Add a description for the pointed_decomposition() doctest.

[sage.d.git] / mjo / cone / cone.py

from sage.all import *

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_ambient_dim=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_ambient_dim=8)
        sage: R = K.lattice().vector_space().base_ring()
        sage: L = zero_matrix(R, K.lattice_dim())
        sage: is_lyapunov_like(L,K)
        True

        Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like
        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() ])
        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)

    A random element of the nonnegative orthant should have all
    components nonnegative::

        sage: set_random_seed()
        sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
        sage: all([ x >= 0 for x in random_element(K) ])
        True

    TESTS:

    Any cone should contain a random element of itself::

        sage: set_random_seed()
        sage: K = random_cone(max_ambient_dim=8)
        sage: K.contains(random_element(K))
        True

    A strictly convex cone contains no lines, and thus no negative
    multiples of any of its elements besides zero::

        sage: set_random_seed()
        sage: K = random_cone(max_ambient_dim=8, strictly_convex=True)
        sage: x = random_element(K)
        sage: x.is_zero() or not K.contains(-x)
        True

    The sum of random elements of a cone lies in the cone::

        sage: set_random_seed()
        sage: K = random_cone(max_ambient_dim=8)
        sage: K.contains(sum([random_element(K) for i in range(10)]))
        True

    """
    V = K.lattice().vector_space()
    scaled_gens = [ V.base_field().random_element().abs()*V(r) for r in K ]

    # Make sure we return a vector. Without the coercion, we might
    # return ``0`` when ``K`` has no rays.
    return V(sum(scaled_gens))


def pointed_decomposition(K):
    """
    Every convex cone is the direct sum of a pointed cone and a linear
    subspace. Return a pair ``(P,S)`` of cones such that ``P`` is
    pointed, ``S`` is a subspace, and ``K`` is the direct sum of ``P``
    and ``S``.

    OUTPUT:

    An ordered pair ``(P,S)`` of closed convex polyhedral cones where
    ``P`` is pointed, ``S`` is a subspace, and ``K`` is the direct sum
    of ``P`` and ``S``.

    TESTS:

    A random point in the cone should belong to either the pointed
    subcone ``P`` or the subspace ``S``. If the point is nonzero, it
    should lie in one but not both of them::

        sage: set_random_seed()
        sage: K = random_cone(max_ambient_dim=8)
        sage: (P,S) = pointed_decomposition(K)
        sage: x = random_element(K)
        sage: P.contains(x) or S.contains(x)
        True
        sage: x.is_zero() or (P.contains(x) != S.contains(x))
        True
    """
    linspace_gens  = [ copy(b) for b in K.linear_subspace().basis() ]
    linspace_gens += [ -b for b in linspace_gens ]

    S = Cone(linspace_gens, K.lattice())

    # Since ``S`` is a subspace, its dual is its orthogonal complement
    # (albeit in the wrong lattice).
    S_perp = Cone(S.dual(), K.lattice())
    P = K.intersection(S_perp)

    return (P,S)

def positive_operator_gens(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_operator_gens(K)
        []

    Positive operators on the nonnegative orthant are nonnegative matrices::

        sage: K = Cone([(1,)])
        sage: positive_operator_gens(K)
        [[1]]

        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]
        ]

    Every operator is positive on the ambient vector space::

        sage: K = Cone([(1,),(-1,)])
        sage: K.is_full_space()
        True
        sage: positive_operator_gens(K)
        [[1], [-1]]

        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]
        ]

    TESTS:

    A positive operator on a cone should send its generators into the cone::

        sage: set_random_seed()
        sage: K = random_cone(max_ambient_dim=5)
        sage: pi_of_K = positive_operator_gens(K)
        sage: all([K.contains(p*x) for p in pi_of_K for x in K.rays()])
        True

    The dimension of the cone of positive operators is given by the
    corollary in my paper::

        sage: set_random_seed()
        sage: K = random_cone(max_ambient_dim=5)
        sage: n = K.lattice_dim()
        sage: m = K.dim()
        sage: l = K.lineality()
        sage: pi_of_K = positive_operator_gens(K)
        sage: L = ToricLattice(n**2)
        sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).dim()
        sage: expected = n**2 - l*(m - l) - (n - m)*m
        sage: actual == expected
        True

    The lineality of the cone of positive operators is given by the
    corollary in my paper::

        sage: set_random_seed()
        sage: K = random_cone(max_ambient_dim=5)
        sage: n = K.lattice_dim()
        sage: pi_of_K = positive_operator_gens(K)
        sage: L = ToricLattice(n**2)
        sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).lineality()
        sage: expected = n**2 - K.dim()*K.dual().dim()
        sage: actual == expected
        True

