from sage.all import *
+from sage.geometry.cone import is_Cone
-def is_lyapunov_like(L,K):
+def is_positive_on(L,K):
r"""
- Determine whether or not ``L`` is Lyapunov-like on ``K``.
+ 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``.
- 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.
+ 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:
- - ``L`` -- A linear transformation or matrix.
+ - ``L`` -- A matrix over either an exact ring or ``SR``.
- ``K`` -- A polyhedral closed convex cone.
OUTPUT:
- ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``,
- and ``False`` otherwise.
+ If the base ring of ``L`` is exact, then ``True`` will be returned if
+ and only if ``L`` is positive on ``K``.
- .. WARNING::
+ If the base ring of ``L`` is ``SR``, then the situation is more
+ complicated:
- 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.
+ - ``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.
- REFERENCES:
+ .. SEEALSO::
- M. Orlitzky. The Lyapunov rank of an improper cone.
- http://www.optimization-online.org/DB_HTML/2015/10/5135.html
+ :func:`is_cross_positive_on`,
+ :func:`is_Z_operator_on`,
+ :func:`is_lyapunov_like_on`
EXAMPLES:
- The identity is always Lyapunov-like in a nontrivial space::
+ Nonnegative matrices are positive operators on the nonnegative
+ orthant::
+
+ 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
+
+ TESTS:
+
+ The identity operator is always positive::
sage: set_random_seed()
- sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
+ sage: K = random_cone(max_ambient_dim=8)
sage: L = identity_matrix(K.lattice_dim())
- sage: is_lyapunov_like(L,K)
+ sage: is_positive_on(L,K)
True
- As is the "zero" transformation::
+ The "zero" operator is always positive::
- sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
+ 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(L,K)
+ sage: is_positive_on(L,K)
True
- Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like
- on ``K``::
+ Everything in ``K.positive_operators_gens()`` should be
+ positive on ``K``::
- 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() ])
+ 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: 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: 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: 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.
+
"""
- return all([(L*x).inner_product(s) == 0
- for (x,s) in K.discrete_complementarity_set()])
+ 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 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() ])
-def motzkin_decomposition(K):
+
+def is_cross_positive_on(L,K):
r"""
- Return the pair of components in the Motzkin decomposition of this cone.
+ Determine whether or not ``L`` is cross-positive on ``K``.
- 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``.
+ 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.
- OUTPUT:
+ 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.
- 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``.
+ INPUT:
- REFERENCES:
+ - ``L`` -- A matrix over either an exact ring or ``SR``.
- .. [Stoer-Witzgall] J. Stoer and C. Witzgall. Convexity and
- Optimization in Finite Dimensions I. Springer-Verlag, New
- York, 1970.
+ - ``K`` -- A polyhedral closed convex cone.
- EXAMPLES:
+ OUTPUT:
- The nonnegative orthant is strictly convex, so it is its own
- strictly convex component and its subspace component is trivial::
+ If the base ring of ``L`` is exact, then ``True`` will be returned if
+ and only if ``L`` is cross-positive on ``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
+ If the base ring of ``L`` is ``SR``, then the situation is more
+ complicated:
- Likewise, full spaces are their own subspace components::
+ - ``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.
- 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
+ .. SEEALSO::
- TESTS:
+ :func:`is_positive_on`,
+ :func:`is_Z_operator_on`,
+ :func:`is_lyapunov_like_on`
+
+ EXAMPLES:
- 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::
+ The identity operator is always cross-positive::
sage: set_random_seed()
sage: K = random_cone(max_ambient_dim=8)
- sage: (P,S) = motzkin_decomposition(K)
- sage: x = K.random_element(ring=QQ)
- sage: P.contains(x) or S.contains(x)
- True
- sage: x.is_zero() or (P.contains(x) != S.contains(x))
+ sage: L = identity_matrix(K.lattice_dim())
+ sage: is_cross_positive_on(L,K)
True
- The strictly convex component should always be strictly convex, and
- the subspace component should always be a subspace::
+ The "zero" operator is always cross-positive::
- 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()
+ sage: R = K.lattice().vector_space().base_ring()
+ sage: L = zero_matrix(R, K.lattice_dim())
+ sage: is_cross_positive_on(L,K)
True
- A strictly convex cone should be equal to its strictly convex component::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8, strictly_convex=True)
- sage: (P,_) = motzkin_decomposition(K)
- sage: K.is_equivalent(P)
- True
+ TESTS:
- The generators of the components are obtained from orthogonal
- projections of the original generators [Stoer-Witzgall]_::
+ Everything in ``K.cross_positive_operators_gens()`` should be
+ cross-positive on ``K``::
- sage: set_random_seed()
- 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)
+ 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: 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)
+ sage: all([ is_cross_positive_on(L.change_ring(SR),K) # long time
+ ....: for L in K.cross_positive_operators_gens() ]) # long time
True
- """
- # 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(), check=False)
- # Since ``S`` is a subspace, the rays of its dual generate its
- # orthogonal complement.
