-# 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 *
+from sage.geometry.cone import is_Cone
-
-def drop_dependent(vs):
- r"""
- Return the largest linearly-independent subset of ``vs``.
- """
- if len(vs) == 0:
- # ...for lazy enough definitions of linearly-independent
- return vs
-
- result = []
- old_V = VectorSpace(vs[0].parent().base_field(), 0)
-
- for v in vs:
- new_V = span(result + [v])
- if new_V.dimension() > old_V.dimension():
- result.append(v)
- old_V = new_V
-
- return result
-
-
-def basically_the_same(K1,K2):
- r"""
- ``True`` if ``K1`` and ``K2`` are basically the same, and ``False``
- otherwise.
- """
- if K1.lattice_dim() != K2.lattice_dim():
- return False
-
- if K1.nrays() != K2.nrays():
- return False
-
- if K1.dim() != K2.dim():
- return False
-
- if lineality(K1) != lineality(K2):
- return False
-
- if K1.is_solid() != K2.is_solid():
- return False
-
- if K1.is_strictly_convex() != K2.is_strictly_convex():
- return False
-
- if len(LL(K1)) != len(LL(K2)):
- return False
-
- C_of_K1 = discrete_complementarity_set(K1)
- C_of_K2 = discrete_complementarity_set(K2)
- if len(C_of_K1) != len(C_of_K2):
- return False
-
- if len(K1.facets()) != len(K2.facets()):
- return False
-
- return True
-
-
-
-def iso_space(K):
- r"""
- Construct the space `W \times W^{\perp}` isomorphic to the ambient space
- of ``K`` where `W` is equal to the span of ``K``.
- """
- V = K.lattice().vector_space()
-
- # Create the space W \times W^{\perp} isomorphic to V.
- # First we get an orthogonal (but not normal) basis...
- M = matrix(V.base_field(), K.rays())
- W_basis = drop_dependent(K.rays())
-
- W = V.subspace_with_basis(W_basis)
- W_perp = W.complement()
-
- return W.cartesian_product(W_perp)
-
-
-def ips_iso(K):
+def is_positive_on(L,K):
r"""
- Construct the IPS isomorphism and its inverse from our paper.
-
- Given a cone ``K``, the returned isomorphism will split its ambient
- vector space `V` into a cartesian product `W \times W^{\perp}` where
- `W` equals the span of ``K``.
- """
- V = K.lattice().vector_space()
- V_iso = iso_space(K)
- (W, W_perp) = V_iso.cartesian_factors()
-
- # A space equivalent to V, but using our basis.
- V_user = V.subspace_with_basis( W.basis() + W_perp.basis() )
-
- def phi(v):
- # Write v in terms of our custom basis, where the first dim(W)
- # coordinates are for the W-part of the basis.
- cs = V_user.coordinates(v)
-
- w1 = sum([ V_user.basis()[idx]*cs[idx]
- for idx in range(0, W.dimension()) ])
- w2 = sum([ V_user.basis()[idx]*cs[idx]
- for idx in range(W.dimension(), V.dimension()) ])
-
- return V_iso( (w1, w2) )
-
-
- def phi_inv( pair ):
- # Crash if the arguments are in the wrong spaces.
- V_iso(pair)
-
- #w = sum([ sub_w[idx]*W.basis()[idx] for idx in range(0,m) ])
- #w_prime = sum([ sub_w_prime[idx]*W_perp.basis()[idx]
- # for idx in range(0,n-m) ])
-
- return sum( pair.cartesian_factors() )
-
-
- return (phi,phi_inv)
-
-
-
-def unrestrict_span(K, K2=None):
- if K2 is None:
- K2 = K
-
- _,phi_inv = ips_iso(K2)
- V_iso = iso_space(K2)
- (W, W_perp) = V_iso.cartesian_factors()
-
- rays = []
- for r in K.rays():
- w = sum([ r[idx]*W.basis()[idx] for idx in range(0,len(r)) ])
- pair = V_iso( (w, W_perp.zero()) )
- rays.append( phi_inv(pair) )
-
- L = ToricLattice(W.dimension() + W_perp.dimension())
-
- return Cone(rays, lattice=L)
+ Determine whether or not ``L`` is positive on ``K``.
+ We say that ``L`` is positive on a closed convex cone ``K`` if
+ `L\left\lparen x \right\rparen` belongs to ``K`` for all `x` in
+ ``K``. This property need only be checked for generators of ``K``.
-
-def restrict_span(K, K2=None):
- r"""
- Restrict ``K`` into its own span, or the span of another cone.
+ To reliably check whether or not ``L`` is positive, its base ring
+ must be either exact (for example, the rationals) or ``SR``. An
+ exact ring is more reliable, but in some cases a matrix whose
+ entries contain symbolic constants like ``e`` and ``pi`` will work.
