-# 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 _restrict_to_space(K, W):
+def is_lyapunov_like(L,K):
r"""
- Restrict this cone (up to linear isomorphism) to a vector subspace.
-
- This operation not only restricts the cone to a subspace of its
- ambient space, but also represents the rays of the cone in a new
- (smaller) lattice corresponding to the subspace. The resulting cone
- will be linearly isomorphic **but not equal** to the desired
- restriction, since it has likely undergone a change of basis.
-
- To explain the difficulty, consider the cone ``K = Cone([(1,1,1)])``
- having a single ray. The span of ``K`` is a one-dimensional subspace
- containing ``K``, yet we have no way to perform operations like
- :meth:`dual` in the subspace. To represent ``K`` in the space
- ``K.span()``, we must perform a change of basis and write its sole
- ray as ``(1,0,0)``. Now the restricted ``Cone([(1,)])`` is linearly
- isomorphic (but of course not equal) to ``K`` interpreted as living
- in ``K.span()``.
+ Determine whether or not ``L`` is Lyapunov-like on ``K``.
+
+ We say that ``L`` is Lyapunov-like on ``K`` if `\left\langle
+ L\left\lparenx\right\rparen,s\right\rangle = 0` for all pairs
+ `\left\langle x,s \right\rangle` in the complementarity set of
+ ``K``. It is known [Orlitzky]_ that this property need only be
+ checked for generators of ``K`` and its dual.
+
+ There are faster ways of checking this property. For example, we
+ could compute a `lyapunov_like_basis` of the cone, and then test
+ whether or not the given matrix is contained in the span of that
+ basis. The value of this function is that it works on symbolic
+ matrices.
INPUT:
- - ``W`` -- The subspace into which this cone will be restricted.
+ - ``L`` -- A linear transformation or matrix.
+
+ - ``K`` -- A polyhedral closed convex cone.
OUTPUT:
- A new cone in a sublattice corresponding to ``W``.
+ ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``,
+ and ``False`` otherwise.
+
+ .. WARNING::
+
+ If this function returns ``True``, then ``L`` is Lyapunov-like
+ on ``K``. However, if ``False`` is returned, that could mean one
+ of two things. The first is that ``L`` is definitely not
+ Lyapunov-like on ``K``. The second is more of an "I don't know"
+ answer, returned (for example) if we cannot prove that an inner
+ product is zero.
REFERENCES:
EXAMPLES:
- Restricting a solid cone to its own span returns a cone linearly
- isomorphic to the original::
-
- sage: K = Cone([(1,2,3),(-1,1,0),(9,0,-2)])
- sage: K.is_solid()
- True
- sage: _restrict_to_space(K, K.span()).rays()
- N(-1, 1, 0),
- N( 1, 0, 0),
- N( 9, -6, -1)
- in 3-d lattice N
-
- A single ray restricted to its own span has the same representation
- regardless of the ambient space::
-
- sage: K2 = Cone([(1,0)])
- sage: K2_S = _restrict_to_space(K2, K2.span()).rays()
- sage: K2_S
- N(1)
- in 1-d lattice N
- sage: K3 = Cone([(1,1,1)])
- sage: K3_S = _restrict_to_space(K3, K3.span()).rays()
- sage: K3_S
- N(1)
- in 1-d lattice N
- sage: K2_S == K3_S
- True
-
- Restricting to a trivial space gives the trivial cone::
-
- sage: K = Cone([(8,3,-1,0),(9,2,2,0),(-4,6,7,0)])
- sage: trivial_space = K.lattice().vector_space().span([])
- sage: _restrict_to_space(K, trivial_space)
- 0-d cone in 0-d lattice N
-
- TESTS:
-
- Restricting a cone to its own span results in a solid cone::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 8)
- sage: K_S = _restrict_to_space(K, K.span())
- sage: K_S.is_solid()
- True
-
- Restricting a cone to its own span should not affect the number of
- rays in the cone::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 8)
- sage: K_S = _restrict_to_space(K, K.span())
- sage: K.nrays() == K_S.nrays()
- True
-
- Restricting a cone to its own span should not affect its dimension::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 8)
- sage: K_S = _restrict_to_space(K, K.span())
- sage: K.dim() == K_S.dim()
- True
-
- Restricting a cone to its own span should not affects its lineality::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 8)
- sage: K_S = _restrict_to_space(K, K.span())
- sage: K.lineality() == K_S.lineality()
- True
-
- Restricting a cone to its own span should not affect the number of
- facets it has::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 8)
- sage: K_S = _restrict_to_space(K, K.span())
- sage: len(K.facets()) == len(K_S.facets())
- True
-
- Restricting a solid cone to its own span is a linear isomorphism and
- should not affect the dimension of its ambient space::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 8, solid = True)
- sage: K_S = _restrict_to_space(K, K.span())
- sage: K.lattice_dim() == K_S.lattice_dim()
- True
-
- Restricting a solid cone to its own span is a linear isomorphism
- that establishes a one-to-one correspondence of discrete
- complementarity sets::
+ The identity is always Lyapunov-like in a nontrivial space::
sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 8, solid = True)
- sage: K_S = _restrict_to_space(K, K.span())
- sage: dcs_K = K.discrete_complementarity_set()
- sage: dcs_K_S = K_S.discrete_complementarity_set()
- sage: len(dcs_K) == len(dcs_K_S)
+ 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
- Restricting a solid cone to its own span is a linear isomorphism
- under which the Lyapunov rank (the length of a Lyapunov-like basis)
- is invariant::
+ As is the "zero" transformation::
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 8, solid = True)
- sage: K_S = _restrict_to_space(K, K.span())
- sage: len(K.lyapunov_like_basis()) == len(K_S.lyapunov_like_basis())
+ 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
- If we restrict a cone to a subspace of its span, the resulting cone
- should have the same dimension as the space we restricted it to::
+ Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like
+ on ``K``::
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 8)
- sage: W_basis = random_sublist(K.rays(), 0.5)
- sage: W = K.lattice().vector_space().span(W_basis)
- sage: K_W = _restrict_to_space(K, W)
- sage: K_W.lattice_dim() == W.dimension()
+ 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
- Through a series of restrictions, any closed convex cone can be
- reduced to a cartesian product with a proper factor [Orlitzky]_::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 8)
- sage: K_S = _restrict_to_space(K, K.span())
- sage: K_SP = _restrict_to_space(K_S, K_S.dual().span())
- sage: K_SP.is_proper()
- True
"""
- # We want to intersect ``K`` with ``W``. An easy way to do this is
- # via cone intersection, so we turn the space ``W`` into a cone.
