Duality

Functions to switch between halfspace and vertex representations.

pypoman.duality.compute_cone_face_matrix(S: ndarray) → ndarray

Compute the face matrix of a polyhedral cone from its span matrix.

Parameters:

S – Span matrix defining the cone as \(x = S \lambda\) with \(\lambda \geq 0\).

Returns:

Face matrix defining the cone equivalently by \(F x \leq 0\).

pypoman.duality.compute_polytope_halfspaces(vertices: List[ndarray]) → ndarray | Tuple[ndarray, ndarray]

Compute the halfspace representation (H-rep) of a polytope.

The polytope is defined as convex hull of a set of vertices:

\[A x \leq b \quad \Leftrightarrow \quad x \in \mathrm{conv}(\mathrm{vertices})\]

Parameters:

vertices – List of polytope vertices.

Returns:

Tuple (A, b) of the halfspace representation, or empty array if it is empty.

pypoman.duality.compute_polytope_vertices(A: ndarray, b: ndarray) → List[ndarray]

Compute the vertices of a polytope.

The polytope is given in halfspace representation by \(A x \leq b\).

Parameters:

• A – Matrix of halfspace representation.

• b – Vector of halfspace representation.

Returns:

List of polytope vertices.

Notes

This method won’t work well if your halfspace representation includes equality constraints \(A x = b\) written as \((A x \leq b \wedge -A x \leq -b)\). If this is your use case, consider using directly the linear set lin_set of equality-constraint generatorsin pycddlib.

pypoman.duality.convex_hull(points: List[ndarray]) → List[ndarray]

Compute the convex hull of a set of points.

Parameters:

points – Set of points.

Returns:

List of polytope vertices.