``.
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.
+ An ordered pair ``(P,S)`` of closed convex polyhedral cones where
+ ``P`` is strictly convex, ``S`` is a subspace, and ``K`` is the
+ direct sum of ``P`` and ``S``.
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::
+ The nonnegative orthant is strictly convex, so it is its own
+ strictly convex component and its subspace component is trivial::
- sage: set_random_seed()
- sage: K = random_cone(max_dim = 10)
- sage: l = lineality(K)
- sage: l in ZZ
+ sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
+ sage: (P,S) = motzkin_decomposition(K)
+ sage: K.is_equivalent(P)
True
- sage: (0 <= l) and (l <= K.lattice_dim())
+ sage: S.is_trivial()
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:
+ Likewise, full spaces are their own subspace components::
- 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::
-
- 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
+ sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
+ sage: K.is_full_space()
+ True
+ sage: (P,S) = motzkin_decomposition(K)
+ sage: K.is_equivalent(S)
+ True
+ sage: P.is_trivial()
+ True
TESTS:
- The codimension of a cone should be an integer between zero and
- the dimension of the ambient space, inclusive::
+ A random point in the cone should belong to either the strictly
+ convex component or the subspace component. If the point is nonzero,
+ it cannot be in both::
sage: set_random_seed()
- sage: K = random_cone(max_dim = 10)
- sage: c = codim(K)
- sage: c in ZZ
+ sage: K = random_cone(max_ambient_dim=8)
+ sage: (P,S) = motzkin_decomposition(K)
+ sage: x = K.random_element()
+ sage: P.contains(x) or S.contains(x)
True
- sage: (0 <= c) and (c <= K.lattice_dim())
+ sage: x.is_zero() or (P.contains(x) != S.contains(x))
True
- A solid cone should have codimension zero::
+ The strictly convex component should always be strictly convex, and
+ the subspace component should always be a subspace::
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())
+ sage: K = random_cone(max_ambient_dim=8)
+ sage: (P,S) = motzkin_decomposition(K)
+ sage: P.is_strictly_convex()
+ True
+ sage: S.lineality() == S.dim()
True
- """
- return (K.lattice_dim() - K.dim())
-
-
-def discrete_complementarity_set(K):
- r"""
- Compute the discrete complementarity set of this cone.
-
- The complementarity set of this cone is the set of all orthogonal
- pairs `(x,s)` such that `x` is in this cone, and `s` is in its
- dual. The discrete complementarity set restricts `x` and `s` to be
- generators of their respective cones.
-
- OUTPUT:
-
- A list of pairs `(x,s)` such that,
-
- * `x` is in this cone.
- * `x` is a generator of this cone.
- * `s` is in this cone's dual.
- * `s` is a generator of this cone's dual.
- * `x` and `s` are orthogonal.
-
- EXAMPLES:
-
- The discrete complementarity set of the nonnegative orthant consists
- of pairs of standard basis vectors::
-
- sage: K = Cone([(1,0),(0,1)])
- sage: discrete_complementarity_set(K)
- [((1, 0), (0, 1)), ((0, 1), (1, 0))]
-
- If the cone consists of a single ray, the second components of the
- discrete complementarity set should generate the orthogonal
- complement of that ray::
-
- sage: K = Cone([(1,0)])
- sage: discrete_complementarity_set(K)
- [((1, 0), (0, 1)), ((1, 0), (0, -1))]
- sage: K = Cone([(1,0,0)])
- sage: discrete_complementarity_set(K)
- [((1, 0, 0), (0, 1, 0)),
- ((1, 0, 0), (0, -1, 0)),
- ((1, 0, 0), (0, 0, 1)),
- ((1, 0, 0), (0, 0, -1))]
-
- When the cone is the entire space, its dual is the trivial cone, so
- the discrete complementarity set is empty::
-
- sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
- sage: discrete_complementarity_set(K)
- []
-
- TESTS:
-
- The complementarity set of the dual can be obtained by switching the
- components of the complementarity set of the original cone::
+ The generators of the strictly convex component are obtained from
+ the orthogonal projections of the original generators onto the
+ orthogonal complement of the subspace component::
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: sorted(actual) == sorted(expected)
+ sage: K = random_cone(max_ambient_dim=8)
+ sage: (P,S) = motzkin_decomposition(K)
+ sage: S_perp = S.linear_subspace().complement()
+ sage: A = S_perp.matrix().transpose()
+ sage: proj = A * (A.transpose()*A).inverse() * A.transpose()
+ sage: expected = Cone([ proj*g for g in K ], K.lattice())
+ sage: P.is_equivalent(expected)
True
-
"""
- V = K.lattice().vector_space()
+ linspace_gens = [ copy(b) for b in K.linear_subspace().basis() ]
+ linspace_gens += [ -b for b in linspace_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()]
+ S = Cone(linspace_gens, K.lattice())
- return [(x,s) for x in xs for s in ss if x.inner_product(s) == 0]
+ # Since ``S`` is a subspace, its dual is its orthogonal complement
+ # (albeit in the wrong lattice).