- W_cone = Cone(W.basis() + [-b for b in W.basis()], lattice=K.lattice())
- K = K.intersection(W_cone)
-
- # We've already intersected K with W, so every generator of K
- # should belong to W now.
- K_W_rays = [ W.coordinate_vector(r) for r in K.rays() ]
-
- L = ToricLattice(W.dimension())
- return Cone(K_W_rays, lattice=L)
+ return all([(L*x).inner_product(s) == 0
+ for (x,s) in K.discrete_complementarity_set()])
-def lyapunov_rank(K):
+def positive_operator_gens(K1, K2 = None):
r"""
- Compute the Lyapunov rank of this cone.
+ Compute generators of the cone of positive operators on this cone. A
+ linear operator on a cone is positive if the image of the cone under
+ the operator is a subset of the cone. This concept can be extended
+ to two cones, where the image of the first cone under a positive
+ operator is a subset of the second cone.
+
+ INPUT:
- The Lyapunov rank of a cone is the dimension of the space of its
- Lyapunov-like transformations -- that is, the length of a
- :meth:`lyapunov_like_basis`. Equivalently, the Lyapunov rank is the
- dimension of the Lie algebra of the automorphism group of the cone.
+ - ``K2`` -- (default: ``K1``) the codomain cone; the image of this
+ cone under the returned operators is a subset of ``K2``.
OUTPUT:
- A nonnegative integer representing the Lyapunov rank of this cone.
-
- If the ambient space is trivial, the Lyapunov rank will be zero.
- Otherwise, if the dimension of the ambient vector space is `n`, then
- the resulting Lyapunov rank will be between `1` and `n` inclusive. A
- Lyapunov rank of `n-1` is not possible [Orlitzky]_.
+ A list of `m`-by-``n`` matrices where ``m == K2.lattice_dim()`` and
+ ``n == K1.lattice_dim()``. Each matrix ``P`` in the list should have
+ the property that ``P*x`` is an element of ``K2`` whenever ``x`` is
+ an element of ``K1``. Moreover, any nonnegative linear combination of
+ these matrices shares the same property.
- ALGORITHM:
+ .. SEEALSO::
- 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}`.
+ :meth:`cross_positive_operator_gens`, :meth:`Z_operator_gens`,
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.
-
- M. Orlitzky. The Lyapunov rank of an improper cone.
- http://www.optimization-online.org/DB_HTML/2015/10/5135.html
-
- 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.
+ .. [Tam]
+ B.-S. Tam.
+ Some results of polyhedral cones and simplicial cones.
+ Linear and Multilinear Algebra, 4:4 (1977) 281--284.
EXAMPLES:
- The nonnegative orthant in `\mathbb{R}^{n}` always has rank `n`
- [Rudolf]_::
-
- 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]_::
-
- 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]_::
+ Positive operators on the nonnegative orthant are nonnegative matrices::
- sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)])
- sage: lyapunov_rank(L31)
- 1
+ sage: K = Cone([(1,)])
+ sage: positive_operator_gens(K)
+ [[1]]
- Likewise for the `L^{3}_{\infty}` cone [Rudolf]_::
+ sage: K = Cone([(1,0),(0,1)])
+ sage: positive_operator_gens(K)
+ [
+ [1 0] [0 1] [0 0] [0 0]
+ [0 0], [0 0], [1 0], [0 1]
+ ]
- sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)])
- sage: lyapunov_rank(L3infty)
- 1
+ The trivial cone in a trivial space has no positive operators::
- A single ray in `n` dimensions should have Lyapunov rank `n^{2} - n
- + 1` [Orlitzky]_::
+ sage: K = Cone([], ToricLattice(0))
+ sage: positive_operator_gens(K)
+ []
- sage: K = Cone([(1,0,0,0,0)])
- sage: lyapunov_rank(K)
- 21
- sage: K.lattice_dim()**2 - K.lattice_dim() + 1
- 21
+ Every operator is positive on the trivial cone::
- A subspace (of dimension `m`) in `n` dimensions should have a
- Lyapunov rank of `n^{2} - m\left(n - m)` [Orlitzky]_::
+ sage: K = Cone([(0,)])
+ sage: positive_operator_gens(K)
+ [[1], [-1]]
- 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
+ sage: K = Cone([(0,0)])
+ sage: K.is_trivial()
+ True
+ sage: positive_operator_gens(K)
+ [
+ [1 0] [-1 0] [0 1] [ 0 -1] [0 0] [ 0 0] [0 0] [ 0 0]
+ [0 0], [ 0 0], [0 0], [ 0 0], [1 0], [-1 0], [0 1], [ 0 -1]
+ ]
- The Lyapunov rank should be additive on a product of proper cones
- [Rudolf]_::
+ 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: 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)
+ sage: K = Cone([(1,),(-1,)])
+ sage: K.is_full_space()
True
+ sage: positive_operator_gens(K)
+ [[1], [-1]]
- Two isomorphic cones should have the same Lyapunov rank [Rudolf]_.