- S_perp = Cone(S.dual(), K.lattice(), check=False)
- P = K.intersection(S_perp)
+ Technically we could test this, but for now only closed convex cones
+ are supported as our ``K`` argument::
- return (P,S)
+ 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.
+ We can't give reliable answers over inexact rings::
-def positive_operator_gens(K):
- r"""
- Compute generators of the cone of positive operators on this cone.
-
- OUTPUT:
+ 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.
- 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.
+ """
+ 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.')
- EXAMPLES:
+ return all([ s*(L*x) >= 0
+ for (x,s) in K.discrete_complementarity_set() ])
- Positive operators on the nonnegative orthant are nonnegative matrices::
+def is_Z_operator_on(L,K):
+ r"""
+ Determine whether or not ``L`` is a Z-operator on ``K``.
- sage: K = Cone([(1,)])
- sage: positive_operator_gens(K)
- [[1]]
+ 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),(0,1)])
- sage: positive_operator_gens(K)
- [
- [1 0] [0 1] [0 0] [0 0]
- [0 0], [0 0], [1 0], [0 1]
- ]
+ A matrix is a Z-operator on ``K`` if and only if its negation is a
+ cross-positive operator on ``K``.
- The trivial cone in a trivial space has no positive operators::
+ 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.
- sage: K = Cone([], ToricLattice(0))
- sage: positive_operator_gens(K)
- []
+ INPUT:
- Every operator is positive on the trivial cone::
+ - ``L`` -- A matrix over either an exact ring or ``SR``.
- sage: K = Cone([(0,)])
- sage: positive_operator_gens(K)
- [[1], [-1]]
+ - ``K`` -- A polyhedral closed convex cone.
- 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]
- ]
+ OUTPUT:
- Every operator is positive on the ambient vector space::
+ If the base ring of ``L`` is exact, then ``True`` will be returned if
+ and only if ``L`` is a Z-operator on ``K``.
- sage: K = Cone([(1,),(-1,)])
- sage: K.is_full_space()
- True
- sage: positive_operator_gens(K)
- [[1], [-1]]
+ If the base ring of ``L`` is ``SR``, then the situation is more
+ complicated:
- 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]
- ]
-
- A non-obvious application is to find the positive operators on the
- right half-plane::
-
- 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]
- ]
+ - ``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.
