Double Description Algorithm for Cones

This module implements the double description algorithm for extremal vertex enumeration in a pointed cone following [FP1996]. With a little bit of preprocessing (see double_description_inhomogeneous) this defines a backend for polyhedral computations. But as far as this module is concerned, inequality always means without a constant term and the origin is always a point of the cone.

EXAMPLES:

sage: from sage.geometry.polyhedron.double_description import StandardAlgorithm
sage: A = matrix(QQ, [(1,0,1), (0,1,1), (-1,-1,1)])
sage: alg = StandardAlgorithm(A);  alg
Pointed cone with inequalities
(1, 0, 1)
(0, 1, 1)
(-1, -1, 1)
sage: DD, _ = alg.initial_pair();  DD
Double description pair (A, R) defined by
    [ 1  0  1]        [ 2/3 -1/3 -1/3]
A = [ 0  1  1],   R = [-1/3  2/3 -1/3]
    [-1 -1  1]        [ 1/3  1/3  1/3]

The implementation works over any exact field that is embedded in \(\RR\), for example:

sage: from sage.geometry.polyhedron.double_description import StandardAlgorithm
sage: A = matrix(AA, [(1,0,1), (0,1,1), (-AA(2).sqrt(),-AA(3).sqrt(),1),  # optional - sage.rings.number_field
....:                 (-AA(3).sqrt(),-AA(2).sqrt(),1)])
sage: alg = StandardAlgorithm(A)                                          # optional - sage.rings.number_field
sage: alg.run().R                                                         # optional - sage.rings.number_field
[(-0.4177376677004119?, 0.5822623322995881?, 0.4177376677004119?),
 (-0.2411809548974793?, -0.2411809548974793?, 0.2411809548974793?),
 (0.07665629029830300?, 0.07665629029830300?, 0.2411809548974793?),
 (0.5822623322995881?, -0.4177376677004119?, 0.4177376677004119?)]

class sage.geometry.polyhedron.double_description.DoubleDescriptionPair(problem, A_rows, R_cols)

Bases: object

Base class for a double description pair \((A, R)\)

Warning

You should use the Problem.initial_pair() or Problem.run() to generate double description pairs for a set of inequalities, and not generate DoubleDescriptionPair instances directly.

INPUT:

• problem – instance of Problem.

• A_rows – list of row vectors of the matrix \(A\). These encode the inequalities.

• R_cols – list of column vectors of the matrix \(R\). These encode the rays.

R_by_sign(a)

Classify the rays into those that are positive, zero, and negative on \(a\).

INPUT:

• a – vector. Coefficient vector of a homogeneous inequality.

OUTPUT:

A triple consisting of the rays (columns of \(R\)) that are positive, zero, and negative on \(a\). In that order.

EXAMPLES:

sage: from sage.geometry.polyhedron.double_description import StandardAlgorithm
sage: A = matrix(QQ, [(1,0,1), (0,1,1), (-1,-1,1)])
sage: DD, _ = StandardAlgorithm(A).initial_pair()
sage: DD.R_by_sign(vector([1,-1,0]))
([(2/3, -1/3, 1/3)], [(-1/3, -1/3, 1/3)], [(-1/3, 2/3, 1/3)])
sage: DD.R_by_sign(vector([1,1,1]))
([(2/3, -1/3, 1/3), (-1/3, 2/3, 1/3)], [], [(-1/3, -1/3, 1/3)])

are_adjacent(r1, r2)

Return whether the two rays are adjacent.

INPUT:

• r1, r2 – two rays.

OUTPUT:

Boolean. Whether the two rays are adjacent.

EXAMPLES:

sage: from sage.geometry.polyhedron.double_description import StandardAlgorithm
sage: A = matrix(QQ, [(0,1,0), (1,0,0), (0,-1,1), (-1,0,1)])
sage: DD = StandardAlgorithm(A).run()
sage: DD.are_adjacent(DD.R[0], DD.R[1])
True
sage: DD.are_adjacent(DD.R[0], DD.R[2])
True
sage: DD.are_adjacent(DD.R[0], DD.R[3])
False

cone()

Return the cone defined by \(A\).

This method is for debugging only. Assumes that the base ring is \(\QQ\).

OUTPUT:

The cone defined by the inequalities as a Polyhedron(), using the PPL backend.

EXAMPLES:

sage: from sage.geometry.polyhedron.double_description import StandardAlgorithm
sage: A = matrix(QQ, [(1,0,1), (0,1,1), (-1,-1,1)])
sage: DD, _ = StandardAlgorithm(A).initial_pair()
sage: DD.cone().Hrepresentation()
(An inequality (-1, -1, 1) x + 0 >= 0,
 An inequality (0, 1, 1) x + 0 >= 0,
 An inequality (1, 0, 1) x + 0 >= 0)

dual()

Return the dual.