- The cone ``K`` in the following example is isomorphic to the nonnegative
- octant in `\mathbb{R}^{3}`::
-
- sage: K = Cone([(1,2,3), (-1,1,0), (1,0,6)])
- sage: lyapunov_rank(K)
- 3
+ sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
+ sage: K.is_full_space()
+ True
+ sage: positive_operator_gens(K)
+ [
+ [1 0] [-1 0] [0 1] [ 0 -1] [0 0] [ 0 0] [0 0] [ 0 0]
+ [0 0], [ 0 0], [0 0], [ 0 0], [1 0], [-1 0], [0 1], [ 0 -1]
+ ]
- The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
- itself [Rudolf]_::
+ A non-obvious application is to find the positive operators on the
+ right half-plane::
- sage: K = Cone([(2,2,4), (-1,9,0), (2,0,6)])
- sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
- True
+ sage: K = Cone([(1,0),(0,1),(0,-1)])
+ sage: positive_operator_gens(K)
+ [
+ [1 0] [0 0] [ 0 0] [0 0] [ 0 0]
+ [0 0], [1 0], [-1 0], [0 1], [ 0 -1]
+ ]
TESTS:
- The Lyapunov rank should be additive on a product of proper cones
- [Rudolf]_::
+ Each positive operator generator should send the generators of one
+ cone into the other cone::
sage: set_random_seed()
- sage: K1 = random_cone(max_ambient_dim=8,
- ....: strictly_convex=True,
- ....: solid=True)
- sage: K2 = random_cone(max_ambient_dim=8,
- ....: strictly_convex=True,
- ....: solid=True)
- sage: K = K1.cartesian_product(K2)
- sage: lyapunov_rank(K) == lyapunov_rank(K1) + lyapunov_rank(K2)
+ sage: K1 = random_cone(max_ambient_dim=4)
+ sage: K2 = random_cone(max_ambient_dim=4)
+ sage: pi_K1_K2 = positive_operator_gens(K1,K2)
+ sage: all([ K2.contains(P*x) for P in pi_K1_K2 for x in K1 ])
True
- The Lyapunov rank is invariant under a linear isomorphism
- [Orlitzky]_::
+ Each positive operator generator should send a random element of one
+ cone into the other cone::
- sage: K1 = random_cone(max_ambient_dim = 8)
- sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular')
- sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice())
- sage: lyapunov_rank(K1) == lyapunov_rank(K2)
+ sage: set_random_seed()
+ sage: K1 = random_cone(max_ambient_dim=4)
+ sage: K2 = random_cone(max_ambient_dim=4)
+ sage: pi_K1_K2 = positive_operator_gens(K1,K2)
+ sage: all([ K2.contains(P*K1.random_element(QQ)) for P in pi_K1_K2 ])
True
- The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
- itself [Rudolf]_::
+ A random element of the positive operator cone should send the
+ generators of one cone into the other cone::
sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8)
- sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
- True
+ sage: K1 = random_cone(max_ambient_dim=4)
+ sage: K2 = random_cone(max_ambient_dim=4)
+ sage: pi_K1_K2 = positive_operator_gens(K1,K2)
+ sage: L = ToricLattice(K1.lattice_dim() * K2.lattice_dim())
+ sage: pi_cone = Cone([ g.list() for g in pi_K1_K2 ],
+ ....: lattice=L,
+ ....: check=False)
+ sage: P = matrix(K2.lattice_dim(),
+ ....: K1.lattice_dim(),
+ ....: pi_cone.random_element(QQ).list())
+ sage: all([ K2.contains(P*x) for x in K1 ])
+ True
+
+ A random element of the positive operator cone should send a random
+ element of one cone into the other cone::
- 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: K1 = random_cone(max_ambient_dim=4)
+ sage: K2 = random_cone(max_ambient_dim=4)
+ sage: pi_K1_K2 = positive_operator_gens(K1,K2)
+ sage: L = ToricLattice(K1.lattice_dim() * K2.lattice_dim())
+ sage: pi_cone = Cone([ g.list() for g in pi_K1_K2 ],
+ ....: lattice=L,
+ ....: check=False)
+ sage: P = matrix(K2.lattice_dim(),
+ ....: K1.lattice_dim(),
+ ....: pi_cone.random_element(QQ).list())
+ sage: K2.contains(P*K1.random_element(ring=QQ))
+ True
+
+ The lineality space of the dual of the cone of positive operators
+ can be computed from the lineality spaces of the cone and its dual::
sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8,
- ....: strictly_convex=True,
- ....: solid=True)
- sage: b = lyapunov_rank(K)
- sage: n = K.lattice_dim()
- sage: (n == 0 or 1 <= b) and b <= n
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: L = ToricLattice(K.lattice_dim()**2)
+ sage: pi_cone = Cone([ g.list() for g in pi_of_K ],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = pi_cone.dual().linear_subspace()
+ sage: U1 = [ vector((s.tensor_product(x)).list())
+ ....: for x in K.lines()
+ ....: for s in K.dual() ]
+ sage: U2 = [ vector((s.tensor_product(x)).list())
+ ....: for x in K
+ ....: for s in K.dual().lines() ]
+ sage: expected = pi_cone.lattice().vector_space().span(U1 + U2)
+ sage: actual == expected
True
- sage: b == n-1
- False
- In fact [Orlitzky]_, no closed convex polyhedral cone can have
- Lyapunov rank `n-1` in `n` dimensions::
+ The lineality of the dual of the cone of positive operators
+ is known from its lineality space::
sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8)
- sage: b = lyapunov_rank(K)