INPUT:
- - ``K2`` -- another cone whose lattice has the same rank as this cone.
-
- OUTPUT:
+ - ``L`` -- A matrix over either an exact ring or ``SR``.
- A new cone in a sublattice.
-
- EXAMPLES::
-
- sage: K = Cone([(1,)])
- sage: restrict_span(K) == K
- True
+ - ``K`` -- A polyhedral closed convex cone.
- sage: K2 = Cone([(1,0)])
- sage: restrict_span(K2).rays()
- N(1)
- in 1-d lattice N
- sage: K3 = Cone([(1,0,0)])
- sage: restrict_span(K3).rays()
- N(1)
- in 1-d lattice N
- sage: restrict_span(K2) == restrict_span(K3)
- True
-
- TESTS:
-
- The projected cone should always be solid::
-
- sage: set_random_seed()
- sage: K = random_cone(max_dim = 8)
- sage: K_S = restrict_span(K)
- sage: K_S.is_solid()
- True
-
- And the resulting cone should live in a space having the same
- dimension as the space we restricted it to::
-
- sage: set_random_seed()
- sage: K = random_cone(max_dim = 8)
- sage: K_S = restrict_span(K, K.dual() )
- sage: K_S.lattice_dim() == K.dual().dim()
- True
-
- This function has ``unrestrict_span()`` as its inverse::
-
- sage: set_random_seed()
- sage: K = random_cone(max_dim = 8, solid=True)
- sage: J = restrict_span(K)
- sage: K == unrestrict_span(J,K)
- True
+ OUTPUT:
- This function should not affect the dimension of a cone::
+ If the base ring of ``L`` is exact, then ``True`` will be returned if
+ and only if ``L`` is positive on ``K``.
- sage: set_random_seed()
- sage: K = random_cone(max_dim = 8)
- sage: K.dim() == restrict_span(K).dim()
- True
+ If the base ring of ``L`` is ``SR``, then the situation is more
+ complicated:
- Nor should it affect the lineality of a cone::
+ - ``True`` will be returned if it can be proven that ``L``
+ is positive on ``K``.
+ - ``False`` will be returned if it can be proven that ``L``
+ is not positive on ``K``.
+ - ``False`` will also be returned if we can't decide; specifically
+ if we arrive at a symbolic inequality that cannot be resolved.
- sage: set_random_seed()
- sage: K = random_cone(max_dim = 8)
- sage: lineality(K) == lineality(restrict_span(K))
- True
+ .. SEEALSO::
- No matter which space we restrict to, the lineality should not
- increase::
+ :func:`is_cross_positive_on`,
+ :func:`is_Z_operator_on`,
+ :func:`is_lyapunov_like_on`
- sage: set_random_seed()
- sage: K = random_cone(max_dim = 8)
- sage: lineality(K) >= lineality(restrict_span(K))
- True
- sage: lineality(K) >= lineality(restrict_span(K, K.dual()))
- True
+ EXAMPLES:
- If we do this according to our paper, then the result is proper::
+ Nonnegative matrices are positive operators on the nonnegative
+ orthant::
- sage: set_random_seed()
- sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=False)
- sage: K_S = restrict_span(K)
- sage: P = restrict_span(K_S.dual()).dual()
- sage: P.is_proper()
- True
- sage: P = restrict_span(K_S, K_S.dual())
- sage: P.is_proper()
+ sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
+ sage: L = random_matrix(QQ,3).apply_map(abs)
+ sage: is_positive_on(L,K)
True
- ::
-
- sage: set_random_seed()
- sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=False)
- sage: K_S = restrict_span(K)
- sage: P = restrict_span(K_S.dual()).dual()
- sage: P.is_proper()
- True
- sage: P = restrict_span(K_S, K_S.dual())
- sage: P.is_proper()
- True
+ TESTS:
- ::
+ The identity operator is always positive::
sage: set_random_seed()
- sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=True)
- sage: K_S = restrict_span(K)
- sage: P = restrict_span(K_S.dual()).dual()
- sage: P.is_proper()
- True
- sage: P = restrict_span(K_S, K_S.dual())
- sage: P.is_proper()
+ sage: K = random_cone(max_ambient_dim=8)
+ sage: L = identity_matrix(K.lattice_dim())
+ sage: is_positive_on(L,K)
True
- ::
+ The "zero" operator is always positive::
- sage: set_random_seed()
- sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=True)
- sage: K_S = restrict_span(K)
- sage: P = restrict_span(K_S.dual()).dual()
- sage: P.is_proper()
- True
- sage: P = restrict_span(K_S, K_S.dual())
- sage: P.is_proper()
