pypoman 0.5.1

Python Polyhedron Manipulation

This library implements common operations over convex polyhedra:

• Polytope projection

• Duality: conversion between halfspace and vertex representations

• Chebyshev center calculation

See the complete API documentation for details.

Installation

For Debian users, make sure you have all system-wide dependencies by:

sudo apt-get install cython libglpk-dev python python-dev python-pip python-scipy

Then, install the module via:

pip install pypoman

Examples

Vertex enumeration

We can compute the list of vertices of a polytope described in halfspace representation by A * x <= b:

import numpy
import pypoman

A = numpy.array([
    [-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,  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,  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,  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,  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,  0],
    [0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, -1],
    [1,  1,  1,  0,  0,  0,  0,  0,  0,  0,  0,  0],
    [0,  0,  0,  1,  1,  1,  0,  0,  0,  0,  0,  0],
    [0,  0,  0,  0,  0,  0,  1,  1,  1,  0,  0,  0],
    [0,  0,  0,  0,  0,  0,  0,  0,  0,  1,  1,  1],
    [1,  0,  0,  1,  0,  0,  1,  0,  0,  1,  0,  0],
    [0,  1,  0,  0,  1,  0,  0,  1,  0,  0,  1,  0],
    [0,  0,  1,  0,  0,  1,  0,  0,  1,  0,  0,  1]])
b = numpy.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 2, 2, 1, 2, 3])
vertices = pypoman.compute_polytope_vertices(A, b)

Polytope projection

Let us project an n-dimensional polytope over x = [x_1 ... x_n] onto its first two coordinates proj(x) = [x_1 x_2]:

import pypoman
from numpy import array, eye, ones, vstack, zeros

n = 10  # dimension of the original polytope
p = 2   # dimension of the projected polytope

# Original polytope:
# - inequality constraints: \forall i, |x_i| <= 1
# - equality constraint: sum_i x_i = 0
A = vstack([+eye(n), -eye(n)])
b = ones(2 * n)
C = ones(n).reshape((1, n))
d = array([0])
ineq = (A, b)  # A * x <= b
eq = (C, d)    # C * x == d

# Projection is proj(x) = [x_0 x_1]
E = zeros((p, n))
E[0, 0] = 1.
E[1, 1] = 1.
f = zeros(p)
proj = (E, f)  # proj(x) = E * x + f

vertices = pypoman.project_polytope(proj, ineq, eq, method='bretl')

if __name__ == "__main__":   # plot projected polytope
    import pylab
    pylab.ion()
    pylab.figure()
    pypoman.plot_polygon(vertices)