+ sage: K = random_cone(max_ambient_dim=4)
sage: n = K.lattice_dim()
- sage: b == n-1
- False
+ sage: m = K.dim()
+ sage: l = K.lineality()
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: L = ToricLattice(n**2)
+ sage: pi_cone = Cone([p.list() for p in pi_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = pi_cone.dual().lineality()
+ sage: expected = l*(m - l) + m*(n - m)
+ sage: actual == expected
+ True
- The calculation of the Lyapunov rank of an improper cone can be
- reduced to that of a proper cone [Orlitzky]_::
+ 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_ambient_dim=8)
- sage: actual = lyapunov_rank(K)
- sage: K_S = _restrict_to_space(K, K.span())
- sage: K_SP = _restrict_to_space(K_S.dual(), K_S.dual().span()).dual()
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: n = K.lattice_dim()
+ sage: m = K.dim()
sage: l = K.lineality()
- sage: c = K.codim()
- sage: expected = lyapunov_rank(K_SP) + K.dim()*(l + c) + c**2
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: L = ToricLattice(n**2)
+ sage: pi_cone = Cone([p.list() for p in pi_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = pi_cone.dim()
+ sage: expected = n**2 - l*(m - l) - (n - m)*m
sage: actual == expected
True
- The Lyapunov rank of a cone is the size of a :meth:`lyapunov_like_basis`::
+ The trivial cone, full space, and half-plane all give rise to the
+ expected dimensions::
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8)
- sage: lyapunov_rank(K) == len(K.lyapunov_like_basis())
+ sage: n = ZZ.random_element().abs()
+ sage: K = Cone([[0] * n], ToricLattice(n))
+ sage: K.is_trivial()
+ True
+ sage: L = ToricLattice(n^2)
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: pi_cone = Cone([p.list() for p in pi_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = pi_cone.dim()
+ sage: actual == n^2
+ True
+ sage: K = K.dual()
+ sage: K.is_full_space()
+ True
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: pi_cone = Cone([p.list() for p in pi_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = pi_cone.dim()
+ sage: actual == n^2
+ True
+ sage: K = Cone([(1,0),(0,1),(0,-1)])
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: actual = Cone([p.list() for p in pi_of_K], check=False).dim()
+ sage: actual == 3
True
- We can make an imperfect cone perfect by adding a slack variable
- (a Theorem in [Orlitzky]_)::
+ The lineality of the cone of positive operators follows from the
+ description of its generators::
sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8,
- ....: strictly_convex=True,
- ....: solid=True)
- sage: L = ToricLattice(K.lattice_dim() + 1)
- sage: K = Cone([ r.list() + [0] for r in K.rays() ], lattice=L)
- sage: lyapunov_rank(K) >= K.lattice_dim()
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: n = K.lattice_dim()
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: L = ToricLattice(n**2)
+ sage: pi_cone = Cone([p.list() for p in pi_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = pi_cone.lineality()
+ sage: expected = n**2 - K.dim()*K.dual().dim()
+ sage: actual == expected
True
- """
- beta = 0 # running tally of the Lyapunov rank
+ The trivial cone, full space, and half-plane all give rise to the
+ expected linealities::
- m = K.dim()
- n = K.lattice_dim()
- l = K.lineality()
+ sage: n = ZZ.random_element().abs()
+ sage: K = Cone([[0] * n], ToricLattice(n))
+ sage: K.is_trivial()
+ True
+ sage: L = ToricLattice(n^2)
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: pi_cone = Cone([p.list() for p in pi_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = pi_cone.lineality()
+ sage: actual == n^2
+ True
+ sage: K = K.dual()
+ sage: K.is_full_space()
+ True
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: pi_cone = Cone([p.list() for p in pi_of_K], lattice=L)
+ sage: pi_cone.lineality() == n^2
+ True
+ sage: K = Cone([(1,0),(0,1),(0,-1)])
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: pi_cone = Cone([p.list() for p in pi_of_K], check=False)
+ sage: actual = pi_cone.lineality()
+ sage: actual == 2
+ True
- if m < n:
- # K is not solid, restrict to its span.
- K = _restrict_to_space(K, K.span())
+ A cone is proper if and only if its cone of positive operators
+ is proper::
- # Non-solid reduction lemma.
- beta += (n - m)*n
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=4)
+ 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,
+ ....: check=False)
+ sage: K.is_proper() == pi_cone.is_proper()
+ True
- if l > 0:
- # K is not pointed, restrict to the span of its dual. Uses a
- # proposition from our paper, i.e. this is equivalent to K =
- # _rho(K.dual()).dual().
- K = _restrict_to_space(K, K.dual().span())
+ The positive operators of a permuted cone can be obtained by
+ conjugation::
- # Non-pointed reduction lemma.