+ sage: K = random_cone(max_ambient_dim=8)
+ sage: R = K.lattice().vector_space().base_ring()
+ sage: L = zero_matrix(R, K.lattice_dim())
+ sage: is_positive_on(L,K)
True
- Test the proposition in our paper concerning the duals, where the
- subspace `W` is the span of `K^{*}`::
+ Everything in ``K.positive_operators_gens()`` should be
+ positive on ``K``::
- sage: set_random_seed()
- sage: K = random_cone(max_dim = 8, solid=False, strictly_convex=False)
- sage: K_W = restrict_span(K, K.dual())
- sage: K_star_W_star = restrict_span(K.dual()).dual()
- sage: basically_the_same(K_W, K_star_W_star)
+ sage: K = random_cone(max_ambient_dim=5)
+ sage: all([ is_positive_on(L,K) # long time
+ ....: for L in K.positive_operators_gens() ]) # long time
True
-
- ::
-
- sage: set_random_seed()
- sage: K = random_cone(max_dim = 8, solid=True, strictly_convex=False)
- sage: K_W = restrict_span(K, K.dual())
- sage: K_star_W_star = restrict_span(K.dual()).dual()
- sage: basically_the_same(K_W, K_star_W_star)
+ sage: all([ is_positive_on(L.change_ring(SR),K) # long time
+ ....: for L in K.positive_operators_gens() ]) # long time
True
- ::
+ Technically we could test this, but for now only closed convex cones
+ are supported as our ``K`` argument::
- sage: set_random_seed()
- sage: K = random_cone(max_dim = 8, solid=False, strictly_convex=True)
- sage: K_W = restrict_span(K, K.dual())
- sage: K_star_W_star = restrict_span(K.dual()).dual()
- sage: basically_the_same(K_W, K_star_W_star)
- True
+ sage: K = [ vector([1,2,3]), vector([5,-1,7]) ]
+ sage: L = identity_matrix(3)
+ sage: is_positive_on(L,K)
+ Traceback (most recent call last):
+ ...
+ TypeError: K must be a Cone.
- ::
+ We can't give reliable answers over inexact rings::
- sage: set_random_seed()
- sage: K = random_cone(max_dim = 8, solid=True, strictly_convex=True)
- sage: K_W = restrict_span(K, K.dual())
- sage: K_star_W_star = restrict_span(K.dual()).dual()
- sage: basically_the_same(K_W, K_star_W_star)
- True
+ sage: K = Cone([(1,2,3), (4,5,6)])
+ sage: L = identity_matrix(RR,3)
+ sage: is_positive_on(L,K)
+ Traceback (most recent call last):
+ ...
+ ValueError: The base ring of L is neither SR nor exact.
"""
- if K2 is None:
- K2 = K
- phi,_ = ips_iso(K2)
- (W, W_perp) = iso_space(K2).cartesian_factors()
+ if not is_Cone(K):
+ raise TypeError('K must be a Cone.')
+ if not L.base_ring().is_exact() and not L.base_ring() is SR:
+ raise ValueError('The base ring of L is neither SR nor exact.')
- ray_pairs = [ phi(r) for r in K.rays() ]
+ if L.base_ring().is_exact():
+ # This should be way faster than computing the dual and
+ # checking a bunch of inequalities, but it doesn't work if
+ # ``L*x`` is symbolic. For example, ``e in Cone([(1,)])``
+ # is true, but returns ``False``.
+ return all([ L*x in K for x in K ])
+ else:
+ # Fall back to inequality-checking when the entries of ``L``
+ # might be symbolic.
+ return all([ s*(L*x) >= 0 for x in K for s in K.dual() ])
- # Shouldn't matter?
- #
- #if any([ w2 != W_perp.zero() for (_, w2) in ray_pairs ]):
- # msg = 'Cone has nonzero components in W-perp!'
- # raise ValueError(msg)
- # Represent the cone in terms of a basis for W, i.e. with smaller
- # vectors.
- ws = [ W.coordinate_vector(w1) for (w1, _) in ray_pairs ]
+def is_cross_positive_on(L,K):
+ r"""
+ Determine whether or not ``L`` is cross-positive on ``K``.
- L = ToricLattice(W.dimension())
+ We say that ``L`` is cross-positive on a closed convex cone``K`` if
+ `\left\langle L\left\lparenx\right\rparen,s\right\rangle \ge 0` for
+ all pairs `\left\langle x,s \right\rangle` in the complementarity
+ set of ``K``. This property need only be checked for generators of
+ ``K`` and its dual.
- return Cone(ws, lattice=L)
+ To reliably check whether or not ``L`` is cross-positive, its base
+ ring must be either exact (for example, the rationals) or ``SR``. An
+ exact ring is more reliable, but in some cases a matrix whose
+ entries contain symbolic constants like ``e`` and ``pi`` will work.
