This library implements common operations over convex polyhedra such as polytope projection and vertex enumeration.
See the complete API documentation for details.
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 itself:
pip install pypoman
We can compute the list of vertices of a polytope described in halfspace
representation by A * x <= b
:
from numpy import array
from pypoman import compute_polytope_vertices
A = 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 = array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 2, 2, 1, 2, 3])
vertices = compute_polytope_vertices(A, b)
The other way round, assume we know the vertices of a polytope, and want to get
its halfspace representation A * x <= b
.
from numpy import array
from pypoman import compute_polytope_halfspaces
vertices = map(array, [[1, 0, 0], [0, 1, 0], [1, 1, 0], [0, 0, 1], [0, 1, 1]])
A, b = compute_polytope_halfspaces(vertices)
Let us project an n-dimensional polytope over x = [x_1 ... x_n]
onto its first two coordinates proj(x) = [x_1 x_2]
:
from numpy import array, eye, ones, vstack, zeros
from pypoman import plot_polygon, project_polytope
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 = project_polytope(proj, ineq, eq, method='bretl')
if __name__ == "__main__": # plot projected polytope
import pylab
pylab.ion()
pylab.figure()
plot_polygon(vertices)
- A short introduction to Polyhedra and polytopes
- Komei Fukuda's Frequently Asked Questions in Polyhedral Computation
- The Polyhedron class in Sage
- StabiliPy: a Python package implementing a more general recursive method for polytope projection