- beta += l * m
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: L = ToricLattice(K.lattice_dim()**2)
+ sage: p = SymmetricGroup(K.lattice_dim()).random_element().matrix()
+ sage: pK = Cone([ p*k for k in K ], K.lattice(), check=False)
+ sage: pi_of_pK = positive_operator_gens(pK)
+ sage: actual = Cone([t.list() for t in pi_of_pK],
+ ....: lattice=L,
+ ....: check=False)
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: expected = Cone([(p*t*p.inverse()).list() for t in pi_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual.is_equivalent(expected)
+ True
+
+ A transformation is positive on a cone if and only if its adjoint is
+ positive on the dual of that cone::
- beta += len(K.lyapunov_like_basis())
- return beta
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: F = K.lattice().vector_space().base_field()
+ sage: n = K.lattice_dim()
+ sage: L = ToricLattice(n**2)
+ sage: W = VectorSpace(F, n**2)
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: pi_of_K_star = positive_operator_gens(K.dual())
+ sage: pi_cone = Cone([p.list() for p in pi_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: pi_star = Cone([p.list() for p in pi_of_K_star],
+ ....: lattice=L,
+ ....: check=False)
+ sage: M = MatrixSpace(F, n)
+ sage: L = M(pi_cone.random_element(ring=QQ).list())
+ sage: pi_star.contains(W(L.transpose().list()))
+ True
+
+ sage: L = W.random_element()
+ sage: L_star = W(M(L.list()).transpose().list())
+ sage: pi_cone.contains(L) == pi_star.contains(L_star)
+ True
+
+ The Lyapunov rank of the positive operator cone is the product of
+ the Lyapunov ranks of the associated cones if they're all proper::
+
+ sage: K1 = random_cone(max_ambient_dim=4,
+ ....: strictly_convex=True,
+ ....: solid=True)
+ sage: K2 = random_cone(max_ambient_dim=4,
+ ....: strictly_convex=True,
+ ....: solid=True)
+ sage: pi_K1_K2 = positive_operator_gens(K1,K2)
+ sage: L = ToricLattice(K1.lattice_dim() * K2.lattice_dim())
+ sage: pi_cone = Cone([ g.list() for g in pi_K1_K2 ],
+ ....: lattice=L,
+ ....: check=False)
+ sage: beta1 = K1.lyapunov_rank()
+ sage: beta2 = K2.lyapunov_rank()
+ sage: pi_cone.lyapunov_rank() == beta1*beta2
+ True
+ The Lyapunov-like operators on a proper polyhedral positive operator
+ cone can be computed from the Lyapunov-like operators on the cones
+ with respect to which the operators are positive::
+ sage: K1 = random_cone(max_ambient_dim=4,
+ ....: strictly_convex=True,
+ ....: solid=True)
+ sage: K2 = random_cone(max_ambient_dim=4,
+ ....: strictly_convex=True,
+ ....: solid=True)
+ sage: pi_K1_K2 = positive_operator_gens(K1,K2)
+ sage: F = K1.lattice().base_field()
+ sage: m = K1.lattice_dim()
+ sage: n = K2.lattice_dim()
+ sage: L = ToricLattice(m*n)
+ sage: M1 = MatrixSpace(F, m, m)
+ sage: M2 = MatrixSpace(F, n, n)
+ sage: LL_K1 = [ M1(x.list()) for x in K1.dual().lyapunov_like_basis() ]
+ sage: LL_K2 = [ M2(x.list()) for x in K2.lyapunov_like_basis() ]
+ sage: tps = [ s.tensor_product(x) for x in LL_K1 for s in LL_K2 ]
+ sage: W = VectorSpace(F, (m**2)*(n**2))
+ sage: expected = span(F, [ W(x.list()) for x in tps ])
+ sage: pi_cone = Cone([p.list() for p in pi_K1_K2],
+ ....: lattice=L,
+ ....: check=False)
+ sage: LL_pi = pi_cone.lyapunov_like_basis()
+ sage: actual = span(F, [ W(x.list()) for x in LL_pi ])
+ sage: actual == expected
+ True
-def is_lyapunov_like(L,K):
- r"""
- Determine whether or not ``L`` is Lyapunov-like on ``K``.
+ """
+ if K2 is None:
+ K2 = K1
+
+ # 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 = K1.lattice().base_field()
+ n = K1.lattice_dim()
+ m = K2.lattice_dim()
+
+ tensor_products = [ s.tensor_product(x) for x in K1 for s in K2.dual() ]
+
+ # Convert those tensor products to long vectors.
+ W = VectorSpace(F, n*m)
+ vectors = [ W(tp.list()) for tp in tensor_products ]
+
+ check = True
+ if K1.is_proper() and K2.is_proper():
+ # All of the generators involved are extreme vectors and
+ # therefore minimal. If this cone is neither solid nor
+ # strictly convex, then the tensor product of ``s`` and ``x``
+ # is the same as that of ``-s`` and ``-x``. However, as a
+ # /set/, ``tensor_products`` may still be minimal.
+ check = False
+
+ # Create the dual cone of the positive operators, expressed as
+ # long vectors.
+ pi_dual = Cone(vectors, ToricLattice(W.dimension()), check=check)
- 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.
+ # Now compute the desired cone from its dual...
+ pi_cone = pi_dual.dual()
- INPUT:
+ # And finally convert its rays back to matrix representations.
+ M = MatrixSpace(F, m, n)
+ return [ M(v.list()) for v in pi_cone ]
- - ``L`` -- A linear transformation or matrix.
- - ``K`` -- A polyhedral closed convex cone.
+def cross_positive_operator_gens(K):
+ r"""
+ Compute generators of the cone of cross-positive operators on this
+ cone.
OUTPUT:
- ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``,
- and ``False`` otherwise.
-
- .. WARNING::
-
- 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.
+ 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 of
+ this cone's :meth:`discrete_complementarity_set`. Moreover, any
+ conic (nonnegative linear) combination of these matrices shares the
+ same property.
- REFERENCES:
+ .. SEEALSO::
- M. Orlitzky. The Lyapunov rank of an improper cone.