OUTPUT:

For the double description pair \((A, R)\) this method returns the dual double description pair \((R^T, A^T)\)

EXAMPLES:

sage: from sage.geometry.polyhedron.double_description import Problem
sage: A = matrix(QQ, [(0,1,0), (1,0,0), (0,-1,1), (-1,0,1)])
sage: DD, _ = Problem(A).initial_pair()
sage: DD
Double description pair (A, R) defined by
    [ 0  1  0]        [0 1 0]
A = [ 1  0  0],   R = [1 0 0]
    [ 0 -1  1]        [1 0 1]
sage: DD.dual()
 Double description pair (A, R) defined by
    [0 1 1]        [ 0  1  0]
A = [1 0 0],   R = [ 1  0 -1]
    [0 0 1]        [ 0  0  1]

first_coordinate_plane()

Restrict to the first coordinate plane.

OUTPUT:

A new double description pair with the constraint \(x_0 = 0\) added.

EXAMPLES:

sage: A = matrix([(1, 1), (-1, 1)])
sage: from sage.geometry.polyhedron.double_description import StandardAlgorithm
sage: DD, _ = StandardAlgorithm(A).initial_pair()
sage: DD
Double description pair (A, R) defined by
A = [ 1  1],   R = [ 1/2 -1/2]
    [-1  1]        [ 1/2  1/2]
sage: DD.first_coordinate_plane()
Double description pair (A, R) defined by
    [ 1  1]
A = [-1  1],   R = [  0]
    [-1  0]        [1/2]
    [ 1  0]

inner_product_matrix()

Return the inner product matrix between the rows of \(A\) and the columns of \(R\).

OUTPUT:

A matrix over the base ring. There is one row for each row of \(A\) and one column for each column of \(R\).

EXAMPLES:

sage: from sage.geometry.polyhedron.double_description import StandardAlgorithm
sage: A = matrix(QQ, [(1,0,1), (0,1,1), (-1,-1,1)])
sage: alg = StandardAlgorithm(A)
sage: DD, _ = alg.initial_pair()
sage: DD.inner_product_matrix()
[1 0 0]
[0 1 0]
[0 0 1]

is_extremal(ray)

Test whether the ray is extremal.

EXAMPLES:

sage: from sage.geometry.polyhedron.double_description import StandardAlgorithm
sage: A = matrix(QQ, [(0,1,0), (1,0,0), (0,-1,1), (-1,0,1)])
sage: DD = StandardAlgorithm(A).run()
sage: DD.is_extremal(DD.R[0])
True

matrix_space(nrows, ncols)

Return a matrix space of size nrows and ncols over the base ring of self.

These matrix spaces are cached to avoid their creation in the very demanding add_inequality() and more precisely are_adjacent().

EXAMPLES:

sage: from sage.geometry.polyhedron.double_description import Problem
sage: A = matrix(QQ, [(1,0,1), (0,1,1), (-1,-1,1)])
sage: DD, _ = Problem(A).initial_pair()
sage: DD.matrix_space(2,2)
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: DD.matrix_space(3,2)
Full MatrixSpace of 3 by 2 dense matrices over Rational Field

sage: K.<sqrt2> = QuadraticField(2)        # optional - sage.rings.number_field
sage: A = matrix([[1,sqrt2],[2,0]])        # optional - sage.rings.number_field
sage: DD, _  = Problem(A).initial_pair()   # optional - sage.rings.number_field
sage: DD.matrix_space(1,2)                 # optional - sage.rings.number_field
Full MatrixSpace of 1 by 2 dense matrices
 over Number Field in sqrt2 with defining polynomial x^2 - 2 with sqrt2 = 1.414213562373095?

verify()

Validate the double description pair.

This method used the PPL backend to check that the double description pair is valid. An assertion is triggered if it is not. Does nothing if the base ring is not \(\QQ\).

EXAMPLES:

sage: from sage.geometry.polyhedron.double_description import \
....:     DoubleDescriptionPair, Problem
sage: A = matrix(QQ, [(1,0,1), (0,1,1), (-1,-1,1)])
sage: alg = Problem(A)
sage: DD = DoubleDescriptionPair(alg,
....:     [(1, 0, 3), (0, 1, 1), (-1, -1, 1)],
....:     [(2/3, -1/3, 1/3), (-1/3, 2/3, 1/3), (-1/3, -1/3, 1/3)])
sage: DD.verify()
Traceback (most recent call last):
...
    assert A_cone == R_cone
AssertionError

zero_set(ray)

Return the zero set (active set) \(Z(r)\).

INPUT:

• ray – a ray vector.

OUTPUT:

A set containing the inequality vectors that are zero on ray.

EXAMPLES:

sage: from sage.geometry.polyhedron.double_description import Problem
sage: A = matrix(QQ, [(1,0,1), (0,1,1), (-1,-1,1)])
sage: DD, _ = Problem(A).initial_pair()
sage: r = DD.R[0];  r
(2/3, -1/3, 1/3)
sage: DD.zero_set(r)
{(-1, -1, 1), (0, 1, 1)}

class sage.geometry.polyhedron.double_description.Problem(A)

Bases: object

Base class for implementations of the double description algorithm

It does not make sense to instantiate the base class directly, it just provides helpers for implementations.

