.. _spkg_sagemath_polyhedra:

============================================================================================================================================
sagemath_polyhedra: Convex polyhedra in arbitrary dimension, mixed integer linear optimization
============================================================================================================================================


This pip-installable distribution ``passagemath-polyhedra`` is a distribution of a part of the Sage Library.  It provides a small subset of the modules of the Sage library ("sagelib", `passagemath-standard`), sufficient for computations with convex polyhedra in arbitrary dimension (in exact rational arithmetic), and linear and mixed integer linear optimization (in floating point arithmetic).


What is included
----------------

* `Combinatorial and Discrete Geometry <https://passagemath.org/docs/latest/html/en/reference/discrete_geometry/index.html>`_: Polyhedra, lattice polyhedra, lattice points in polyhedra, triangulations, fans, polyhedral complexes, hyperplane arrrangements

* `Parma Polyhedra Library (PPL) backends for rational polyhedra <https://passagemath.org/docs/latest/html/en/reference/discrete_geometry/sage/geometry/polyhedron/backend_ppl.html>`_, `lattice polygons <https://passagemath.org/docs/latest/html/en/reference/discrete_geometry/sage/geometry/polyhedron/ppl_lattice_polygon.html>`_, `lattice polytopes <https://passagemath.org/docs/latest/html/en/reference/discrete_geometry/sage/geometry/polyhedron/ppl_lattice_polytope.html>`_; via `pplpy <https://passagemath.org/docs/latest/html/en/reference/spkg/pplpy.html#spkg-pplpy>`_

* `Python backend for polyhedra over general ordered fields <https://passagemath.org/docs/latest/html/en/reference/discrete_geometry/sage/geometry/polyhedron/backend_field.html>`_

* `Linear, Mixed Integer Linear, and Semidefinite Optimization frontends <https://passagemath.org/docs/latest/html/en/reference/numerical/index.html#numerical-optimization>`_

* `GNU Linear Programming Kit (GLPK) backend for large-scale linear and mixed integer linear optimization (floating point arithmetic) <https://passagemath.org/docs/latest/html/en/reference/numerical/sage/numerical/backends/glpk_backend.html>`_

* `Interactive Simplex Method <https://passagemath.org/docs/latest/html/en/reference/numerical/sage/numerical/interactive_simplex_method.html>`_

* see https://github.com/passagemath/passagemath/blob/main/pkgs/sagemath-polyhedra/MANIFEST.in


Examples
--------

A quick way to try it out interactively::

    $ pipx run --pip-args="--prefer-binary" --spec "passagemath-polyhedra[test]" ipython

    In [1]: from passagemath_polyhedra import *

    In [2]: P = Polyhedron(ieqs=[[0, 1, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 1], [0, 0, 1, -1, -1, 1, 0], [0, 0, -1, 1, -1, 1, 0]], eqns=[[-31, 1, 1, 1, 1, 1, 1]]); P
    Out[2]: A 5-dimensional polyhedron in QQ^6 defined as the convex hull of 7 vertices

    In [3]: P.Vrepresentation()
    Out[4]:
    (A vertex at (31, 0, 0, 0, 0, 0),
     A vertex at (0, 0, 0, 0, 0, 31),
     A vertex at (0, 0, 0, 0, 31, 0),
     A vertex at (0, 0, 31/2, 0, 31/2, 0),
     A vertex at (0, 31/2, 31/2, 0, 0, 0),
     A vertex at (0, 31/2, 0, 0, 31/2, 0),
     A vertex at (0, 0, 0, 31/2, 31/2, 0))


Available as extras, from other distributions
---------------------------------------------