+ INPUT:
+ - ``L`` -- A matrix over either an exact ring or ``SR``.
-def lineality(K):
- r"""
- Compute the lineality of this cone.
-
- The lineality of a cone is the dimension of the largest linear
- subspace contained in that cone.
+ - ``K`` -- A polyhedral closed convex cone.
OUTPUT:
- A nonnegative integer; the dimension of the largest subspace
- contained within this cone.
-
- REFERENCES:
+ If the base ring of ``L`` is exact, then ``True`` will be returned if
+ and only if ``L`` is cross-positive on ``K``.
- .. [Rockafellar] R.T. Rockafellar. Convex Analysis. Princeton
- University Press, Princeton, 1970.
+ If the base ring of ``L`` is ``SR``, then the situation is more
+ complicated:
- EXAMPLES:
+ - ``True`` will be returned if it can be proven that ``L``
+ is cross-positive on ``K``.
+ - ``False`` will be returned if it can be proven that ``L``
+ is not cross-positive on ``K``.
+ - ``False`` will also be returned if we can't decide; specifically
+ if we arrive at a symbolic inequality that cannot be resolved.
- The lineality of the nonnegative orthant is zero, since it clearly
- contains no lines::
+ .. SEEALSO::
- sage: K = Cone([(1,0,0), (0,1,0), (0,0,1)])
- sage: lineality(K)
- 0
+ :func:`is_positive_on`,
+ :func:`is_Z_operator_on`,
+ :func:`is_lyapunov_like_on`
- However, if we add another ray so that the entire `x`-axis belongs
- to the cone, then the resulting cone will have lineality one::
-
- sage: K = Cone([(1,0,0), (-1,0,0), (0,1,0), (0,0,1)])
- sage: lineality(K)
- 1
+ EXAMPLES:
- If our cone is all of `\mathbb{R}^{2}`, then its lineality is equal
- to the dimension of the ambient space (i.e. two)::
+ The identity operator is always cross-positive::
- sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)])
- sage: lineality(K)
- 2
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=8)
+ sage: L = identity_matrix(K.lattice_dim())
+ sage: is_cross_positive_on(L,K)
+ True
- Per the definition, the lineality of the trivial cone in a trivial
- space is zero::
+ The "zero" operator is always cross-positive::
- sage: K = Cone([], lattice=ToricLattice(0))
- sage: lineality(K)
- 0
+ sage: K = random_cone(max_ambient_dim=8)
+ sage: R = K.lattice().vector_space().base_ring()
+ sage: L = zero_matrix(R, K.lattice_dim())
+ sage: is_cross_positive_on(L,K)
+ True
TESTS:
- The lineality of a cone should be an integer between zero and the
- dimension of the ambient space, inclusive::
+ Everything in ``K.cross_positive_operators_gens()`` should be
+ cross-positive on ``K``::
- sage: set_random_seed()
- sage: K = random_cone(max_dim = 8)
- sage: l = lineality(K)
- sage: l in ZZ
+ sage: K = random_cone(max_ambient_dim=5)
+ sage: all([ is_cross_positive_on(L,K) # long time
+ ....: for L in K.cross_positive_operators_gens() ]) # long time
True
- sage: (0 <= l) and (l <= K.lattice_dim())
+ sage: all([ is_cross_positive_on(L.change_ring(SR),K) # long time
+ ....: for L in K.cross_positive_operators_gens() ]) # long time
True
- A strictly convex cone should have lineality zero::
+ Technically we could test this, but for now only closed convex cones
+ are supported as our ``K`` argument::
- sage: set_random_seed()
- sage: K = random_cone(max_dim = 8, strictly_convex = True)
- sage: lineality(K)
- 0
-
- """
- return K.linear_subspace().dimension()
-
-
-def discrete_complementarity_set(K):
- r"""
- Compute the discrete complementarity set of this cone.
+ sage: L = identity_matrix(3)
+ sage: K = [ vector([8,2,-8]), vector([5,-5,7]) ]
+ sage: is_cross_positive_on(L,K)
+ Traceback (most recent call last):
+ ...
+ TypeError: K must be a Cone.
- The complementarity set of this cone is the set of all orthogonal
- pairs `(x,s)` such that `x` is in this cone, and `s` is in its
- dual. The discrete complementarity set restricts `x` and `s` to be
- generators of their respective cones.
-
- OUTPUT:
-
- A list of pairs `(x,s)` such that,
-
- * `x` is in this cone.
- * `x` is a generator of this cone.
- * `s` is in this cone's dual.
- * `s` is a generator of this cone's dual.
- * `x` and `s` are orthogonal.
