-# 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 _basically_the_same(K1, K2):
+def is_lyapunov_like(L,K):
r"""
- Test whether or not ``K1`` and ``K2`` are "basically the same."
+ Determine whether or not ``L`` is Lyapunov-like on ``K``.
- This is a hack to get around the fact that it's difficult to tell
- when two cones are linearly isomorphic. We have a proposition that
- equates two cones, but represented over `\mathbb{Q}`, they are
- merely linearly isomorphic (not equal). So rather than test for
- equality, we test a list of properties that should be preserved
- under an invertible linear transformation.
+ 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.
- OUTPUT:
+ INPUT:
- ``True`` if ``K1`` and ``K2`` are basically the same, and ``False``
- otherwise.
+ - ``L`` -- A linear transformation or matrix.
- EXAMPLES:
+ - ``K`` -- A polyhedral closed convex cone.
- Any proper cone with three generators in `\mathbb{R}^{3}` is
- basically the same as the nonnegative orthant::
+ OUTPUT:
- sage: K1 = Cone([(1,0,0), (0,1,0), (0,0,1)])
- sage: K2 = Cone([(1,2,3), (3, 18, 4), (66, 51, 0)])
- sage: _basically_the_same(K1, K2)
- True
+ ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``,
+ and ``False`` otherwise.
- Negating a cone gives you another cone that is basically the same::
+ .. WARNING::
- sage: K = Cone([(0,2,-5), (-6, 2, 4), (0, 51, 0)])
- sage: _basically_the_same(K, -K)
- True
+ 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.
- TESTS:
+ REFERENCES:
- Any cone is basically the same as itself::
+ M. Orlitzky. The Lyapunov rank of an improper cone.
+ http://www.optimization-online.org/DB_HTML/2015/10/5135.html
- sage: K = random_cone(max_ambient_dim = 8)
- sage: _basically_the_same(K, K)
- True
+ EXAMPLES:
- After applying an invertible matrix to the rows of a cone, the
- result should be basically the same as the cone we started with::
+ The identity is always Lyapunov-like in a nontrivial space::
- 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: _basically_the_same(K1, K2)
+ sage: set_random_seed()
+ 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 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
+ As is the "zero" transformation::
- 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
+ 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 len(K1.facets()) != len(K2.facets()):
- return False
+ Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like
+ on ``K``::
- return True
+ 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
+ """
+ return all([(L*x).inner_product(s) == 0
+ for (x,s) in K.discrete_complementarity_set()])
-def _restrict_to_space(K, W):
+def motzkin_decomposition(K):
r"""
- Restrict this cone a subspace of its ambient space.
+ Return the pair of components in the Motzkin decomposition of this cone.
- INPUT:
-
- - ``W`` -- The subspace into which this cone will be restricted.
+ Every convex cone is the direct sum of a strictly convex cone and a
+ linear subspace [Stoer-Witzgall]_. 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 new cone in a sublattice corresponding to ``W``.
+ 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``.
+
+ REFERENCES:
+
+ .. [Stoer-Witzgall] J. Stoer and C. Witzgall. Convexity and
+ Optimization in Finite Dimensions I. Springer-Verlag, New
+ York, 1970.
