picos.expressions.set_simplex¶

Implements Simplex.

Classes

class picos.expressions.set_simplex.Simplex(radius=Constant(1), truncated=False, symmetrized=False)[source]¶

Bases: Set

A (truncated, symmetrized) real simplex.

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

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

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