if K1.is_strictly_convex() != K2.is_strictly_convex():
return False
- if len(LL(K1)) != len(LL(K2)):
+ if len(K1.LL()) != len(K2.LL()):
return False
- C_of_K1 = discrete_complementarity_set(K1)
- C_of_K2 = discrete_complementarity_set(K2)
+ C_of_K1 = K1.discrete_complementarity_set()
+ C_of_K2 = K2.discrete_complementarity_set()
if len(C_of_K1) != len(C_of_K2):
return False
return Cone(K_W_rays, lattice=L)
-
-def discrete_complementarity_set(K):
- r"""
- Compute a discrete complementarity set of this cone.
-
- A discrete complementarity set of `K` is the set of all orthogonal
- pairs `(x,s)` such that `x \in G_{1}` and `s \in G_{2}` for some
- generating sets `G_{1}` of `K` and `G_{2}` of its dual. Polyhedral
- convex cones are input in terms of their generators, so "the" (this
- particular) discrete complementarity set corresponds to ``G1
- == K.rays()`` and ``G2 == K.dual().rays()``.
-
- OUTPUT:
-
- A list of pairs `(x,s)` such that,
-
- * Both `x` and `s` are vectors (not rays).
- * `x` is one of ``K.rays()``.
- * `s` is one of ``K.dual().rays()``.
- * `x` and `s` are orthogonal.
-
- REFERENCES:
-
- .. [Orlitzky/Gowda] M. Orlitzky and M. S. Gowda. The Lyapunov Rank of an
- Improper Cone. Work in-progress.
-
- EXAMPLES:
-
- The discrete complementarity set of the nonnegative orthant consists
- of pairs of standard basis vectors::
-
- sage: K = Cone([(1,0),(0,1)])
- sage: discrete_complementarity_set(K)
- [((1, 0), (0, 1)), ((0, 1), (1, 0))]
-
- If the cone consists of a single ray, the second components of the
- discrete complementarity set should generate the orthogonal
- complement of that ray::
-
- sage: K = Cone([(1,0)])
- sage: discrete_complementarity_set(K)
- [((1, 0), (0, 1)), ((1, 0), (0, -1))]
- sage: K = Cone([(1,0,0)])
- sage: discrete_complementarity_set(K)
- [((1, 0, 0), (0, 1, 0)),
- ((1, 0, 0), (0, -1, 0)),
- ((1, 0, 0), (0, 0, 1)),
- ((1, 0, 0), (0, 0, -1))]
-
- When the cone is the entire space, its dual is the trivial cone, so
- the discrete complementarity set is empty::
-
- sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
- sage: discrete_complementarity_set(K)
- []
-
- Likewise when this cone is trivial (its dual is the entire space)::
-
- sage: L = ToricLattice(0)
- sage: K = Cone([], ToricLattice(0))
- sage: discrete_complementarity_set(K)
- []
-
- TESTS:
-
- The complementarity set of the dual can be obtained by switching the
- components of the complementarity set of the original cone::
-
- sage: set_random_seed()
- sage: K1 = random_cone(max_ambient_dim=6)
- sage: K2 = K1.dual()
- sage: expected = [(x,s) for (s,x) in discrete_complementarity_set(K2)]
- sage: actual = discrete_complementarity_set(K1)
- sage: sorted(actual) == sorted(expected)
- True
-
- The pairs in the discrete complementarity set are in fact
- complementary::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=6)
- sage: dcs = discrete_complementarity_set(K)
- sage: sum([x.inner_product(s).abs() for (x,s) in dcs])
- 0
-
- """
- V = K.lattice().vector_space()
-
- # Convert rays to vectors so that we can compute inner products.
- xs = [V(x) for x in K.rays()]
-
- # We also convert the generators of the dual cone so that we
- # return pairs of vectors and not (vector, ray) pairs.
- ss = [V(s) for s in K.dual().rays()]
-
- return [(x,s) for x in xs for s in ss if x.inner_product(s) == 0]
-
-
-def LL(K):
- r"""
- Compute the space `\mathbf{LL}` of all Lyapunov-like transformations
- on this cone.
-
- OUTPUT:
-
- A list of matrices forming a basis for the space of all
- Lyapunov-like transformations on the given cone.