-
- EXAMPLES:
+ We can't give reliable answers over inexact rings::
- The discrete complementarity set of the nonnegative orthant consists
- of pairs of standard basis vectors::
+ sage: K = Cone([(1,2,3), (4,5,6)])
+ sage: L = identity_matrix(RR,3)
+ sage: is_cross_positive_on(L,K)
+ Traceback (most recent call last):
+ ...
+ ValueError: The base ring of L is neither SR nor exact.
- sage: K = Cone([(1,0),(0,1)])
- sage: discrete_complementarity_set(K)
- [((1, 0), (0, 1)), ((0, 1), (1, 0))]
+ """
+ if not is_Cone(K):
+ raise TypeError('K must be a Cone.')
+ if not L.base_ring().is_exact() and not L.base_ring() is SR:
+ raise ValueError('The base ring of L is neither SR nor exact.')
- If the cone consists of a single ray, the second components of the
- discrete complementarity set should generate the orthogonal
- complement of that ray::
+ return all([ s*(L*x) >= 0
+ for (x,s) in K.discrete_complementarity_set() ])
- 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))]
+def is_Z_operator_on(L,K):
+ r"""
+ Determine whether or not ``L`` is a Z-operator on ``K``.
- When the cone is the entire space, its dual is the trivial cone, so
- the discrete complementarity set is empty::
+ We say that ``L`` is a Z-operator on a closed convex cone``K`` if
+ `\left\langle L\left\lparenx\right\rparen,s\right\rangle \le 0` for
+ all pairs `\left\langle x,s \right\rangle` in the complementarity
+ set of ``K``. It is known that this property need only be checked
+ for generators of ``K`` and its dual.
- sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
- sage: discrete_complementarity_set(K)
- []
+ A matrix is a Z-operator on ``K`` if and only if its negation is a
+ cross-positive operator on ``K``.
- TESTS:
+ To reliably check whether or not ``L`` is a Z operator, its base
+ ring must be either exact (for example, the rationals) or ``SR``. An
+ exact ring is more reliable, but in some cases a matrix whose
+ entries contain symbolic constants like ``e`` and ``pi`` will work.
- The complementarity set of the dual can be obtained by switching the
- components of the complementarity set of the original cone::
+ INPUT:
- sage: set_random_seed()
- sage: K1 = random_cone(max_dim=6)
- sage: K2 = K1.dual()
- sage: expected = [(x,s) for (s,x) in discrete_complementarity_set(K2)]
- sage: actual = discrete_complementarity_set(K1)
- sage: sorted(actual) == sorted(expected)
- True
+ - ``L`` -- A matrix over either an exact ring or ``SR``.
- """
- V = K.lattice().vector_space()
+ - ``K`` -- A polyhedral closed convex cone.
- # 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()]
+ OUTPUT:
- return [(x,s) for x in xs for s in ss if x.inner_product(s) == 0]
+ If the base ring of ``L`` is exact, then ``True`` will be returned if
+ and only if ``L`` is a Z-operator on ``K``.
+ If the base ring of ``L`` is ``SR``, then the situation is more
+ complicated:
-def LL(K):
- r"""
- Compute the space `\mathbf{LL}` of all Lyapunov-like transformations
- on this cone.
+ - ``True`` will be returned if it can be proven that ``L``
+ is a Z-operator on ``K``.
+ - ``False`` will be returned if it can be proven that ``L``
+ is not a Z-operator on ``K``.
+ - ``False`` will also be returned if we can't decide; specifically
+ if we arrive at a symbolic inequality that cannot be resolved.
- OUTPUT:
+ .. SEEALSO::
- A list of matrices forming a basis for the space of all
- Lyapunov-like transformations on the given cone.