- TESTS:
+ .. SEEALSO::
- Each positive operator generator should send the generators of the
- cone into the cone::
+ :func:`is_positive_on`,
+ :func:`is_cross_positive_on`,
+ :func:`is_lyapunov_like_on`
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- 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
+ EXAMPLES:
- Each positive operator generator should send a random element of the
- cone into the cone::
+ The identity operator is always a Z-operator::
sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: pi_of_K = positive_operator_gens(K)
- sage: all([ K.contains(P*K.random_element(QQ)) for P in pi_of_K ])
+ sage: K = random_cone(max_ambient_dim=8)
+ sage: L = identity_matrix(K.lattice_dim())
+ sage: is_Z_operator_on(L,K)
True
- A random element of the positive operator cone should send the
- generators of the cone into the cone::
+ The "zero" operator is always a Z-operator::
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: pi_of_K = positive_operator_gens(K)
- sage: L = ToricLattice(K.lattice_dim()**2)
- sage: pi_cone = Cone([ g.list() for g in pi_of_K ],
- ....: lattice=L,
- ....: check=False)
- sage: P = matrix(K.lattice_dim(), pi_cone.random_element(QQ).list())
- sage: all([ K.contains(P*x) for x in 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
- 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=4)
- sage: pi_of_K = positive_operator_gens(K)
- sage: L = ToricLattice(K.lattice_dim()**2)
- sage: pi_cone = Cone([ g.list() for g in pi_of_K ],
- ....: lattice=L,
- ....: check=False)
- sage: P = matrix(K.lattice_dim(), pi_cone.random_element(QQ).list())
- sage: K.contains(P*K.random_element(ring=QQ))
- True
+ TESTS:
- The lineality space of the dual of the cone of positive operators
- can be computed from the lineality spaces of the cone and its dual::
+ Everything in ``K.Z_operators_gens()`` should be a Z-operator
+ on ``K``::
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: pi_of_K = positive_operator_gens(K)
- sage: L = ToricLattice(K.lattice_dim()**2)
- sage: pi_cone = Cone([ g.list() for g in pi_of_K ],
- ....: lattice=L,
- ....: check=False)
- sage: actual = pi_cone.dual().linear_subspace()
- sage: U1 = [ vector((s.tensor_product(x)).list())
- ....: for x in K.lines()
- ....: for s in K.dual() ]
- sage: U2 = [ vector((s.tensor_product(x)).list())
- ....: for x in K
- ....: for s in K.dual().lines() ]
- sage: expected = pi_cone.lattice().vector_space().span(U1 + U2)
- sage: actual == expected
+ 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
-
- The lineality of the dual of the cone of positive operators
- is known from its lineality space::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- 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: pi_cone = Cone([p.list() for p in pi_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: actual = pi_cone.dual().lineality()
- sage: expected = l*(m - l) + m*(n - m)
- sage: actual == expected
+ sage: all([ is_Z_operator_on(L.change_ring(SR),K) # long time
+ ....: for L in K.Z_operators_gens() ]) # long time
True
- The dimension of the cone of positive operators is given by the
- corollary in my paper::
+ 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_ambient_dim=4)
- 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: pi_cone = Cone([p.list() for p in pi_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: actual = pi_cone.dim()
- sage: expected = n**2 - l*(m - l) - (n - m)*m
- sage: actual == expected
- True
+ 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.
- The trivial cone, full space, and half-plane all give rise to the
- expected dimensions::
- sage: n = ZZ.random_element().abs()
- sage: K = Cone([[0] * n], ToricLattice(n))
- sage: K.is_trivial()
- True
- sage: L = ToricLattice(n^2)
- sage: pi_of_K = positive_operator_gens(K)
- sage: pi_cone = Cone([p.list() for p in pi_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: actual = pi_cone.dim()
- sage: actual == n^2
- True
- sage: K = K.dual()
- sage: K.is_full_space()
- True
- sage: pi_of_K = positive_operator_gens(K)
- sage: pi_cone = Cone([p.list() for p in pi_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: actual = pi_cone.dim()
- sage: actual == n^2
- True
- sage: K = Cone([(1,0),(0,1),(0,-1)])
- sage: pi_of_K = positive_operator_gens(K)
- sage: actual = Cone([p.list() for p in pi_of_K], check=False).dim()
- sage: actual == 3
- True
+ We can't give reliable answers over inexact rings::
- The lineality of the cone of positive operators follows from the
- description of its generators::
+ 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.