EXAMPLES:
- When this cone is solid, restricting it into its own span should do
- nothing::
+ The nonnegative orthant is strictly convex, so it is its own
+ strictly convex component and its subspace component is trivial::
- sage: K = Cone([(1,)])
- sage: _restrict_to_space(K, K.span()) == K
+ sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
+ sage: (P,S) = motzkin_decomposition(K)
+ sage: K.is_equivalent(P)
True
-
- A single ray restricted into its own span gives the same output
- 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([(1,0,0)])
- 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
+ sage: S.is_trivial()
True
- TESTS:
-
- The projected cone should always be solid::
+ Likewise, full spaces are their own subspace components::
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 8)
- sage: _restrict_to_space(K, K.span()).is_solid()
+ sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
+ sage: K.is_full_space()
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_ambient_dim = 8)
- sage: K_P = _restrict_to_space(K, K.dual().span())
- sage: K_P.lattice_dim() == K.dual().dim()
+ sage: (P,S) = motzkin_decomposition(K)
+ sage: K.is_equivalent(S)
True
-
- This function should not affect the dimension of a cone::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 8)
- sage: K.dim() == _restrict_to_space(K,K.span()).dim()
+ sage: P.is_trivial()
True
- Nor should it affect the lineality of a cone::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 8)
- sage: K.lineality() == _restrict_to_space(K, K.span()).lineality()
- True
+ TESTS:
- No matter which space we restrict to, the lineality should not
- increase::
+ 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_ambient_dim = 8)
- sage: S = K.span(); P = K.dual().span()
- sage: K.lineality() >= _restrict_to_space(K,S).lineality()
+ 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: K.lineality() >= _restrict_to_space(K,P).lineality()
+ sage: x.is_zero() or (P.contains(x) != S.contains(x))
True
- If we do this according to our paper, then the result is proper::
+ 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_ambient_dim = 8)
- 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_SP.is_proper()
+ sage: K = random_cone(max_ambient_dim=8)
+ sage: (P,S) = motzkin_decomposition(K)
+ sage: P.is_strictly_convex()
True
- sage: K_SP = _restrict_to_space(K_S, K_S.dual().span())
- sage: K_SP.is_proper()
+ sage: S.lineality() == S.dim()
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
- _restrict_to_space::
+ The generators of the components are obtained from orthogonal
+ projections of the original generators [Stoer-Witzgall]_::
sage: set_random_seed()
- sage: J = random_cone(max_ambient_dim = 8)
- sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
- sage: K_W_star = _restrict_to_space(K, J.span()).dual()
- sage: K_star_W = _restrict_to_space(K.dual(), J.span())
- sage: _basically_the_same(K_W_star, K_star_W)
+ sage: K = random_cone(max_ambient_dim=8)
+ sage: (P,S) = motzkin_decomposition(K)
+ sage: A = S.linear_subspace().complement().matrix()
+ sage: proj_S_perp = A.transpose() * (A*A.transpose()).inverse() * A
+ sage: expected_P = Cone([ proj_S_perp*g for g in K ], K.lattice())
+ sage: P.is_equivalent(expected_P)
True
-
- """
- # First we want to intersect ``K`` with ``W``. The easiest way to
- # do this is via cone intersection, so we turn the subspace ``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 the span of K2, 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)
-
-
-
-def discrete_complementarity_set(K):
- r"""
- Compute a discrete complementarity set of this cone.
-
- A discrete complementarity set of `K` is the set of all orthogonal
- pairs `(x,s)` such that `x \in G_{1}` and `s \in G_{2}` for some
- generating sets `G_{1}` of `K` and `G_{2}` of its dual. Polyhedral
- convex cones are input in terms of their generators, so "the" (this
- particular) discrete complementarity set corresponds to ``G1
- == K.rays()`` and ``G2 == K.dual().rays()``.
-
- OUTPUT:
-
- A list of pairs `(x,s)` such that,
-
- * Both `x` and `s` are vectors (not rays).
- * `x` is one of ``K.rays()``.
- * `s` is one of ``K.dual().rays()``.
- * `x` and `s` are orthogonal.
-
- REFERENCES:
-
- .. [Orlitzky/Gowda] M. Orlitzky and M. S. Gowda. The Lyapunov Rank of an
- Improper Cone. Work in-progress.
-
- 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)
- []
-
- Likewise when this cone is trivial (its dual is the entire space)::
-
- sage: L = ToricLattice(0)
- sage: K = Cone([], ToricLattice(0))
- 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::
-
- sage: set_random_seed()
- sage: K1 = random_cone(max_ambient_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: A = S.linear_subspace().matrix()
+ sage: proj_S = A.transpose() * (A*A.transpose()).inverse() * A
+ sage: expected_S = Cone([ proj_S*g for g in K ], K.lattice())
+ sage: S.is_equivalent(expected_S)
True
-
- The pairs in the discrete complementarity set are in fact
- complementary::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=6)
- sage: dcs = discrete_complementarity_set(K)
- sage: sum([x.inner_product(s).abs() for (x,s) in dcs])
- 0
-
"""
- V = K.lattice().vector_space()
+ # The lines() method only returns one generator per line. For a true
+ # line, we also need a generator pointing in the opposite direction.