+ :func:`is_positive_on`,
+ :func:`is_cross_positive_on`,
+ :func:`is_lyapunov_like_on`
EXAMPLES:
- The trivial cone has no Lyapunov-like transformations::
+ The identity operator is always a Z-operator::
- sage: L = ToricLattice(0)
- sage: K = Cone([], lattice=L)
- sage: LL(K)
- []
-
- The Lyapunov-like transformations on the nonnegative orthant are
- simply diagonal matrices::
-
- sage: K = Cone([(1,)])
- sage: LL(K)
- [[1]]
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=8)
+ sage: L = identity_matrix(K.lattice_dim())
+ sage: is_Z_operator_on(L,K)
+ True
- sage: K = Cone([(1,0),(0,1)])
- sage: LL(K)
- [
- [1 0] [0 0]
- [0 0], [0 1]
- ]
+ The "zero" operator is always a Z-operator::
- 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.]_::
-
- 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: 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]
- ]
-
- If our cone is the entire space, then every transformation on it is
- Lyapunov-like::
-
- sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)])
- sage: M = MatrixSpace(QQ,2)
- sage: M.basis() == LL(K)
+ sage: K = random_cone(max_ambient_dim=8)
+ sage: R = K.lattice().vector_space().base_ring()
+ sage: L = zero_matrix(R, K.lattice_dim())
+ sage: is_Z_operator_on(L,K)
True
TESTS:
- The inner product `\left< L\left(x\right), s \right>` is zero for
- every pair `\left( x,s \right)` in the discrete complementarity set
- of the cone::
-
- sage: set_random_seed()
- sage: K = random_cone(max_dim=8)
- sage: C_of_K = discrete_complementarity_set(K)
- sage: l = [ (L*x).inner_product(s) for (x,s) in C_of_K for L in LL(K) ]
- sage: sum(map(abs, l))
- 0
-
- 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)`
+ Everything in ``K.Z_operators_gens()`` should be a Z-operator
+ on ``K``::
- sage: set_random_seed()
- sage: K = random_cone(max_dim=8)
- sage: LL2 = [ L.transpose() for L in LL(K.dual()) ]
- sage: V = VectorSpace( K.lattice().base_field(), K.lattice_dim()^2)
- sage: LL1_vecs = [ V(m.list()) for m in LL(K) ]
- sage: LL2_vecs = [ V(m.list()) for m in LL2 ]
- sage: V.span(LL1_vecs) == V.span(LL2_vecs)
+ sage: K = random_cone(max_ambient_dim=5)
+ sage: all([ is_Z_operator_on(L,K) # long time
+ ....: for L in K.Z_operators_gens() ]) # long time
+ True
+ sage: all([ is_Z_operator_on(L.change_ring(SR),K) # long time
+ ....: for L in K.Z_operators_gens() ]) # long time
True
- """
- 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 ]
+ Technically we could test this, but for now only closed convex cones
+ are supported as our ``K`` argument::
- # Vector space representation of Lyapunov-like matrices
- # (i.e. vec(L) where L is Luapunov-like).
- LL_vector = W.span(vectors).complement()
+ sage: L = identity_matrix(3)
+ sage: K = [ vector([-4,20,3]), vector([1,-5,2]) ]
+ sage: is_Z_operator_on(L,K)
+ Traceback (most recent call last):
+ ...
+ TypeError: K must be a Cone.
- # 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() ]
+ We can't give reliable answers over inexact rings::
- return matrix_basis
+ sage: K = Cone([(1,2,3), (4,5,6)])
+ sage: L = identity_matrix(RR,3)
+ sage: is_Z_operator_on(L,K)
+ Traceback (most recent call last):
+ ...
+ ValueError: The base ring of L is neither SR nor exact.
+ """
+ return is_cross_positive_on(-L,K)
-def lyapunov_rank(K):
+def is_lyapunov_like_on(L,K):
r"""
- Compute the Lyapunov (or bilinearity) rank of this cone.
+ Determine whether or not ``L`` is Lyapunov-like on ``K``.
- The Lyapunov rank of a cone can be thought of in (mainly) two ways:
+ We say that ``L`` is Lyapunov-like on a closed convex cone ``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``. This property need only be checked for generators of
+ ``K`` and its dual.
- 1. The dimension of the Lie algebra of the automorphism group of the
- cone.
+ An operator is Lyapunov-like on ``K`` if and only if both the
+ operator itself and its negation are cross-positive on ``K``.
- 2. The dimension of the linear space of all Lyapunov-like
- transformations on the cone.
+ To reliably check whether or not ``L`` is Lyapunov-like, its base
+ ring must be either exact (for example, the rationals) or ``SR``. An
+ exact ring is more reliable, but in some cases a matrix whose
+ entries contain symbolic constants like ``e`` and ``pi`` will work.
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 the first reference).
-
- .. note::
-
- In the references, the cones are always assumed to be proper. We
- do not impose this restriction.
-
- .. seealso::
+ - ``L`` -- A matrix over either an exact ring or ``SR``.
- :meth:`is_proper`
+ - ``K`` -- A polyhedral closed convex cone.
- ALGORITHM:
+ OUTPUT:
- 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}`.
+ If the base ring of ``L`` is exact, then ``True`` will be returned if
+ and only if ``L`` is Lyapunov-like on ``K``.
- REFERENCES:
+ If the base ring of ``L`` is ``SR``, then the situation is more
+ complicated:
- .. [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.
+ - ``True`` will be returned if it can be proven that ``L``
+ is Lyapunov-like on ``K``.
+ - ``False`` will be returned if it can be proven that ``L``
+ is not Lyapunov-like on ``K``.
+ - ``False`` will also be returned if we can't decide; specifically
+ if we arrive at a symbolic inequality that cannot be resolved.
- .. [Orlitzky/Gowda] M. Orlitzky and M. S. Gowda. The Lyapunov Rank of an
- Improper Cone. Work in-progress.
+ .. SEEALSO::
- .. [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.