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: n = K.lattice_dim()
- sage: pi_of_K = positive_operator_gens(K)
- sage: L = ToricLattice(n**2)
- sage: pi_cone = Cone([p.list() for p in pi_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: actual = pi_cone.lineality()
- sage: expected = n**2 - K.dim()*K.dual().dim()
- sage: actual == expected
- True
+ """
+ return is_cross_positive_on(-L,K)
- The trivial cone, full space, and half-plane all give rise to the
- expected linealities::
- sage: n = ZZ.random_element().abs()
- sage: K = Cone([[0] * n], ToricLattice(n))
- sage: K.is_trivial()
- True
- sage: L = ToricLattice(n^2)
- sage: pi_of_K = positive_operator_gens(K)
- sage: pi_cone = Cone([p.list() for p in pi_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: actual = pi_cone.lineality()
- sage: actual == n^2
- True
- sage: K = K.dual()
- sage: K.is_full_space()
- True
- sage: pi_of_K = positive_operator_gens(K)
- sage: pi_cone = Cone([p.list() for p in pi_of_K], lattice=L)
- sage: pi_cone.lineality() == n^2
- True
- sage: K = Cone([(1,0),(0,1),(0,-1)])
- sage: pi_of_K = positive_operator_gens(K)
- sage: pi_cone = Cone([p.list() for p in pi_of_K], check=False)
- sage: actual = pi_cone.lineality()
- sage: actual == 2
- True
+def is_lyapunov_like_on(L,K):
+ r"""
+ Determine whether or not ``L`` is Lyapunov-like on ``K``.
- A cone is proper if and only if its cone of positive operators
- is proper::
+ 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.
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: pi_of_K = positive_operator_gens(K)
- sage: L = ToricLattice(K.lattice_dim()**2)
- sage: pi_cone = Cone([p.list() for p in pi_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: K.is_proper() == pi_cone.is_proper()
- True
+ An operator is Lyapunov-like on ``K`` if and only if both the
+ operator itself and its negation are cross-positive on ``K``.
- The positive operators of a permuted cone can be obtained by
- conjugation::
+ 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.
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: L = ToricLattice(K.lattice_dim()**2)
- sage: p = SymmetricGroup(K.lattice_dim()).random_element().matrix()
- sage: pK = Cone([ p*k for k in K ], K.lattice(), check=False)
- sage: pi_of_pK = positive_operator_gens(pK)
- sage: actual = Cone([t.list() for t in pi_of_pK],
- ....: lattice=L,
- ....: check=False)
- sage: pi_of_K = positive_operator_gens(K)
- sage: expected = Cone([(p*t*p.inverse()).list() for t in pi_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: actual.is_equivalent(expected)
- True
+ INPUT:
- A transformation is positive on a cone if and only if its adjoint is
- positive on the dual of that cone::
+ - ``L`` -- A matrix over either an exact ring or ``SR``.
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: F = K.lattice().vector_space().base_field()
- sage: n = K.lattice_dim()
- sage: L = ToricLattice(n**2)
- sage: W = VectorSpace(F, n**2)
- sage: pi_of_K = positive_operator_gens(K)
- sage: pi_of_K_star = positive_operator_gens(K.dual())
- sage: pi_cone = Cone([p.list() for p in pi_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: pi_star = Cone([p.list() for p in pi_of_K_star],
- ....: lattice=L,
- ....: check=False)
- sage: M = MatrixSpace(F, n)
- sage: L = M(pi_cone.random_element(ring=QQ).list())
- sage: pi_star.contains(W(L.transpose().list()))
- True
-
- sage: L = W.random_element()
- sage: L_star = W(M(L.list()).transpose().list())
- sage: pi_cone.contains(L) == pi_star.contains(L_star)
- True
- """
- # 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()
-
- tensor_products = [ s.tensor_product(x) for x in K for s in K.dual() ]
-
- # Convert those tensor products to long vectors.
- W = VectorSpace(F, n**2)
- vectors = [ W(tp.list()) for tp in tensor_products ]
-
- check = True
- if K.is_solid() or K.is_strictly_convex():
- # The lineality space of either ``K`` or ``K.dual()`` is
- # trivial and it's easy to show that our generating set is
- # minimal. I would love a proof that this works when ``K`` is
- # neither pointed nor solid.
- #
- # Note that in that case we can get *duplicates*, since the
- # tensor product of (x,s) is the same as that of (-x,-s).
- check = False
-
- # Create the dual cone of the positive operators, expressed as
- # long vectors.
- pi_dual = Cone(vectors, ToricLattice(W.dimension()), check=check)
-
- # 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 ]
-
-
-def Z_transformation_gens(K):
- r"""
- Compute generators of the cone of Z-transformations on this cone.
+ - ``K`` -- A polyhedral closed convex cone.
OUTPUT:
- A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``.