    The cone ``K`` is proper if and only if the cone of positive
    operators on ``K`` is proper::

        sage: set_random_seed()
        sage: K = random_cone(max_ambient_dim=5)
        sage: pi_of_K = positive_operator_gens(K)
        sage: L = ToricLattice(K.lattice_dim()**2)
        sage: pi_cone = Cone([p.list() for p in pi_of_K], lattice=L)
        sage: K.is_proper() == pi_cone.is_proper()
        True
    """
    # Matrices are not vectors in Sage, so we have to convert them
    # to vectors explicitly before we can find a basis. We need these
    # two values to construct the appropriate "long vector" space.
    F = K.lattice().base_field()
    n = K.lattice_dim()

    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 ]

    # Create the *dual* cone of the positive operators, expressed as
    # long vectors..
    pi_dual = Cone(vectors, ToricLattice(W.dimension()))

    # 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.rays() ]


def Z_transformation_gens(K):
    r"""
    Compute generators of the cone of Z-transformations on this 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.

    EXAMPLES:

    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

    The trivial cone in a trivial space has no Z-transformations::

        sage: K = Cone([], ToricLattice(0))
        sage: Z_transformation_gens(K)
        []

    Z-transformations on a subspace are Lyapunov-like and vice-versa::

        sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
        sage: K.is_full_space()
        True
        sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
        sage: zs  = span([ vector(z.list()) for z in Z_transformation_gens(K) ])
        sage: zs == lls
        True

    TESTS:

    The Z-property is possessed by every Z-transformation::

        sage: set_random_seed()
        sage: K = random_cone(max_ambient_dim=6)
        sage: Z_of_K = Z_transformation_gens(K)
        sage: dcs = K.discrete_complementarity_set()
        sage: all([(z*x).inner_product(s) <= 0 for z in Z_of_K
        ....:                                  for (x,s) in dcs])
        True

    The lineality space of Z is LL::

        sage: set_random_seed()
        sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6)
        sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
        sage: z_cone  = Cone([ z.list() for z in Z_transformation_gens(K) ])
        sage: z_cone.linear_subspace() == lls
        True

    And thus, the lineality of Z is the Lyapunov rank::

        sage: set_random_seed()
        sage: K = random_cone(max_ambient_dim=6)
        sage: Z_of_K = Z_transformation_gens(K)
        sage: L = ToricLattice(K.lattice_dim()**2)
        sage: z_cone  = Cone([ z.list() for z in Z_of_K ], lattice=L)
        sage: z_cone.lineality() == K.lyapunov_rank()
        True

    The lineality spaces of pi-star and Z-star are equal:

        sage: set_random_seed()
        sage: K = random_cone(max_ambient_dim=5)
        sage: pi_of_K = positive_operator_gens(K)
        sage: Z_of_K = Z_transformation_gens(K)
        sage: L = ToricLattice(K.lattice_dim()**2)
        sage: pi_star = Cone([p.list() for p in pi_of_K], lattice=L).dual()
        sage: z_star  = Cone([ z.list() for z in Z_of_K], lattice=L).dual()
        sage: pi_star.linear_subspace() == z_star.linear_subspace()
        True
    """
    # Matrices are not vectors in Sage, so we have to convert them
    # to vectors explicitly before we can find a basis. We need these
    # two values to construct the appropriate "long vector" space.
    F = K.lattice().base_field()
    n = K.lattice_dim()

    # 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 ]

    # Create the *dual* cone of the cross-positive operators,
    # expressed as long vectors..
    Sigma_dual = Cone(vectors, lattice=ToricLattice(W.dimension()))

    # Now compute the desired cone from its dual...
    Sigma_cone = Sigma_dual.dual()

    # And finally convert its rays back to matrix representations.
    # But first, make them negative, so we get Z-transformations and
    # not cross-positive ones.
    M = MatrixSpace(F, n)
    return [ -M(v.list()) for v in Sigma_cone.rays() ]


def Z_cone(K):
    gens = Z_transformation_gens(K)
    L = None
    if len(gens) == 0:
        L = ToricLattice(0)
    return Cone([ g.list() for g in gens ], lattice=L)

def pi_cone(K):
    gens = positive_operator_gens(K)
    L = None
    if len(gens) == 0:
        L = ToricLattice(0)
    return Cone([ g.list() for g in gens ], lattice=L)