Additional features
~~~~~~~~~~~~~~~~~~~

``pip install "passagemath-polyhedra[graphs]"``
 Face lattices, combinatorial polyhedra, graph-theoretic constructions

 ::

    $ pipx run --pip-args="--prefer-binary" --spec "passagemath-polyhedra[graphs,test]" ipython

    In [1]: from passagemath_polyhedra import *

    In [2]: c5_10 = Polyhedron(vertices = [[i, i**2, i**3, i**4, i**5] for i in range(1, 11)]); c5_10
    Out[2]: A 5-dimensional polyhedron in ZZ^5 defined as the convex hull of 10 vertices

    In [3]: c5_10_fl = c5_10.face_lattice(); [len(x) for x in c5_10_fl.level_sets()]
    Out[3]: [1, 10, 45, 100, 105, 42, 1]

``pip install "passagemath-polyhedra[graphs,groups]"``
 Constructing symmetric polyhedra, computing automorphisms, lattice point counting modulo group actions

 ::

    $ pipx run --pip-args="--prefer-binary" --spec "passagemath-polyhedra[graphs,groups,test]" ipython

    In [1]: from passagemath_polyhedra import *

    In [2]: P24 = polytopes.twenty_four_cell(); P24
    Out[2]: A 4-dimensional polyhedron in QQ^4 defined as the convex hull of 24 vertices

    In [3]: AutP24 = P24.restricted_automorphism_group(); AutP24.order()
    Out[3]: 1152

``pip install "passagemath-polyhedra[toric]"``
 `Toric varieties <https://passagemath.org/docs/latest/html/en/reference/schemes/index.html#toric-varieties>`_

 ::

    $ pipx run --pip-args="--prefer-binary" --spec "passagemath-polyhedra[graphs,toric,test]" ipython

    In [1]: from passagemath_polyhedra import *

    In [2]: TV3 = ToricVariety(NormalFan(lattice_polytope.cross_polytope(3))); TV3
    Out[2]: 3-d toric variety covered by 6 affine patches

    In [3]: TV3.is_orbifold()
    Out[3]: False

``pip install "passagemath-polyhedra[latte]"``
 Installs `LattE integrale <https://passagemath.org/docs/latest/html/en/reference/spkg/latte_int.html#spkg-latte-int>`_
 for lattice point counting and volume computation using generating function techniques.

 ::

   $ pipx run --pip-args="--prefer-binary" --spec "passagemath-polyhedra[latte,test]" ipython

   In [1]: from passagemath_polyhedra import *

   In [2]: P = polytopes.cube()

   In [3]: P.integral_points_count()
   Out[3]:
   27

   In [4]: (1000000000*P).integral_points_count(verbose=True)
   This is LattE integrale...
   ...
   Total time:...
   Out[4]:
   8000000012000000006000000001


Additional backends for polyhedral computations
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

``pip install "passagemath-polyhedra[normaliz]"``
 `Normaliz <https://passagemath.org/docs/latest/html/en/reference/spkg/normaliz.html#spkg-normaliz>`_, via `PyNormaliz <https://passagemath.org/docs/latest/html/en/reference/spkg/pynormaliz.html#spkg-pynormaliz>`_,
 provides very fast computations in particular for polyhedra with data in algebraic number fields.

 ::

    $ pipx run --pip-args="--prefer-binary" --spec "passagemath-polyhedra[normaliz,test]" ipython

    In [1]: from passagemath_polyhedra import *

    In [2]: gap_norm = polytopes.grand_antiprism(backend='normaliz'); gap_norm

    In [3]: gap_norm.f_vector()

``pip install "passagemath-polyhedra[cddlib]"``
 cddlib provides support for computations with polyhedra in floating-point arithmetic.

 ::

    $ pipx run --pip-args="--prefer-binary" --spec "passagemath-polyhedra[cddlib,test]" ipython

    In [1]: from passagemath_polyhedra import *

    In [2]: P1 = polytopes.regular_polygon(5, exact=False); P1
    Out[2]: A 2-dimensional polyhedron in RDF^2 defined as the convex hull of 5 vertices

``pip install "passagemath-polyhedra[lrslib]"``
