+ TESTS:
+
+ The Lyapunov rank should be additive on a product of proper cones
+ [Rudolf]_::
+
+ sage: set_random_seed()
+ sage: K1 = random_cone(max_ambient_dim=8,
+ ....: strictly_convex=True,
+ ....: solid=True)
+ sage: K2 = random_cone(max_ambient_dim=8,
+ ....: strictly_convex=True,
+ ....: solid=True)
+ sage: K = K1.cartesian_product(K2)
+ sage: lyapunov_rank(K) == lyapunov_rank(K1) + lyapunov_rank(K2)
+ True
+
+ The Lyapunov rank is invariant under a linear isomorphism
+ [Orlitzky]_::
+
+ sage: K1 = random_cone(max_ambient_dim = 8)
+ sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular')
+ sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice())
+ sage: lyapunov_rank(K1) == lyapunov_rank(K2)
+ True
+
+ The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
+ itself [Rudolf]_::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=8)
+ sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+ 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_ambient_dim=8,
+ ....: strictly_convex=True,
+ ....: solid=True)
+ sage: b = lyapunov_rank(K)
+ sage: n = K.lattice_dim()
+ sage: (n == 0 or 1 <= b) and b <= n
+ True
+ sage: b == n-1
+ False
+
+ In fact [Orlitzky]_, no closed convex polyhedral cone can have
+ Lyapunov rank `n-1` in `n` dimensions::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=8)
+ sage: b = lyapunov_rank(K)
+ sage: n = K.lattice_dim()
+ sage: b == n-1
+ False
+
+ The calculation of the Lyapunov rank of an improper cone can be
+ reduced to that of a proper cone [Orlitzky]_::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=8)
+ sage: actual = lyapunov_rank(K)
+ sage: K_S = _restrict_to_space(K, K.span())
+ sage: K_SP = _restrict_to_space(K_S.dual(), K_S.dual().span()).dual()
+ sage: l = K.lineality()
+ sage: c = K.codim()
+ sage: expected = lyapunov_rank(K_SP) + K.dim()*(l + c) + c**2
+ sage: actual == expected
+ True
+
+ The Lyapunov rank of a cone is the size of a :meth:`lyapunov_like_basis`::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=8)
+ sage: lyapunov_rank(K) == len(K.lyapunov_like_basis())
+ True
+
+ We can make an imperfect cone perfect by adding a slack variable
+ (a Theorem in [Orlitzky]_)::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=8,
+ ....: strictly_convex=True,
+ ....: solid=True)
+ sage: L = ToricLattice(K.lattice_dim() + 1)
+ sage: K = Cone([ r.list() + [0] for r in K.rays() ], lattice=L)
+ sage: lyapunov_rank(K) >= K.lattice_dim()
+ True
+
+ """
+ beta = 0 # running tally of the Lyapunov rank
+
+ m = K.dim()
+ n = K.lattice_dim()
+ l = K.lineality()
+
+ if m < n:
+ # K is not solid, restrict to its span.
+ K = _restrict_to_space(K, K.span())
+
+ # Non-solid reduction lemma.
+ beta += (n - m)*n
+
+ if l > 0:
+ # K is not pointed, restrict to the span of its dual. Uses a
+ # proposition from our paper, i.e. this is equivalent to K =
+ # _rho(K.dual()).dual().
+ K = _restrict_to_space(K, K.dual().span())
+
+ # Non-pointed reduction lemma.
+ beta += l * m
+
+ beta += len(K.lyapunov_like_basis())
+ 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:
+
+ M. Orlitzky. The Lyapunov rank of an improper cone.
+ http://www.optimization-online.org/DB_HTML/2015/10/5135.html
+
+ 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.lyapunov_like_basis()`` 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.lyapunov_like_basis() ])
+ 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
+