- http://www.optimization-online.org/DB_HTML/2015/10/5135.html
+ :meth:`positive_operator_gens`, :meth:`Z_operator_gens`,
EXAMPLES:
- The identity is always Lyapunov-like in a nontrivial space::
+ Cross-positive operators on the nonnegative orthant are negations
+ of Z-matrices; that is, matrices whose off-diagonal elements are
+ nonnegative::
- sage: set_random_seed()
- sage: K = random_cone(min_ambient_dim = 1, max_rays = 8)
- sage: L = identity_matrix(K.lattice_dim())
- sage: is_lyapunov_like(L,K)
+ sage: K = Cone([(1,0),(0,1)])
+ sage: cross_positive_operator_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([ c[i][j] >= 0 for c in cross_positive_operator_gens(K)
+ ....: for i in range(c.nrows())
+ ....: for j in range(c.ncols())
+ ....: if i != j ])
True
- As is the "zero" transformation::
+ The trivial cone in a trivial space has no cross-positive operators::
- sage: K = random_cone(min_ambient_dim = 1, max_rays = 5)
- sage: R = K.lattice().vector_space().base_ring()
- sage: L = zero_matrix(R, K.lattice_dim())
- sage: is_lyapunov_like(L,K)
- True
+ sage: K = Cone([], ToricLattice(0))
+ sage: cross_positive_operator_gens(K)
+ []
- Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like
- on ``K``::
+ Every operator is a cross-positive operator on the ambient vector
+ space::
- sage: K = random_cone(min_ambient_dim = 1, max_rays = 5)
- sage: all([ is_lyapunov_like(L,K) for L in K.lyapunov_like_basis() ])
+ sage: K = Cone([(1,),(-1,)])
+ sage: K.is_full_space()
True
+ sage: cross_positive_operator_gens(K)
+ [[1], [-1]]
- """
- return all([(L*x).inner_product(s) == 0
- for (x,s) in K.discrete_complementarity_set()])
-
-
-def random_element(K):
- r"""
- Return a random element of ``K`` from its ambient vector space.
-
- ALGORITHM:
-
- The cone ``K`` is specified in terms of its generators, so that
- ``K`` is equal to the convex conic combination of those generators.
- To choose a random element of ``K``, we assign random nonnegative
- coefficients to each generator of ``K`` and construct a new vector
- from the scaled rays.
+ sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
+ sage: K.is_full_space()
+ True
+ sage: cross_positive_operator_gens(K)
+ [
+ [1 0] [-1 0] [0 1] [ 0 -1] [0 0] [ 0 0] [0 0] [ 0 0]
+ [0 0], [ 0 0], [0 0], [ 0 0], [1 0], [-1 0], [0 1], [ 0 -1]
+ ]
- A vector, rather than a ray, is returned so that the element may
- have non-integer coordinates. Thus the element may have an
- arbitrarily small norm.
+ A non-obvious application is to find the cross-positive operators
+ on the right half-plane::
- EXAMPLES:
+ sage: K = Cone([(1,0),(0,1),(0,-1)])
+ sage: cross_positive_operator_gens(K)
+ [
+ [1 0] [-1 0] [0 0] [ 0 0] [0 0] [ 0 0]
+ [0 0], [ 0 0], [1 0], [-1 0], [0 1], [ 0 -1]
+ ]
- A random element of the trivial cone is zero::
+ Cross-positive operators on a subspace are Lyapunov-like and
+ vice-versa::
- sage: set_random_seed()
- sage: K = Cone([], ToricLattice(0))
- sage: random_element(K)
- ()
- sage: K = Cone([(0,)])
- sage: random_element(K)
- (0)
- sage: K = Cone([(0,0)])
- sage: random_element(K)
- (0, 0)
- sage: K = Cone([(0,0,0)])
- sage: random_element(K)
- (0, 0, 0)
+ sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
+ sage: K.is_full_space()
+ True
+ sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
+ sage: cs = span([ vector(c.list()) for c in cross_positive_operator_gens(K) ])
+ sage: cs == lls
+ True
TESTS:
- Any cone should contain an element of itself::
+ The cross-positive property is possessed by every cross-positive
+ operator::
sage: set_random_seed()
- sage: K = random_cone(max_rays = 8)
- sage: K.contains(random_element(K))
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: Sigma_of_K = cross_positive_operator_gens(K)
+ sage: dcs = K.discrete_complementarity_set()
+ sage: all([(c*x).inner_product(s) >= 0 for c in Sigma_of_K
+ ....: for (x,s) in dcs])
True
- """
- V = K.lattice().vector_space()
- F = V.base_ring()
- coefficients = [ F.random_element().abs() for i in range(K.nrays()) ]
- vector_gens = map(V, K.rays())
- scaled_gens = [ coefficients[i]*vector_gens[i]
- for i in range(len(vector_gens)) ]
-
- # Make sure we return a vector. Without the coercion, we might
- # return ``0`` when ``K`` has no rays.
- v = V(sum(scaled_gens))
- return v
-
-
-def positive_operators(K):
- r"""
- Compute generators of the cone of positive operators on this cone.