- Each matrix ``L`` in the list should have the property that
- ``(L*x).inner_product(s) <= 0`` whenever ``(x,s)`` is an element the
- discrete complementarity set of ``K``. Moreover, any nonnegative
- linear combination of these matrices shares the same property.
+ If the base ring of ``L`` is exact, then ``True`` will be returned if
+ and only if ``L`` is Lyapunov-like on ``K``.
- EXAMPLES:
+ If the base ring of ``L`` is ``SR``, then the situation is more
+ complicated:
- Z-transformations on the nonnegative orthant are just Z-matrices.
- That is, matrices whose off-diagonal elements are nonnegative::
-
- sage: K = Cone([(1,0),(0,1)])
- sage: Z_transformation_gens(K)
- [
- [ 0 -1] [ 0 0] [-1 0] [1 0] [ 0 0] [0 0]
- [ 0 0], [-1 0], [ 0 0], [0 0], [ 0 -1], [0 1]
- ]
- sage: K = Cone([(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)])
- sage: all([ z[i][j] <= 0 for z in Z_transformation_gens(K)
- ....: for i in range(z.nrows())
- ....: for j in range(z.ncols())
- ....: if i != j ])
- True
+ - ``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.
- The trivial cone in a trivial space has no Z-transformations::
+ .. SEEALSO::
- sage: K = Cone([], ToricLattice(0))
- sage: Z_transformation_gens(K)
- []
+ :func:`is_positive_on`,
+ :func:`is_cross_positive_on`,
+ :func:`is_Z_operator_on`
- Every operator is a Z-transformation on the ambient vector space::
+ EXAMPLES:
- sage: K = Cone([(1,),(-1,)])
- sage: K.is_full_space()
- True
- sage: Z_transformation_gens(K)
- [[-1], [1]]
+ Diagonal matrices are Lyapunov-like operators on the nonnegative
+ orthant::
- sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
- sage: K.is_full_space()
- True
- sage: Z_transformation_gens(K)
- [
- [-1 0] [1 0] [ 0 -1] [0 1] [ 0 0] [0 0] [ 0 0] [0 0]
- [ 0 0], [0 0], [ 0 0], [0 0], [-1 0], [1 0], [ 0 -1], [0 1]
- ]
-
- A non-obvious application is to find the Z-transformations on the
- right half-plane::
-
- sage: K = Cone([(1,0),(0,1),(0,-1)])
- sage: Z_transformation_gens(K)
- [
- [-1 0] [1 0] [ 0 0] [0 0] [ 0 0] [0 0]
- [ 0 0], [0 0], [-1 0], [1 0], [ 0 -1], [0 1]
- ]
-
- 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
+ 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 Z-property is possessed by every Z-transformation::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- 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 lineality space of the cone of Z-transformations is the space of
- Lyapunov-like transformations::
+ The identity operator is always Lyapunov-like::
sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: L = ToricLattice(K.lattice_dim()**2)
- sage: Z_cone = Cone([ z.list() for z in Z_transformation_gens(K) ],
- ....: lattice=L,
- ....: check=False)
- sage: ll_basis = [ vector(l.list()) for l in K.lyapunov_like_basis() ]
- sage: lls = L.vector_space().span(ll_basis)
- sage: Z_cone.linear_subspace() == lls
+ sage: K = random_cone(max_ambient_dim=8)
+ sage: L = identity_matrix(K.lattice_dim())
+ sage: is_lyapunov_like_on(L,K)
True
- The lineality of the Z-transformations on a cone is the Lyapunov
- rank of that cone::
+ The "zero" operator is always Lyapunov-like::
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- 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,
- ....: check=False)
- sage: Z_cone.lineality() == K.lyapunov_rank()
+ 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
- The lineality spaces of the duals of the positive operator and
- Z-transformation cones are equal. From this it follows that the
- dimensions of the Z-transformation cone and positive operator cone
- are equal::
+ Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like
+ on ``K``::
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- 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_cone = Cone([p.list() for p in pi_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: Z_cone = Cone([ z.list() for z in Z_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: pi_cone.dim() == Z_cone.dim()
+ 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
- sage: pi_star = pi_cone.dual()
- sage: z_star = Z_cone.dual()
- sage: pi_star.linear_subspace() == z_star.linear_subspace()
+ sage: all([ is_lyapunov_like_on(L.change_ring(SR),K) # long time
+ ....: for L in K.lyapunov_like_basis() ]) # long time
True
- The trivial cone, full space, and half-plane all give rise to the
- expected dimensions::
+ Technically we could test this, but for now only closed convex cones
+ are supported as our ``K`` argument::
- sage: n = ZZ.random_element().abs()
- sage: K = Cone([[0] * n], ToricLattice(n))
- sage: K.is_trivial()
- True
- sage: L = ToricLattice(n^2)
- sage: Z_of_K = Z_transformation_gens(K)
- sage: Z_cone = Cone([z.list() for z in Z_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: actual = Z_cone.dim()
- sage: actual == n^2
- True
- sage: K = K.dual()
- sage: K.is_full_space()
- True
- sage: Z_of_K = Z_transformation_gens(K)
- sage: Z_cone = Cone([z.list() for z in Z_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: actual = Z_cone.dim()
- sage: actual == n^2
- True
- sage: K = Cone([(1,0),(0,1),(0,-1)])
- sage: Z_of_K = Z_transformation_gens(K)
- sage: Z_cone = Cone([z.list() for z in Z_of_K], check=False)
- sage: Z_cone.dim() == 3
- True
+ 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.
