-# 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 discrete_complementarity_set(K):
+def is_lyapunov_like(L,K):
r"""
- Compute the discrete complementarity set of this cone.
+ 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.
- 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.
+ - ``K`` -- A polyhedral closed convex cone.
OUTPUT:
- A list of pairs `(x,s)` such that,
+ ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``,
+ and ``False`` otherwise.
- * `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.
+ .. WARNING::
- EXAMPLES:
+ 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.
- The discrete complementarity set of the nonnegative orthant consists
- of pairs of standard basis vectors::
+ REFERENCES:
- 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::
+ M. Orlitzky. The Lyapunov rank of an improper cone.
+ http://www.optimization-online.org/DB_HTML/2015/10/5135.html
- sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
- sage: discrete_complementarity_set(K)
- []
+ EXAMPLES:
- TESTS:
+ The identity is always Lyapunov-like in a nontrivial space::
- 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: 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
- sage: K1 = random_cone(max_dim=10, max_rays=10)
- sage: K2 = K1.dual()
- sage: expected = [(x,s) for (s,x) in discrete_complementarity_set(K2)]
- sage: actual = discrete_complementarity_set(K1)
- sage: actual == expected
+ As is the "zero" transformation::
+
+ 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
- """
- V = K.lattice().vector_space()
+ Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like
+ on ``K``::
- # 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()]
+ 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 [(x,s) for x in xs for s in ss if x.inner_product(s) == 0]
+ """
+ return all([(L*x).inner_product(s) == 0
+ for (x,s) in K.discrete_complementarity_set()])
-def LL(K):
+def motzkin_decomposition(K):
r"""
- Compute the space `\mathbf{LL}` of all Lyapunov-like transformations
- on this cone.
+ Return the pair of components in the motzkin decomposition of this 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 ``MatrixSpace`` object `M` such that every matrix `L \in M` is
- Lyapunov-like on this cone.
+ 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``.
- """
- V = K.lattice().vector_space()
+ EXAMPLES:
- C_of_K = discrete_complementarity_set(K)
+ The nonnegative orthant is strictly convex, so it is its own
+ strictly convex component and its subspace component is trivial::
- matrices = [x.tensor_product(s) for (x,s) in C_of_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
+ sage: S.is_trivial()
+ 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)
+ Likewise, full spaces are their own subspace components::
- # Turn our matrices into long vectors...
- vectors = [ W(m.list()) for m in matrices ]
+ 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:
- # Vector space representation of Lyapunov-like matrices
- # (i.e. vec(L) where L is Luapunov-like).
- LL_vector = W.span(vectors).complement()
+ 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::
- # Now construct an ambient MatrixSpace in which to stick our
- # transformations.
- M = MatrixSpace(V.base_ring(), V.dimension())
+ sage: set_random_seed()
+ 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: x.is_zero() or (P.contains(x) != S.contains(x))
+ True
+
+ 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: (P,S) = motzkin_decomposition(K)
+ sage: P.is_strictly_convex()
+ True
+ sage: S.lineality() == S.dim()
+ True
- matrices = [ M(v.list()) for v in LL_vector.basis() ]
+ 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: 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
+ """
+ linspace_gens = [ copy(b) for b in K.linear_subspace().basis() ]
+ linspace_gens += [ -b for b in linspace_gens ]
- return matrices
+ S = Cone(linspace_gens, K.lattice())
+ # 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 lyapunov_rank(K):
+def positive_operator_gens(K):
r"""
- Compute the Lyapunov (or bilinearity) rank of this cone.
+ Compute generators of the cone of positive operators on this cone.
+
+ OUTPUT:
- The Lyapunov rank of a cone can be thought of in (mainly) two ways:
+ 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.
- 1. The dimension of the Lie algebra of the automorphism group of the
- cone.
+ EXAMPLES:
- 2. The dimension of the linear space of all Lyapunov-like
- transformations on the cone.
+ The trivial cone in a trivial space has no positive operators::
- INPUT:
+ sage: K = Cone([], ToricLattice(0))
+ sage: positive_operator_gens(K)
+ []
- A closed, convex polyhedral cone.
+ Positive operators on the nonnegative orthant are nonnegative matrices::
- OUTPUT:
+ sage: K = Cone([(1,)])
+ sage: positive_operator_gens(K)
+ [[1]]
- 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).
+ 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]
+ ]
- .. note::
+ Every operator is positive on the ambient vector space::
- In the references, the cones are always assumed to be proper. We
- do not impose this restriction.