INPUT:

• A – a matrix. The rows of the matrix are interpreted as homogeneous inequalities \(A x \geq 0\). Must have maximal rank.

A()

Return the rows of the defining matrix \(A\).

OUTPUT:

The matrix \(A\) whose rows are the inequalities.

EXAMPLES:

sage: A = matrix([(1, 1), (-1, 1)])
sage: from sage.geometry.polyhedron.double_description import Problem
sage: Problem(A).A()
((1, 1), (-1, 1))

A_matrix()

Return the defining matrix \(A\).

OUTPUT:

Matrix whose rows are the inequalities.

EXAMPLES:

sage: A = matrix([(1, 1), (-1, 1)])
sage: from sage.geometry.polyhedron.double_description import Problem
sage: Problem(A).A_matrix()
[ 1  1]
[-1  1]

base_ring()

Return the base field.

OUTPUT:

A field.

EXAMPLES:

sage: A = matrix(AA, [(1, 1), (-1, 1)])  # optional - sage.rings.number_field
sage: from sage.geometry.polyhedron.double_description import Problem
sage: Problem(A).base_ring()             # optional - sage.rings.number_field
Algebraic Real Field

dim()

Return the ambient space dimension.

OUTPUT:

Integer. The ambient space dimension of the cone.

EXAMPLES:

sage: A = matrix(QQ, [(1, 1), (-1, 1)])
sage: from sage.geometry.polyhedron.double_description import Problem
sage: Problem(A).dim()
2

initial_pair()

Return an initial double description pair.

Picks an initial set of rays by selecting a basis. This is probably the most efficient way to select the initial set.

INPUT:

• pair_class – subclass of DoubleDescriptionPair.

OUTPUT:

A pair consisting of a DoubleDescriptionPair instance and the tuple of remaining unused inequalities.

EXAMPLES:

sage: A = matrix([(-1, 1), (-1, 2), (1/2, -1/2), (1/2, 2)])
sage: from sage.geometry.polyhedron.double_description import Problem
sage: DD, remaining = Problem(A).initial_pair()
sage: DD.verify()
sage: remaining
[(1/2, -1/2), (1/2, 2)]

pair_class

alias of DoubleDescriptionPair

class sage.geometry.polyhedron.double_description.StandardAlgorithm(A)

Bases: Problem

Standard implementation of the double description algorithm

See [FP1996] for the definition of the “Standard Algorithm”.

EXAMPLES:

sage: A = matrix(QQ, [(1, 1), (-1, 1)])
sage: from sage.geometry.polyhedron.double_description import StandardAlgorithm
sage: DD = StandardAlgorithm(A).run()
sage: DD.R    # the extremal rays
[(1/2, 1/2), (-1/2, 1/2)]

pair_class

alias of StandardDoubleDescriptionPair

run()

Run the Standard Algorithm.

OUTPUT:

A double description pair \((A, R)\) of all inequalities as a DoubleDescriptionPair. By virtue of the double description algorithm, the columns of \(R\) are the extremal rays.

EXAMPLES:

sage: from sage.geometry.polyhedron.double_description import StandardAlgorithm
sage: A = matrix(QQ, [(0,1,0), (1,0,0), (0,-1,1), (-1,0,1)])
sage: StandardAlgorithm(A).run()
Double description pair (A, R) defined by
    [ 0  1  0]        [0 0 1 1]
A = [ 1  0  0],   R = [1 0 1 0]
    [ 0 -1  1]        [1 1 1 1]
    [-1  0  1]

class sage.geometry.polyhedron.double_description.StandardDoubleDescriptionPair(problem, A_rows, R_cols)

Bases: DoubleDescriptionPair

Double description pair for the “Standard Algorithm”.

See StandardAlgorithm.

add_inequality(a)

Add the inequality a to the matrix \(A\) of the double description.

INPUT:

• a – vector. An inequality.

EXAMPLES:

sage: A = matrix([(-1, 1, 0), (-1, 2, 1), (1/2, -1/2, -1)])
sage: from sage.geometry.polyhedron.double_description import StandardAlgorithm
sage: DD, _ = StandardAlgorithm(A).initial_pair()
sage: DD.add_inequality(vector([1,0,0]))
sage: DD
Double description pair (A, R) defined by
    [  -1    1    0]        [   1    1    0    0]
A = [  -1    2    1],   R = [   1    1    1    1]
    [ 1/2 -1/2   -1]        [   0   -1 -1/2   -2]
    [   1    0    0]

sage.geometry.polyhedron.double_description.random_inequalities(d, n)

Random collections of inequalities for testing purposes.

INPUT:

• d – integer. The dimension.

• n – integer. The number of random inequalities to generate.

OUTPUT:

A random set of inequalities as a StandardAlgorithm instance.

EXAMPLES:

sage: from sage.geometry.polyhedron.double_description import random_inequalities
sage: P = random_inequalities(5, 10)
sage: P.run().verify()