- The Z-transformations of a permuted cone can be obtained by
- conjugation::
+ We can't give reliable answers over inexact rings::
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: L = ToricLattice(K.lattice_dim()**2)
- sage: p = SymmetricGroup(K.lattice_dim()).random_element().matrix()
- sage: pK = Cone([ p*k for k in K ], K.lattice(), check=False)
- sage: Z_of_pK = Z_transformation_gens(pK)
- sage: actual = Cone([t.list() for t in Z_of_pK],
- ....: lattice=L,
- ....: check=False)
- sage: Z_of_K = Z_transformation_gens(K)
- sage: expected = Cone([(p*t*p.inverse()).list() for t in Z_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: actual.is_equivalent(expected)
- 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.
- A transformation is a Z-transformation on a cone if and only if its
- adjoint is a Z-transformation on the dual of that cone::
+ """
+ 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.')
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: F = K.lattice().vector_space().base_field()
- sage: n = K.lattice_dim()
- sage: L = ToricLattice(n**2)
- sage: W = VectorSpace(F, n**2)
- sage: Z_of_K = Z_transformation_gens(K)
- sage: Z_of_K_star = Z_transformation_gens(K.dual())
- sage: Z_cone = Cone([p.list() for p in Z_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: Z_star = Cone([p.list() for p in Z_of_K_star],
- ....: lattice=L,
- ....: check=False)
- sage: M = MatrixSpace(F, n)
- sage: L = M(Z_cone.random_element(ring=QQ).list())
- sage: Z_star.contains(W(L.transpose().list()))
- True
+ return all([ s*(L*x) == 0
+ for (x,s) in K.discrete_complementarity_set() ])
- sage: L = W.random_element()
- sage: L_star = W(M(L.list()).transpose().list())
- sage: Z_cone.contains(L) == Z_star.contains(L_star)
- 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()
-
- # 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 ]
-
- check = True
- if K.is_solid() or K.is_strictly_convex():
- # The lineality space of either ``K`` or ``K.dual()`` is
- # trivial and it's easy to show that our generating set is
- # minimal. I would love a proof that this works when ``K`` is
- # neither pointed nor solid.
- #
- # Note that in that case we can get *duplicates*, since the
- # tensor product of (x,s) is the same as that of (-x,-s).
- check = False
-
- # Create the dual cone of the cross-positive operators,
- # expressed as long vectors.
- Sigma_dual = Cone(vectors, lattice=ToricLattice(W.dimension()), check=check)
-
- # Now compute the desired cone from its dual...
- 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 ]
+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)
+
+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)
def Z_cone(K):
- gens = Z_transformation_gens(K)
+ gens = K.Z_operators_gens()
L = ToricLattice(K.lattice_dim()**2)
return Cone([ g.list() for g in gens ], lattice=L, check=False)
-def pi_cone(K):
- gens = positive_operator_gens(K)
- L = ToricLattice(K.lattice_dim()**2)
+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