-
- EXAMPLES:
-
- The trivial cone has no Lyapunov-like transformations::
-
- sage: L = ToricLattice(0)
- sage: K = Cone([], lattice=L)
- sage: LL(K)
- []
-
- The Lyapunov-like transformations on the nonnegative orthant are
- simply diagonal matrices::
-
- sage: K = Cone([(1,)])
- sage: LL(K)
- [[1]]
-
- sage: K = Cone([(1,0),(0,1)])
- sage: LL(K)
- [
- [1 0] [0 0]
- [0 0], [0 1]
- ]
-
- sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
- sage: LL(K)
- [
- [1 0 0] [0 0 0] [0 0 0]
- [0 0 0] [0 1 0] [0 0 0]
- [0 0 0], [0 0 0], [0 0 1]
- ]
-
- Only the identity matrix is Lyapunov-like on the `L^{3}_{1}` and
- `L^{3}_{\infty}` cones [Rudolf et al.]_::
-
- 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)
- 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_ambient_dim=8)
- sage: C_of_K = discrete_complementarity_set(K)
- sage: l = [ (L*x).inner_product(s) for (x,s) in C_of_K for L in LL(K) ]
- sage: sum(map(abs, l))
- 0
-
- The Lyapunov-like transformations on a cone and its dual are related
- by transposition, but we're not guaranteed to compute transposed
- elements of `LL\left( K \right)` as our basis for `LL\left( K^{*}
- \right)`
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8)
- sage: LL2 = [ L.transpose() for L in LL(K.dual()) ]
- sage: V = VectorSpace( K.lattice().base_field(), K.lattice_dim()^2)
- sage: LL1_vecs = [ V(m.list()) for m in LL(K) ]
- sage: LL2_vecs = [ V(m.list()) for m in LL2 ]
- sage: V.span(LL1_vecs) == V.span(LL2_vecs)
- 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 ]
-
- # Vector space representation of Lyapunov-like matrices
- # (i.e. vec(L) where L is Luapunov-like).
- LL_vector = W.span(vectors).complement()
-
- # Now construct an ambient MatrixSpace in which to stick our
- # transformations.
- M = MatrixSpace(V.base_ring(), V.dimension())
-
- matrix_basis = [ M(v.list()) for v in LL_vector.basis() ]
-
- return matrix_basis
-
-
-
def lyapunov_rank(K):
r"""
Compute the Lyapunov rank (or bilinearity rank) of this cone.
sage: actual == expected
True
- The Lyapunov rank of any cone is just the dimension of ``LL(K)``::
+ The Lyapunov rank of any cone is just the dimension of ``K.LL()``::
sage: set_random_seed()
sage: K = random_cone(max_ambient_dim=8)
- sage: lyapunov_rank(K) == len(LL(K))
+ sage: lyapunov_rank(K) == len(K.LL())
True
We can make an imperfect cone perfect by adding a slack variable
# Non-pointed reduction lemma.
beta += l * m
- beta += len(LL(K))
+ beta += len(K.LL())
return beta
+
+
+
+def is_lyapunov_like(L,K):
+ r"""
+ Determine whether or not ``L`` is Lyapunov-like on ``K``.
+
+ We say that ``L`` is Lyapunov-like on ``K`` if `\left\langle
+ L\left\lparenx\right\rparen,s\right\rangle = 0` for all pairs
+ `\left\langle x,s \right\rangle` in the complementarity set of
+ ``K``. It is known [Orlitzky]_ that this property need only be
+ checked for generators of ``K`` and its dual.
+
+ INPUT:
+
+ - ``L`` -- A linear transformation or matrix.
+
+ - ``K`` -- A polyhedral closed convex cone.
+
+ OUTPUT:
+
+ ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``,
+ and ``False`` otherwise.
+
+ .. WARNING::
+
+ If this function returns ``True``, then ``L`` is Lyapunov-like
+ on ``K``. However, if ``False`` is returned, that could mean one
+ of two things. The first is that ``L`` is definitely not
+ Lyapunov-like on ``K``. The second is more of an "I don't know"
+ answer, returned (for example) if we cannot prove that an inner
+ product is zero.
+
+ REFERENCES:
+
+ .. [Orlitzky] M. Orlitzky. The Lyapunov rank of an
+ improper cone (preprint).
+
+ EXAMPLES:
+
+ The identity is always Lyapunov-like in a nontrivial space::
+
+ sage: set_random_seed()
+ sage: K = random_cone(min_ambient_dim = 1, max_rays = 8)
+ sage: L = identity_matrix(K.lattice_dim())
+ sage: is_lyapunov_like(L,K)
+ True
+
+ As is the "zero" transformation::
+
+ sage: K = random_cone(min_ambient_dim = 1, max_rays = 5)
+ sage: R = K.lattice().vector_space().base_ring()
+ sage: L = zero_matrix(R, K.lattice_dim())
+ sage: is_lyapunov_like(L,K)
+ True
+
+ Everything in ``K.LL()`` should be Lyapunov-like on ``K``::
+
+ sage: K = random_cone(min_ambient_dim = 1, max_rays = 5)
+ sage: all([is_lyapunov_like(L,K) for L in K.LL()])
+ True
+
+ """
+ return all([(L*x).inner_product(s) == 0
+ for (x,s) in K.discrete_complementarity_set()])
+
+
+def random_element(K):
+ r"""
+ Return a random element of ``K`` from its ambient vector space.