+ S_gens = [ direction*gen for direction in [1,-1] for gen in K.lines() ]
+ S = Cone(S_gens, K.lattice())
- # Convert rays to vectors so that we can compute inner products.
- xs = [V(x) for x in K.rays()]
+ # Since ``S`` is a subspace, the rays of its dual generate its
+ # orthogonal complement.
+ S_perp = Cone(S.dual(), K.lattice())
+ P = K.intersection(S_perp)
- # We also convert the generators of the dual cone so that we
- # return pairs of vectors and not (vector, ray) pairs.
- ss = [V(s) for s in K.dual().rays()]
+ return (P,S)
- return [(x,s) for x in xs for s in ss if x.inner_product(s) == 0]
-
-def LL(K):
+def positive_operator_gens(K):
r"""
- Compute a basis of 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::
-
- sage: L = ToricLattice(0)
- sage: K = Cone([], lattice=L)
- sage: LL(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.]_::
+ The trivial cone in a trivial space has no positive operators::
- 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([], ToricLattice(0))
+ sage: positive_operator_gens(K)
+ []
- sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)])
- sage: LL(L3infty)
- [
- [1 0 0]
- [0 1 0]
- [0 0 1]
- ]
+ Every operator is positive on the trivial cone::
- If our cone is the entire space, then every transformation on it is
- Lyapunov-like::
+ sage: K = Cone([(0,)])
+ sage: positive_operator_gens(K)
+ [[1], [-1]]
- sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)])
- sage: M = MatrixSpace(QQ,2)
- sage: M.basis() == LL(K)
+ 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]
+ ]
- TESTS:
-
- The inner product `\left< L\left(x\right), s \right>` is zero for
- every pair `\left( x,s \right)` in the discrete complementarity set
- of the cone::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_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
-
- 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)`
+ Every operator is positive on the ambient vector space::
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_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)
+ sage: K = Cone([(1,),(-1,)])
+ sage: K.is_full_space()
True
+ sage: positive_operator_gens(K)
+ [[1], [-1]]
- """
- V = K.lattice().vector_space()
-
- C_of_K = discrete_complementarity_set(K)
-
- tensor_products = [ s.tensor_product(x) for (x,s) in C_of_K ]
-
- # Sage doesn't think matrices are vectors, so we have to convert
- # our matrices to vectors explicitly before we can figure out how
- # many are linearly-indepenedent.
- #
- # The space W has the same base ring as V, but dimension
- # dim(V)^2. So it has the same dimension as the space of linear
- # transformations on V. In other words, it's just the right size
- # to create an isomorphism between it and our matrices.
- W = VectorSpace(V.base_ring(), V.dimension()**2)
-
- # Turn our matrices into long vectors...
- vectors = [ W(m.list()) for m in tensor_products ]
-
- # Vector space representation of Lyapunov-like matrices
- # (i.e. vec(L) where L is Luapunov-like).
- LL_vector = W.span(vectors).complement()
-
- # Now construct an ambient MatrixSpace in which to stick our
- # transformations.
- M = MatrixSpace(V.base_ring(), V.dimension())
-
- matrix_basis = [ M(v.list()) for v in LL_vector.basis() ]
-
- return matrix_basis
-
-
-
-def lyapunov_rank(K):
- r"""
- Compute the Lyapunov rank (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.
-
- 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 [Orlitzky/Gowda]_).
-
- 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.
-
- 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()*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)])
- 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([(1,0),(-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]
+ ]
- Two isomorphic cones should have the same Lyapunov rank [Rudolf et al.]_.