+ The lineality space of the cone of cross-positive operators is the
+ space of Lyapunov-like operators::
- OUTPUT:
-
- 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 in a trivial space has no positive operators::
-
- sage: K = Cone([], ToricLattice(0))
- sage: positive_operators(K)
- []
-
- Positive operators on the nonnegative orthant are nonnegative matrices::
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: L = ToricLattice(K.lattice_dim()**2)
+ sage: Sigma_cone = Cone([ c.list() for c in cross_positive_operator_gens(K) ],
+ ....: lattice=L,
+ ....: check=False)
+ sage: ll_basis = [ vector(l.list()) for l in K.lyapunov_like_basis() ]
+ sage: lls = L.vector_space().span(ll_basis)
+ sage: Sigma_cone.linear_subspace() == lls
+ True
- sage: K = Cone([(1,)])
- sage: positive_operators(K)
- [[1]]
+ The lineality of the cross-positive operators on a cone is the
+ Lyapunov rank of that cone::
- sage: K = Cone([(1,0),(0,1)])
- sage: positive_operators(K)
- [
- [1 0] [0 1] [0 0] [0 0]
- [0 0], [0 0], [1 0], [0 1]
- ]
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: Sigma_of_K = cross_positive_operator_gens(K)
+ sage: L = ToricLattice(K.lattice_dim()**2)
+ sage: Sigma_cone = Cone([ c.list() for c in Sigma_of_K ],
+ ....: lattice=L,
+ ....: check=False)
+ sage: Sigma_cone.lineality() == K.lyapunov_rank()
+ True
- Every operator is positive on the ambient vector space::
+ The lineality spaces of the duals of the positive and cross-positive
+ operator cones are equal. From this it follows that the dimensions of
+ the cross-positive operator cone and positive operator cone are equal::
- sage: K = Cone([(1,),(-1,)])
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: Sigma_of_K = cross_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,
+ ....: check=False)
+ sage: Sigma_cone = Cone([ c.list() for c in Sigma_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: pi_cone.dim() == Sigma_cone.dim()
+ True
+ sage: pi_star = pi_cone.dual()
+ sage: sigma_star = Sigma_cone.dual()
+ sage: pi_star.linear_subspace() == sigma_star.linear_subspace()
+ True
+
+ The trivial cone, full space, and half-plane all give rise to the
+ expected dimensions::
+
+ sage: n = ZZ.random_element().abs()
+ sage: K = Cone([[0] * n], ToricLattice(n))
+ sage: K.is_trivial()
+ True
+ sage: L = ToricLattice(n^2)
+ sage: Sigma_of_K = cross_positive_operator_gens(K)
+ sage: Sigma_cone = Cone([c.list() for c in Sigma_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = Sigma_cone.dim()
+ sage: actual == n^2
+ True
+ sage: K = K.dual()
sage: K.is_full_space()
True
- sage: positive_operators(K)
- [[1], [-1]]
-
- sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
- sage: K.is_full_space()
+ sage: Sigma_of_K = cross_positive_operator_gens(K)
+ sage: Sigma_cone = Cone([ c.list() for c in Sigma_of_K ],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = Sigma_cone.dim()
+ sage: actual == n^2
+ True
+ sage: K = Cone([(1,0),(0,1),(0,-1)])
+ sage: Sigma_of_K = cross_positive_operator_gens(K)
+ sage: Sigma_cone = Cone([ c.list() for c in Sigma_of_K ], check=False)
+ sage: Sigma_cone.dim() == 3
True
- sage: positive_operators(K)
- [
- [1 0] [-1 0] [0 1] [ 0 -1] [0 0] [ 0 0] [0 0] [ 0 0]
- [0 0], [ 0 0], [0 0], [ 0 0], [1 0], [-1 0], [0 1], [ 0 -1]
- ]
- TESTS:
+ The cross-positive operators of a permuted cone can be obtained by
+ conjugation::
- A positive operator on a cone should send its generators into the cone::
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: L = ToricLattice(K.lattice_dim()**2)
+ sage: p = SymmetricGroup(K.lattice_dim()).random_element().matrix()
+ sage: pK = Cone([ p*k for k in K ], K.lattice(), check=False)
+ sage: Sigma_of_pK = cross_positive_operator_gens(pK)
+ sage: actual = Cone([t.list() for t in Sigma_of_pK],
+ ....: lattice=L,
+ ....: check=False)
+ sage: Sigma_of_K = cross_positive_operator_gens(K)
+ sage: expected = Cone([ (p*t*p.inverse()).list() for t in Sigma_of_K ],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual.is_equivalent(expected)
+ True
+
+ An operator is cross-positive on a cone if and only if its
+ adjoint is cross-positive on the dual of that cone::
- sage: K = random_cone(max_ambient_dim = 6)
- sage: pi_of_K = positive_operators(K)
- sage: all([K.contains(p*x) for p in pi_of_K for x in K.rays()])
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: F = K.lattice().vector_space().base_field()
+ sage: n = K.lattice_dim()
+ sage: L = ToricLattice(n**2)
+ sage: W = VectorSpace(F, n**2)
+ sage: Sigma_of_K = cross_positive_operator_gens(K)
+ sage: Sigma_of_K_star = cross_positive_operator_gens(K.dual())
+ sage: Sigma_cone = Cone([ p.list() for p in Sigma_of_K ],
+ ....: lattice=L,
+ ....: check=False)
+ sage: Sigma_star = Cone([ p.list() for p in Sigma_of_K_star ],
+ ....: lattice=L,
+ ....: check=False)
+ sage: M = MatrixSpace(F, n)
+ sage: L = M(Sigma_cone.random_element(ring=QQ).list())
+ sage: Sigma_star.contains(W(L.transpose().list()))
+ True
+
+ sage: L = W.random_element()
+ sage: L_star = W(M(L.list()).transpose().list())
+ sage: Sigma_cone.contains(L) == Sigma_star.contains(L_star)
True
-
"""
- # Sage doesn't think matrices are vectors, so we have to convert
- # our matrices to vectors explicitly before we can figure out how
- # many are linearly-indepenedent.
- #
- # The space W has the same base ring as V, but dimension
- # dim(V)^2. So it has the same dimension as the space of linear
- # transformations on V. In other words, it's just the right size
- # to create an isomorphism between it and our matrices.
- V = K.lattice().vector_space()
- W = VectorSpace(V.base_ring(), V.dimension()**2)
-
- tensor_products = [ s.tensor_product(x) for x in K for s in K.dual() ]
+ # 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()
+
+ # These tensor products contain generators for the dual cone of
+ # the cross-positive operators.
+ tensor_products = [ s.tensor_product(x)
+ for (x,s) in K.discrete_complementarity_set() ]
# Turn our matrices into long vectors...
+ W = VectorSpace(F, n**2)
vectors = [ W(m.list()) for m in tensor_products ]
- # Create the *dual* cone of the positive operators, expressed as
- # long vectors..