+ S_perp = Cone(S.dual(), K.lattice())
+ P = K.intersection(S_perp)
+ return (P,S)
-def LL(K):
+def positive_operator_gens(K):
r"""
- Compute the space `\mathbf{LL}` of all Lyapunov-like transformations
- on this cone.
+ Compute generators of the cone of positive operators on this cone.
OUTPUT:
- A list of matrices forming a basis for the space of all
- Lyapunov-like transformations on the given 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.
EXAMPLES:
- The trivial cone has no Lyapunov-like transformations::
+ The trivial cone in a trivial space has no positive operators::
- sage: L = ToricLattice(0)
- sage: K = Cone([], lattice=L)
- sage: LL(K)
+ sage: K = Cone([], ToricLattice(0))
+ sage: positive_operator_gens(K)
[]
- The Lyapunov-like transformations on the nonnegative orthant are
- simply diagonal matrices::
+ Positive operators on the nonnegative orthant are nonnegative matrices::
sage: K = Cone([(1,)])
- sage: LL(K)
+ sage: positive_operator_gens(K)
[[1]]
sage: K = Cone([(1,0),(0,1)])
- sage: LL(K)
+ sage: positive_operator_gens(K)
[
- [1 0] [0 0]
- [0 0], [0 1]
+ [1 0] [0 1] [0 0] [0 0]
+ [0 0], [0 0], [1 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.]_::
+ Every operator is positive on the ambient vector space::
- 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: K = Cone([(1,),(-1,)])
+ sage: K.is_full_space()
+ True
+ sage: positive_operator_gens(K)
+ [[1], [-1]]
- sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)])
- sage: LL(L3infty)
+ sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
+ sage: K.is_full_space()
+ True
+ sage: positive_operator_gens(K)
[
- [1 0 0]
- [0 1 0]
- [0 0 1]
+ [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]
]
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::
+ A positive operator on a cone should send its generators into 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
+ sage: K = random_cone(max_ambient_dim=5)
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: all([K.contains(p*x) for p in pi_of_K for x in K.rays()])
+ True
- 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)`
+ The dimension of the cone of positive operators is given by the
+ corollary in my paper::
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)
+ sage: K = random_cone(max_ambient_dim=5)
+ sage: n = K.lattice_dim()
+ sage: m = K.dim()
+ sage: l = K.lineality()
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: L = ToricLattice(n**2)
+ sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).dim()
+ sage: expected = n**2 - l*(m - l) - (n - m)*m
+ sage: actual == expected
True
- """
- V = K.lattice().vector_space()
-
- C_of_K = discrete_complementarity_set(K)
+ The lineality of the cone of positive operators is given by the
+ corollary in my paper::
- tensor_products = [ s.tensor_product(x) for (x,s) in C_of_K ]
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=5)
+ sage: n = K.lattice_dim()
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: L = ToricLattice(n**2)
+ sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).lineality()
+ sage: expected = n**2 - K.dim()*K.dual().dim()
+ sage: actual == expected
+ 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.