- The cone ``K`` in the following example is isomorphic to the nonnegative
- octant in `\mathbb{R}^{3}`::
-
- sage: K = Cone([(1,2,3), (-1,1,0), (1,0,6)])
- sage: lyapunov_rank(K)
- 3
-
- The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
- itself [Rudolf et al.]_::
+ 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 et al.]_::
+ Each positive operator generator should send the generators of the
+ cone into the 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: 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 ])
True
- The Lyapunov rank is invariant under a linear isomorphism
- [Orlitzky/Gowda]_::
+ Each positive operator generator should send a random element of the
+ cone into the 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: K = random_cone(max_ambient_dim=5)
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: all([ K.contains(P*K.random_element()) for P in pi_of_K ])
True
- The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
- itself [Rudolf et al.]_::
+ A random element of the positive operator cone should send the
+ generators of the cone into the cone::
sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8)
- sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+ 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([ g.list() for g in pi_of_K ], lattice=L)
+ sage: P = matrix(K.lattice_dim(), pi_cone.random_element().list())
+ sage: all([ K.contains(P*x) for x in K ])
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::
+ A random element of the positive operator cone should send a random
+ element of the cone into the cone::
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=5)
+ 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)
+ sage: P = matrix(K.lattice_dim(), pi_cone.random_element().list())
+ sage: K.contains(P*K.random_element())
True
- sage: b == n-1
- False
- In fact [Orlitzky/Gowda]_, no closed convex polyhedral cone can have
- Lyapunov rank `n-1` in `n` dimensions::
+ 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: b = lyapunov_rank(K)
+ sage: K = random_cone(max_ambient_dim=5)
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_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: 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: 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
- The Lyapunov rank of any cone is just the dimension of ``LL(K)``::
+ The lineality 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: lyapunov_rank(K) == len(LL(K))
+ 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
- We can make an imperfect cone perfect by adding a slack variable
- (a Theorem in [Orlitzky/Gowda]_)::
+ The cone ``K`` is proper if and only if the cone of positive
+ operators on ``K`` is proper::
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=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
-
"""
- 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 = K.lineality()
-
- if m < n:
- # K is not solid, restrict to its span.
- K = _restrict_to_space(K, K.span())
- # Non-solid reduction lemma.
- beta += (n - m)*n
+ tensor_products = [ s.tensor_product(x) for x in K for s in K.dual() ]
- 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())
+ # Convert those tensor products to long vectors.
+ W = VectorSpace(F, n**2)
+ vectors = [ W(tp.list()) for tp in tensor_products ]
- # Non-pointed reduction lemma.
- beta += l * m
+ # Create the *dual* cone of the positive operators, expressed as
+ # long vectors..
+ pi_dual = Cone(vectors, ToricLattice(W.dimension()))
- beta += len(LL(K))
- return beta
+ # 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 is_lyapunov_like(L,K):
+def Z_transformation_gens(K):
r"""
- 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.
-
- INPUT:
-
- - ``L`` -- A linear transformation or matrix.
-
- - ``K`` -- A polyhedral closed convex cone.
+ Compute generators of the cone of Z-transformations 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.
-
- REFERENCES:
-
- .. [Orlitzky] M. Orlitzky. The Lyapunov rank of an
- improper cone (preprint).
+ 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 identity is always Lyapunov-like in a nontrivial space::
+ Z-transformations on the nonnegative orthant are just 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: 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
- As is the "zero" transformation::
+ The trivial cone in a trivial space has no Z-transformations::
- 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: Z_transformation_gens(K)
+ []
- Everything in ``LL(K)`` should be Lyapunov-like on ``K``::
+ Z-transformations on a subspace are Lyapunov-like and vice-versa::
- sage: K = random_cone(min_ambient_dim = 1, max_rays = 5)
- sage: all([is_lyapunov_like(L,K) for L in LL(K)])
+ 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_transformation_gens(K) ])
+ sage: zs == lls
True
- """
- return all([(L*x).inner_product(s) == 0
- for (x,s) in discrete_complementarity_set(K)])
-
-
-def random_element(K):
- r"""
- Return a random element of ``K`` from its ambient vector space.
+ TESTS:
- ALGORITHM:
+ The Z-property is possessed by every Z-transformation::
- 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: set_random_seed()
+ 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
- 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.
+ The lineality space of Z is LL::
- EXAMPLES:
+ 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_transformation_gens(K) ])
+ sage: z_cone.linear_subspace() == lls
+ True
- A random element of the trivial cone is zero::
+ And thus, the lineality of Z is the Lyapunov rank::
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)
-
- TESTS:
+ 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
- Any cone should contain an element of itself::
+ The lineality spaces of pi-star and Z-star are equal:
sage: set_random_seed()
- sage: K = random_cone(max_rays = 8)
- sage: K.contains(random_element(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
-
"""
- 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
+ # 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 transformations.
+ 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 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)