 `lrslib <https://passagemath.org/docs/latest/html/en/reference/spkg/lrslib.html#spkg-lrslib>`_
 can be used for polytope volume computations and for enumerating Nash equilibria.

 ::

    $ pipx run --pip-args="--prefer-binary" --spec "passagemath-polyhedra[flint,lrslib,test]" ipython

    In [1]: from passagemath_polyhedra import *

    In [2]: A = matrix([[2, 1], [1, 5/2]]); B = matrix([[-1, 3], [2, 1]])

    In [3]: g = NormalFormGame([A, B]); g.obtain_nash(algorithm='lrs')
    Out[3]: [[(1/5, 4/5), (3/5, 2/5)]]

``pip install "passagemath-polyhedra[polymake]"``
 `Polymake <https://passagemath.org/docs/latest/html/en/reference/spkg/polymake.html#spkg-polymake>`_, via passagemath-polymake


Optional backends for optimization
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

``pip install "passagemath-polyhedra[cbc]"``
 `COIN/OR CBC <https://passagemath.org/docs/latest/html/en/reference/spkg/cbc.html#spkg-cbc>`_ Mixed Integer Linear Optimization solver,
 via `passagemath-coin-or-cbc <https://passagemath.org/docs/latest/html/en/reference/spkg/sage_numerical_backends_coin.html#spkg-sage-numerical-backends-coin>`_

``pip install "passagemath-polyhedra[cplex]"``
 CPLEX Mixed Integer Optimization solver (proprietary; requires licensed installation),
 via `passagemath-cplex <https://passagemath.org/docs/latest/html/en/reference/spkg/sage_numerical_backends_cplex.html#spkg-sage-numerical-backends-cplex>`_

``pip install "passagemath-polyhedra[cvxpy]"``
 `CVXPy <https://passagemath.org/docs/latest/html/en/reference/spkg/cvxpy.html#spkg-cvxpy>`_ as middle-end for `various backends <https://www.cvxpy.org/install/>`_

``pip install "passagemath-polyhedra[gurobi]"``
 Gurobi Mixed Integer Optimization solver (proprietary; requires licensed installation), via `passagemath-gurobi <https://passagemath.org/docs/latest/html/en/reference/spkg/sage_numerical_backends_gurobi.html#spkg-sage-numerical-backends-gurobi>`_

``pip install "passagemath-polyhedra[scip]"``
 `SCIP <https://passagemath.org/docs/latest/html/en/reference/spkg/scip.html#spkg-scip>`_ Mixed Integer Optimization solver,
 via `PySCIPOpt <https://passagemath.org/docs/latest/html/en/reference/spkg/pyscipopt.html#spkg-pyscipopt>`_


Development
-----------

::

    $ git clone --origin passagemath https://github.com/passagemath/passagemath.git
    $ cd passagemath
    passagemath $ ./bootstrap
    passagemath $ python3 -m venv polyhedra-venv
    passagemath $ source polyhedra-venv/bin/activate
    (polyhedra-venv) passagemath $ pip install -v -e pkgs/sagemath-polyhedra


Type
----

standard


Dependencies
------------

- $(PYTHON)
- $(PYTHON_TOOLCHAIN)
- :ref:`spkg_cysignals`
- :ref:`spkg_cython`
- :ref:`spkg_gmp`
- :ref:`spkg_gmpy2`
- :ref:`spkg_memory_allocator`
- :ref:`spkg_mpc`
- :ref:`spkg_mpfr`
- :ref:`spkg_pkgconf`
- :ref:`spkg_pkgconfig`
- :ref:`spkg_pplpy`
- :ref:`spkg_sage_conf`
- :ref:`spkg_sage_setup`
- :ref:`spkg_sagemath_environment`
- :ref:`spkg_sagemath_glpk`
- :ref:`spkg_sagemath_modules`
- :ref:`spkg_setuptools`

Version Information
-------------------

package-version.txt::

    10.8.1.rc0

version_requirements.txt::

    passagemath-polyhedra == 10.8.1rc0

Installation commands
---------------------

.. tab:: PyPI:

   .. CODE-BLOCK:: bash

       $ pip install passagemath-polyhedra==10.8.1rc0

.. tab:: Sage distribution:

   .. CODE-BLOCK:: bash

       $ sage -i sagemath_polyhedra


However, these system packages will not be used for building Sage
because ``spkg-configure.m4`` has not been written for this package;
see :issue:`27330` for more information.
