picos.expressions.set_simplex

Implements Simplex.

Outline

Classes

Simplex

A (truncated, symmetrized) real simplex.

Classes

Simplex

class picos.expressions.set_simplex.Simplex(**kwargs)[source]

Bases: picos.expressions.set.Set

A (truncated, symmetrized) real simplex.

Definition

Let r \in \mathbb{R}_{\geq 0} the specified radius and n \in \mathbb{Z}_{\geq 1} an arbitrary dimensionality.

  1. Without truncation and symmetrization, this is the nonnegative simplex

    \{x \in \mathbb{R}^n_{\geq 0} \mid \sum_{i = 1}^n x_i \leq r\}.

    For r = 1, this is the standard (unit) n-simplex.

  2. With truncation but without symmetrization, this is the nonnegative simplex intersected with the \infty-norm unit ball

    \{
    x \in \mathbb{R}^n_{\geq 0}
\mid
    \sum_{i = 1}^n x_i \leq r \land x \leq 1
\}.

    For r \leq 1, this equals case (1).

  3. With symmetrization but without truncation, this is the 1-norm ball of radius r

    \{x \in \mathbb{R}^n \mid \sum_{i = 1}^n |x_i| \leq r\}.

  4. With both symmetrization and truncation, this is the convex polytope

    \{
    x \in \mathbb{R}
\mid
    \sum_{i = 1}^n |x_i| \leq r \land 0 \leq x \leq 1
\}.

    For r \leq 1, this equals case (3).

__init__(radius=<1×1 Real Constant: 1>, truncated=False, symmetrized=False)[source]

Construct a Simplex.

Parameters

radius (float or AffineExpression) – The radius of the simplex.

property radius

The radius of the simplex.

property symmetrized

Wether the simplex is mirrored onto all orthants.

property truncated

Whether this is intersected with the unit \infty-ball.