+ :func:`is_positive_on`,
+ :func:`is_cross_positive_on`,
+ :func:`is_Z_operator_on`
EXAMPLES:
- The nonnegative orthant in `\mathbb{R}^{n}` always has rank `n`
- [Rudolf et al.]_::
-
- sage: positives = Cone([(1,)])
- sage: lyapunov_rank(positives)
- 1
- sage: quadrant = Cone([(1,0), (0,1)])
- sage: lyapunov_rank(quadrant)
- 2
- sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)])
- sage: lyapunov_rank(octant)
- 3
-
- The full space `\mathbb{R}^{n}` has Lyapunov rank `n^{2}`
- [Orlitzky/Gowda]_::
-
- sage: R5 = VectorSpace(QQ, 5)
- sage: gs = R5.basis() + [ -r for r in R5.basis() ]
- sage: K = Cone(gs)
- sage: lyapunov_rank(K)
- 25
-
- The `L^{3}_{1}` cone is known to have a Lyapunov rank of one
- [Rudolf et al.]_::
-
- sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)])
- sage: lyapunov_rank(L31)
- 1
-
- Likewise for the `L^{3}_{\infty}` cone [Rudolf et al.]_::
-
- sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)])
- sage: lyapunov_rank(L3infty)
- 1
-
- A single ray in `n` dimensions should have Lyapunov rank `n^{2} - n
- + 1` [Orlitzky/Gowda]_::
-
- sage: K = Cone([(1,0,0,0,0)])
- sage: lyapunov_rank(K)
- 21
- sage: K.lattice_dim()**2 - K.lattice_dim() + 1
- 21
-
- A subspace (of dimension `m`) in `n` dimensions should have a
- Lyapunov rank of `n^{2} - m\left(n - m)` [Orlitzky/Gowda]_::
-
- sage: e1 = (1,0,0,0,0)
- sage: neg_e1 = (-1,0,0,0,0)
- sage: e2 = (0,1,0,0,0)
- sage: neg_e2 = (0,-1,0,0,0)
- sage: z = (0,0,0,0,0)
- sage: K = Cone([e1, neg_e1, e2, neg_e2, z, z, z])
- sage: lyapunov_rank(K)
- 19
- sage: K.lattice_dim()**2 - K.dim()*codim(K)
- 19
-
- The Lyapunov rank should be additive on a product of proper cones
- [Rudolf et al.]_::
-
- sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)])
- sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)])
- sage: K = L31.cartesian_product(octant)
- sage: lyapunov_rank(K) == lyapunov_rank(L31) + lyapunov_rank(octant)
- 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}`::
+ Diagonal matrices are Lyapunov-like operators on the nonnegative
+ orthant::
- 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.]_::
-
- 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),(0,1,0),(0,0,1)])
+ sage: L = diagonal_matrix(random_vector(QQ,3))
+ sage: is_lyapunov_like_on(L,K)
True
TESTS:
- The Lyapunov rank should be additive on a product of proper cones
- [Rudolf et al.]_::
-
- sage: set_random_seed()
- sage: K1 = random_cone(max_dim=8, strictly_convex=True, solid=True)
- sage: K2 = random_cone(max_dim=8, strictly_convex=True, solid=True)
- sage: K = K1.cartesian_product(K2)
- sage: lyapunov_rank(K) == lyapunov_rank(K1) + lyapunov_rank(K2)
- True
-
- The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
- itself [Rudolf et al.]_::
-
- sage: set_random_seed()
- sage: K = random_cone(max_dim=8)
- sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
- True
-
- Make sure we exercise the non-strictly-convex/non-solid case::
-
- sage: set_random_seed()
- sage: K = random_cone(max_dim=8, strictly_convex=False, solid=False)
- sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
- True
-
- Let's check the other permutations as well, just to be sure::
+ The identity operator is always Lyapunov-like::
sage: set_random_seed()
- sage: K = random_cone(max_dim=8, strictly_convex=False, solid=True)
- sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+ sage: K = random_cone(max_ambient_dim=8)
+ sage: L = identity_matrix(K.lattice_dim())
+ sage: is_lyapunov_like_on(L,K)
True
- ::
+ The "zero" operator is always Lyapunov-like::
- sage: set_random_seed()
- sage: K = random_cone(max_dim=8, strictly_convex=True, solid=False)
- sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+ sage: K = random_cone(max_ambient_dim=8)
+ sage: R = K.lattice().vector_space().base_ring()
+ sage: L = zero_matrix(R, K.lattice_dim())
+ sage: is_lyapunov_like_on(L,K)
True
- ::
+ Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like
+ on ``K``::
- sage: set_random_seed()
- sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True)
- sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+ sage: K = random_cone(max_ambient_dim=5)
+ sage: all([ is_lyapunov_like_on(L,K) # long time
+ ....: for L in K.lyapunov_like_basis() ]) # long time
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::
-
- sage: set_random_seed()
- sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True)
- sage: b = lyapunov_rank(K)
- sage: n = K.lattice_dim()
- sage: (n == 0 or 1 <= b) and b <= n
+ sage: all([ is_lyapunov_like_on(L.change_ring(SR),K) # long time
+ ....: for L in K.lyapunov_like_basis() ]) # long time
True
- sage: b == n-1
- False
-
- In fact [Orlitzky/Gowda]_, no closed convex polyhedral cone can have
- Lyapunov rank `n-1` in `n` dimensions::
- sage: set_random_seed()
- sage: K = random_cone(max_dim=8)
- sage: b = lyapunov_rank(K)
- sage: n = K.lattice_dim()
- sage: b == n-1
- False
+ Technically we could test this, but for now only closed convex cones
+ are supported as our ``K`` argument::
- The calculation of the Lyapunov rank of an improper cone can be
- reduced to that of a proper cone [Orlitzky/Gowda]_::
+ sage: L = identity_matrix(3)
+ sage: K = [ vector([2,2,-1]), vector([5,4,-3]) ]
+ sage: is_lyapunov_like_on(L,K)
+ Traceback (most recent call last):
+ ...