+
+ ALGORITHM:
+
+ The cone ``K`` is specified in terms of its generators, so that
+ ``K`` is equal to the convex conic combination of those generators.
+ To choose a random element of ``K``, we assign random nonnegative
+ coefficients to each generator of ``K`` and construct a new vector
+ from the scaled rays.
+
+ A vector, rather than a ray, is returned so that the element may
+ have non-integer coordinates. Thus the element may have an
+ arbitrarily small norm.
+
+ EXAMPLES:
+
+ A random element of the trivial cone is zero::
+
+ sage: set_random_seed()
+ sage: K = Cone([], ToricLattice(0))
+ sage: random_element(K)
+ ()
+ sage: K = Cone([(0,)])
+ sage: random_element(K)
+ (0)
+ sage: K = Cone([(0,0)])
+ sage: random_element(K)
+ (0, 0)
+ sage: K = Cone([(0,0,0)])
+ sage: random_element(K)
+ (0, 0, 0)
+
+ TESTS:
+
+ Any cone should contain an element of itself::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_rays = 8)
+ sage: K.contains(random_element(K))
+ True
+
+ """
+ V = K.lattice().vector_space()
+ F = V.base_ring()
+ coefficients = [ F.random_element().abs() for i in range(K.nrays()) ]
+ vector_gens = map(V, K.rays())
+ scaled_gens = [ coefficients[i]*vector_gens[i]
+ for i in range(len(vector_gens)) ]
+
+ # Make sure we return a vector. Without the coercion, we might
+ # return ``0`` when ``K`` has no rays.
+ v = V(sum(scaled_gens))
+ return v
+
+
+def positive_operators(K):
+ r"""
+ Compute generators of the cone of positive operators on this cone.
+
+ OUTPUT:
+
+ A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``.
+ Each matrix ``P`` in the list should have the property that ``P*x``
+ is an element of ``K`` whenever ``x`` is an element of
+ ``K``. Moreover, any nonnegative linear combination of these
+ matrices shares the same property.
+
+ EXAMPLES:
+
+ The trivial cone in a trivial space has no positive operators::
+
+ sage: K = Cone([], ToricLattice(0))
+ sage: positive_operators(K)
+ []
+
+ Positive operators on the nonnegative orthant are nonnegative matrices::
+
+ sage: K = Cone([(1,)])
+ sage: positive_operators(K)
+ [[1]]
+
+ sage: K = Cone([(1,0),(0,1)])
+ sage: positive_operators(K)
+ [
+ [1 0] [0 1] [0 0] [0 0]
+ [0 0], [0 0], [1 0], [0 1]
+ ]
+
+ Every operator is positive on the ambient vector space::
+
+ sage: K = Cone([(1,),(-1,)])
+ sage: K.is_full_space()
+ True
+ sage: positive_operators(K)
+ [[1], [-1]]
+
+ sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
+ sage: K.is_full_space()
+ True
+ sage: positive_operators(K)
+ [
+ [1 0] [-1 0] [0 1] [ 0 -1] [0 0] [ 0 0] [0 0] [ 0 0]
+ [0 0], [ 0 0], [0 0], [ 0 0], [1 0], [-1 0], [0 1], [ 0 -1]
+ ]
+
+ TESTS:
+
+ A positive operator on a cone should send its generators into the cone::
+
+ sage: K = random_cone(max_ambient_dim = 6)
+ sage: pi_of_K = positive_operators(K)
+ sage: all([K.contains(p*x) for p in pi_of_K for x in K.rays()])
+ True
+
+ """
+ # Sage doesn't think matrices are vectors, so we have to convert
+ # our matrices to vectors explicitly before we can figure out how
+ # many are linearly-indepenedent.
+ #
+ # The space W has the same base ring as V, but dimension
+ # dim(V)^2. So it has the same dimension as the space of linear
+ # transformations on V. In other words, it's just the right size
+ # to create an isomorphism between it and our matrices.
+ V = K.lattice().vector_space()
+ W = VectorSpace(V.base_ring(), V.dimension()**2)
+
+ tensor_products = [ s.tensor_product(x) for x in K for s in K.dual() ]
+
+ # Turn our matrices into long vectors...
+ vectors = [ W(m.list()) for m in tensor_products ]
+
+ # Create the *dual* cone of the positive operators, expressed as
+ # long vectors..
+ L = ToricLattice(W.dimension())
+ pi_dual = Cone(vectors, lattice=L)
+
+ # Now compute the desired cone from its dual...
+ pi_cone = pi_dual.dual()
+
+ # And finally convert its rays back to matrix representations.
+ M = MatrixSpace(V.base_ring(), V.dimension())
+
+ return [ M(v.list()) for v in pi_cone.rays() ]
projects / sage.d.git / blobdiff