+ sage: K = Cone([(1,),(-1,)])
+ sage: K.is_full_space()
+ True
+ sage: positive_operator_gens(K)
+ [[1], [-1]]
- .. seealso::
+ 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]
+ ]
- :meth:`is_proper`
+ TESTS:
- ALGORITHM:
+ A positive operator on a cone should send its generators into the cone::
- 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}`.
+ 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*x) for p in pi_of_K for x in K.rays()])
+ True
- REFERENCES:
+ The dimension of the cone of positive operators is given by the
+ corollary in my paper::
- .. [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.
+ sage: set_random_seed()
+ 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
- .. [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.
+ The lineality of the cone of positive operators is given by the
+ corollary in my paper::
- EXAMPLES:
+ 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
- The nonnegative orthant in `\mathbb{R}^{n}` always has rank `n`
- [Rudolf et al.]_::
+ The cone ``K`` is proper if and only if the cone of positive
+ operators on ``K`` is proper::
- 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
+ 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()
- The `L^{3}_{1}` cone is known to have a Lyapunov rank of one
- [Rudolf et al.]_::
+ tensor_products = [ s.tensor_product(x) for x in K for s in K.dual() ]
- sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)])
- sage: lyapunov_rank(L31)
- 1
+ # Convert those tensor products to long vectors.
+ W = VectorSpace(F, n**2)
+ vectors = [ W(tp.list()) for tp in tensor_products ]
- Likewise for the `L^{3}_{\infty}` cone [Rudolf et al.]_::
+ # Create the *dual* cone of the positive operators, expressed as
+ # long vectors..
+ pi_dual = Cone(vectors, ToricLattice(W.dimension()))
- sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)])
- sage: lyapunov_rank(L3infty)
- 1
+ # Now compute the desired cone from its dual...
+ pi_cone = pi_dual.dual()
- The Lyapunov rank should be additive on a product of cones
- [Rudolf et al.]_::
+ # And finally convert its rays back to matrix representations.
+ M = MatrixSpace(F, n)
+ return [ M(v.list()) for v in pi_cone.rays() ]
- 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)
- 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}`::
+def Z_transformation_gens(K):
+ r"""
+ Compute generators of the cone of Z-transformations on this cone.
+
+ OUTPUT:
- sage: K = Cone([(1,2,3), (-1,1,0), (1,0,6)])
- sage: lyapunov_rank(K)
- 3
+ 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 dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
- itself [Rudolf et al.]_::
+ Z-transformations on the nonnegative orthant are just Z-matrices.
+ That is, matrices whose off-diagonal elements are nonnegative::
- 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),(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
- TESTS:
+ The trivial cone in a trivial space has no Z-transformations::
- The Lyapunov rank should be additive on a product of cones
- [Rudolf et al.]_::
+ sage: K = Cone([], ToricLattice(0))
+ sage: Z_transformation_gens(K)
+ []
- sage: K1 = random_cone(max_dim=10, max_rays=10)
- sage: K2 = random_cone(max_dim=10, max_rays=10)
- sage: K = K1.cartesian_product(K2)
- sage: lyapunov_rank(K) == lyapunov_rank(K1) + lyapunov_rank(K2)
+ 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_transformation_gens(K) ])
+ sage: zs == lls
+ True
+
+ TESTS:
- The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
- itself [Rudolf et al.]_::
+ The Z-property is possessed by every Z-transformation::
- sage: K = random_cone(max_dim=10, max_rays=10)
- sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+ 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
- 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, this holds for the trivial cone in a
- trivial space as well)::
+ The lineality space of Z is LL::
- sage: K = random_cone(max_dim=10, strictly_convex=True, solid=True)
- sage: b = lyapunov_rank(K)
- sage: n = K.lattice_dim()
- sage: 1 <= b and b <= n
+ 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
- sage: b == n-1
- False
-
- """
- V = K.lattice().vector_space()
- C_of_K = discrete_complementarity_set(K)
+ And thus, the lineality of Z is the Lyapunov rank::
- matrices = [x.tensor_product(s) for (x,s) in C_of_K]
+ sage: set_random_seed()
+ 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
- # 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 lineality spaces of pi-star and Z-star are equal:
- def phi(m):
- r"""
- Convert a matrix to a vector isomorphically.
- """
- return W(m.list())
+ sage: set_random_seed()
+ 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
+ """
+ # 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()
- vectors = [phi(m) for m in matrices]
+ # 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() ]
- return (W.dimension() - W.span(vectors).rank())
+ # 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)
projects / sage.d.git / blobdiff