+ TypeError: K must be a Cone.
- sage: set_random_seed()
- sage: K = random_cone(max_dim=8)
- sage: actual = lyapunov_rank(K)
- sage: K_S = restrict_span(K)
- sage: P = restrict_span(K_S.dual()).dual()
- sage: l = lineality(K)
- sage: c = codim(K)
- sage: expected = lyapunov_rank(P) + K.dim()*(l + c) + c**2
- sage: actual == expected
- True
-
- The Lyapunov rank of a proper cone is just the dimension of ``LL(K)``::
-
- sage: set_random_seed()
- sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True)
- sage: lyapunov_rank(K) == len(LL(K))
- True
+ We can't give reliable answers over inexact rings::
- In fact the same can be said of any cone. These additional tests
- just increase our confidence that the reduction scheme works::
-
- sage: set_random_seed()
- sage: K = random_cone(max_dim=8, strictly_convex=True, solid=False)
- sage: lyapunov_rank(K) == len(LL(K))
- True
-
- ::
-
- sage: set_random_seed()
- sage: K = random_cone(max_dim=8, strictly_convex=False, solid=True)
- sage: lyapunov_rank(K) == len(LL(K))
- True
-
- ::
-
- sage: set_random_seed()
- sage: K = random_cone(max_dim=8, strictly_convex=False, solid=False)
- sage: lyapunov_rank(K) == len(LL(K))
- True
+ sage: K = Cone([(1,2,3), (4,5,6)])
+ sage: L = identity_matrix(RR,3)
+ sage: is_lyapunov_like_on(L,K)
+ Traceback (most recent call last):
+ ...
+ ValueError: The base ring of L is neither SR nor exact.
"""
- K_orig = K
- beta = 0
+ if not is_Cone(K):
+ raise TypeError('K must be a Cone.')
+ if not L.base_ring().is_exact() and not L.base_ring() is SR:
+ raise ValueError('The base ring of L is neither SR nor exact.')
- m = K.dim()
- n = K.lattice_dim()
- l = lineality(K)
+ return all([ s*(L*x) == 0
+ for (x,s) in K.discrete_complementarity_set() ])
- if m < n:
- # K is not solid, project onto its span.
- K = restrict_span(K)
- # Lemma 2
- beta += m*(n - m) + (n - m)**2
+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)
- if l > 0:
- # K is not pointed, project its dual onto its span.
- # Uses a proposition from our paper, i.e. this is
- # equivalent to K = restrict_span(K.dual()).dual()
- #K = restrict_span(intersect_span(K,K.dual()), K.dual())
- K = restrict_span(K, K.dual())
+def Sigma_cone(K):
+ gens = K.cross_positive_operators_gens()
+ L = ToricLattice(K.lattice_dim()**2)
+ return Cone([ g.list() for g in gens ], lattice=L, check=False)
- # Lemma 3
- beta += m * l
+def Z_cone(K):
+ gens = K.Z_operators_gens()
+ L = ToricLattice(K.lattice_dim()**2)
+ return Cone([ g.list() for g in gens ], lattice=L, check=False)
- beta += len(LL(K))
- return beta
+def pi_cone(K1, K2=None):
+ if K2 is None:
+ K2 = K1
+ gens = K1.positive_operators_gens(K2)
+ L = ToricLattice(K1.lattice_dim()*K2.lattice_dim())
+ return Cone([ g.list() for g in gens ], lattice=L, check=False)
projects / sage.d.git / blobdiff