- W = VectorSpace(V.base_ring(), V.dimension()**2)
+ The cone ``K`` is proper if and only if the cone of positive
+ operators on ``K`` is proper::
- # Turn our matrices into long vectors...
- vectors = [ W(m.list()) for m in tensor_products ]
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=5)
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: L = ToricLattice(K.lattice_dim()**2)
+ sage: pi_cone = Cone([p.list() for p in pi_of_K], lattice=L)
+ sage: K.is_proper() == pi_cone.is_proper()
+ True
+ """
+ # Matrices are not vectors in Sage, so we have to convert them
+ # to vectors explicitly before we can find a basis. We need these
+ # two values to construct the appropriate "long vector" space.
+ F = K.lattice().base_field()
+ n = K.lattice_dim()
- # Vector space representation of Lyapunov-like matrices
- # (i.e. vec(L) where L is Luapunov-like).
- LL_vector = W.span(vectors).complement()
+ tensor_products = [ s.tensor_product(x) for x in K for s in K.dual() ]
- # Now construct an ambient MatrixSpace in which to stick our
- # transformations.
- M = MatrixSpace(V.base_ring(), V.dimension())
+ # Convert those tensor products to long vectors.
+ W = VectorSpace(F, n**2)
+ vectors = [ W(tp.list()) for tp in tensor_products ]
- matrix_basis = [ M(v.list()) for v in LL_vector.basis() ]
+ # Create the *dual* cone of the positive operators, expressed as
+ # long vectors..
+ pi_dual = Cone(vectors, ToricLattice(W.dimension()))
- return matrix_basis
+ # Now compute the desired cone from its dual...
+ pi_cone = pi_dual.dual()
+ # And finally convert its rays back to matrix representations.
+ M = MatrixSpace(F, n)
+ return [ M(v.list()) for v in pi_cone.rays() ]
-def lyapunov_rank(K):
+def Z_transformation_gens(K):
r"""
- Compute the Lyapunov (or bilinearity) rank of this cone.
-
- The Lyapunov rank of a cone can be thought of in (mainly) two ways:
-
- 1. The dimension of the Lie algebra of the automorphism group of the
- cone.
-
- 2. The dimension of the linear space of all Lyapunov-like
- transformations on the cone.
-
- INPUT:
-
- A closed, convex polyhedral cone.
+ Compute generators of the cone of Z-transformations on this cone.
OUTPUT:
- An integer representing the Lyapunov rank of the cone. If the
- dimension of the ambient vector space is `n`, then the Lyapunov rank
- will be between `1` and `n` inclusive; however a rank of `n-1` is
- not possible (see the first reference).
-
- .. note::
-
- In the references, the cones are always assumed to be proper. We
- do not impose this restriction.
-
- .. seealso::
-
- :meth:`is_proper`
-
- ALGORITHM:
-
- The codimension formula from the second reference is used. We find
- all pairs `(x,s)` in the complementarity set of `K` such that `x`
- and `s` are rays of our cone. It is known that these vectors are
- sufficient to apply the codimension formula. Once we have all such
- pairs, we "brute force" the codimension formula by finding all
- linearly-independent `xs^{T}`.
-
- REFERENCES:
-
- .. [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.
-
- .. [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.
+ 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.
EXAMPLES:
- The nonnegative orthant in `\mathbb{R}^{n}` always has rank `n`
- [Rudolf et al.]_::
-
- sage: positives = Cone([(1,)])
- sage: lyapunov_rank(positives)
- 1
- sage: quadrant = Cone([(1,0), (0,1)])
- sage: lyapunov_rank(quadrant)
- 2
- sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)])
- sage: lyapunov_rank(octant)
- 3
-
- The 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 [Rudolf et al.]_::
-
- 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/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: K = L31.cartesian_product(octant)
- sage: lyapunov_rank(K) == lyapunov_rank(L31) + lyapunov_rank(octant)
+ 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_transformation_gens(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_transformation_gens(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 [Rudolf et al.]_.