- L = ToricLattice(W.dimension())
- pi_dual = Cone(vectors, lattice=L)
+ check = True
+ if K.is_proper():
+ # All of the generators involved are extreme vectors and
+ # therefore minimal. If this cone is neither solid nor
+ # strictly convex, then the tensor product of ``s`` and ``x``
+ # is the same as that of ``-s`` and ``-x``. However, as a
+ # /set/, ``tensor_products`` may still be minimal.
+ check = False
+
+ # Create the dual cone of the cross-positive operators,
+ # expressed as long vectors.
+ Sigma_dual = Cone(vectors, lattice=ToricLattice(W.dimension()), check=check)
# Now compute the desired cone from its dual...
- pi_cone = pi_dual.dual()
+ Sigma_cone = Sigma_dual.dual()
# And finally convert its rays back to matrix representations.
- M = MatrixSpace(V.base_ring(), V.dimension())
+ M = MatrixSpace(F, n)
+ return [ M(v.list()) for v in Sigma_cone ]
- return [ M(v.list()) for v in pi_cone.rays() ]
-
-def Z_transformations(K):
+def Z_operator_gens(K):
r"""
- Compute generators of the cone of Z-transformations on this cone.
+ Compute generators of the cone of Z-operators on this cone.
+
+ The Z-operators on a cone generalize the Z-matrices over the
+ nonnegative orthant. They are simply negations of the
+ :meth:`cross_positive_operators`.
OUTPUT:
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.
+ ``(L*x).inner_product(s) <= 0`` whenever ``(x,s)`` is an element of
+ this cone's :meth:`discrete_complementarity_set`. Moreover, any
+ conic (nonnegative linear) combination of these matrices shares the
+ same property.
- EXAMPLES:
-
- Z-transformations on the nonnegative orthant are just Z-matrices.
- That is, matrices whose off-diagonal elements are nonnegative::
-
- sage: K = Cone([(1,0),(0,1)])
- sage: Z_transformations(K)
- [
- [ 0 -1] [ 0 0] [-1 0] [1 0] [ 0 0] [0 0]
- [ 0 0], [-1 0], [ 0 0], [0 0], [ 0 -1], [0 1]
- ]
- sage: K = Cone([(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)])
- sage: all([ z[i][j] <= 0 for z in Z_transformations(K)
- ....: for i in range(z.nrows())
- ....: for j in range(z.ncols())
- ....: if i != j ])
- True
+ .. SEEALSO::
- The trivial cone in a trivial space has no Z-transformations::
-
- sage: K = Cone([], ToricLattice(0))
- sage: Z_transformations(K)
- []
-
- Z-transformations on a subspace are Lyapunov-like and vice-versa::
-
- sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
- sage: K.is_full_space()
- True
- sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
- sage: zs = span([ vector(z.list()) for z in Z_transformations(K) ])
- sage: zs == lls
- True
+ :meth:`positive_operator_gens`, :meth:`cross_positive_operator_gens`,
TESTS:
- The Z-property is possessed by every Z-transformation::
+ The Z-property is possessed by every Z-operator::
sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 6)
- sage: Z_of_K = Z_transformations(K)
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: Z_of_K = Z_operator_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 lineality space of Z is LL::
-
- sage: set_random_seed()
- sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 6)
- sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
- sage: z_cone = Cone([ z.list() for z in Z_transformations(K) ])
- sage: z_cone.linear_subspace() == lls
- True
-
"""
- # Sage doesn't think matrices are vectors, so we have to convert
- # our matrices to vectors explicitly before we can figure out how
- # many are linearly-indepenedent.
- #
- # The space W has the same base ring as V, but dimension
- # dim(V)^2. So it has the same dimension as the space of linear
- # transformations on V. In other words, it's just the right size
- # to create an isomorphism between it and our matrices.
- V = K.lattice().vector_space()
- W = VectorSpace(V.base_ring(), V.dimension()**2)
-
- C_of_K = K.discrete_complementarity_set()
- tensor_products = [ s.tensor_product(x) for (x,s) in C_of_K ]
+ return [ -cp for cp in cross_positive_operator_gens(K) ]
- # Turn our matrices into long vectors...
- vectors = [ W(m.list()) for m in tensor_products ]
- # Create the *dual* cone of the cross-positive operators,
- # expressed as long vectors..
- L = ToricLattice(W.dimension())
- Sigma_dual = Cone(vectors, lattice=L)
+def LL_cone(K):
+ gens = K.lyapunov_like_basis()
+ L = ToricLattice(K.lattice_dim()**2)
+ return Cone([ g.list() for g in gens ], lattice=L, check=False)
- # Now compute the desired cone from its dual...
- Sigma_cone = Sigma_dual.dual()
+def Sigma_cone(K):
+ gens = cross_positive_operator_gens(K)
+ L = ToricLattice(K.lattice_dim()**2)
+ return Cone([ g.list() for g in gens ], lattice=L, check=False)
- # And finally convert its rays back to matrix representations.
- # But first, make them negative, so we get Z-transformations and
- # not cross-positive ones.
- M = MatrixSpace(V.base_ring(), V.dimension())
+def Z_cone(K):
+ gens = Z_operator_gens(K)
+ L = ToricLattice(K.lattice_dim()**2)
+ return Cone([ g.list() for g in gens ], lattice=L, check=False)
- return [ -M(v.list()) for v in Sigma_cone.rays() ]
+def pi_cone(K):
+ gens = positive_operator_gens(K)
+ L = ToricLattice(K.lattice_dim()**2)
+ return Cone([ g.list() for g in gens ], lattice=L, check=False)
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