- 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_transformation_gens(K)
+ []
- The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
- itself [Rudolf et al.]_::
+ 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
-
- TESTS:
-
- The Lyapunov rank should be additive on a product of proper cones
- [Rudolf et al.]_::
-
- 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)
+ sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
+ sage: zs = span([ vector(z.list()) for z in Z_transformation_gens(K) ])
+ sage: zs == lls
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
+ TESTS:
- Make sure we exercise the non-strictly-convex/non-solid case::
+ The Z-property is possessed by every Z-transformation::
sage: set_random_seed()
- sage: K = random_cone(max_dim=10, strictly_convex=False, solid=False)
- sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+ sage: K = random_cone(max_ambient_dim=6)
+ sage: Z_of_K = Z_transformation_gens(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 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::
+ The lineality space of Z is LL::
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
+ 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_transformation_gens(K) ])
+ sage: z_cone.linear_subspace() == lls
True
- sage: b == n-1
- False
- In fact [Orlitzky/Gowda]_, no closed convex polyhedral cone can have
- Lyapunov rank `n-1` in `n` dimensions::
+ And thus, the lineality of Z is the Lyapunov rank::
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
+ sage: K = random_cone(max_ambient_dim=6)
+ sage: Z_of_K = Z_transformation_gens(K)
+ sage: L = ToricLattice(K.lattice_dim()**2)
+ sage: z_cone = Cone([ z.list() for z in Z_of_K ], lattice=L)
+ sage: z_cone.lineality() == K.lyapunov_rank()
True
- The Lyapunov rank of a proper cone is just the dimension of ``LL(K)``::
+ The lineality spaces of pi-star and Z-star are equal:
sage: set_random_seed()
- sage: K = random_cone(max_dim=10, strictly_convex=True, solid=True)
- sage: lyapunov_rank(K) == len(LL(K))
+ sage: K = random_cone(max_ambient_dim=5)
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: Z_of_K = Z_transformation_gens(K)
+ sage: L = ToricLattice(K.lattice_dim()**2)
+ sage: pi_star = Cone([p.list() for p in pi_of_K], lattice=L).dual()
+ sage: z_star = Cone([ z.list() for z in Z_of_K], lattice=L).dual()
+ sage: pi_star.linear_subspace() == z_star.linear_subspace()
True
-
"""
- K_orig = K
- beta = 0
-
- m = K.dim()
+ # Matrices are not vectors in Sage, so we have to convert them
+ # to vectors explicitly before we can find a basis. We need these
+ # two values to construct the appropriate "long vector" space.
+ F = K.lattice().base_field()
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()) ]
+ # These tensor products contain generators for the dual cone of
+ # the cross-positive transformations.
+ tensor_products = [ s.tensor_product(x)
+ for (x,s) in K.discrete_complementarity_set() ]
- #if Ks != Js:
- # print [ list(r) for r in K_orig.rays() ]
-
- # Lemma 3
- beta += m * l
+ # Turn our matrices into long vectors...
+ W = VectorSpace(F, n**2)
+ vectors = [ W(m.list()) for m in tensor_products ]
- beta += len(LL(K))
- return beta
+ # Create the *dual* cone of the cross-positive operators,
+ # expressed as long vectors..
+ Sigma_dual = Cone(vectors, lattice=ToricLattice(W.dimension()))
+
+ # Now compute the desired cone from its dual...
+ Sigma_cone = Sigma_dual.dual()
+
+ # 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(F, n)
+ return [ -M(v.list()) for v in Sigma_cone.rays() ]
+
+
+def Z_cone(K):
+ gens = Z_transformation_gens(K)
+ L = None
+ if len(gens) == 0:
+ L = ToricLattice(0)
+ return Cone([ g.list() for g in gens ], lattice=L)
+
+def pi_cone(K):
+ gens = positive_operator_gens(K)
+ L = None
+ if len(gens) == 0:
+ L = ToricLattice(0)
+ return Cone([ g.list() for g in gens ], lattice=L)
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