Bloating

Bloating{N, S<:LazySet{N}} <: LazySet{N}

Type that represents a uniform expansion of a set in a given norm (also known as bloating).

Fields

• X – set
• ε – (usually positive) bloating factor
• p – $p$-norm (should be $≥ 1$; default: $2$)

Notes

The Bloating operation preserves convexity: if X is convex, then any bloating of X is convex as well.

If ε is positive, then Bloating(X, ε, p) is equivalent to the Minkowski sum of X and a ball in the p-norm of radius ε centered in the origin O (i.e., X ⊕ Ballp(p, O, ε)).

Some operations require, or silently assume, that ε is positive. Check the documentation for further information.

dim(B::Bloating)

Return the dimension of a bloated set.

Input

• B – bloated set

Output

The ambient dimension of the bloated set.

σ(d::AbstractVector, B::Bloating)

Return the support vector of a bloated set in a given direction.

Input

• d – direction
• B – bloated set

Output

The support vector of the bloated set in the given direction.

ρ(d::AbstractVector, B::Bloating)

Return the support function of a bloated set in a given direction.

Input

• d – direction
• B – bloated set

Output

The support function of the bloated set in the given direction.

isbounded(B::Bloating)

Determine whether a bloated set is bounded.

Input

• B – bloated set

Output

true iff the wrapped set is bounded.

isempty(B::Bloating)

Determine whether a bloated set is empty.

Input

• B – bloated set

Output

true iff the wrapped set is empty.

Notes

This implementation disregards negative bloating, which could potentially turn a non-empty set into an empty set.

an_element(B::Bloating)

Return some element of a bloated set.

Input

• B – bloated set

Output

An element in the bloated set.

Algorithm

This implementation disregards negative bloating and returns the result of an_element for the wrapped set.

constraints_list(B::Bloating)

Return the list of constraints of a bloated set.

Input

• B – bloated set

Output

The list of constraints of the bloated set.

Notes

The constraints list is only available for non-negative bloating in the p-norm for $p = 1$ or $p = ∞$ and if constraints_list is available for the unbloated set.

Algorithm

We call constraints_list on the lazy Minkowski sum with the bloating ball.

center(B::Bloating)

Return the center of a bloated set.

Input

• B – bloated set

Output

The center of the wrapped set.

Notes

This implementation disregards negative bloating, which could potentially remove the center from the set.

ispolyhedral(B::Bloating)

Check whether a bloated set is polyhedral.

Input

• B – bloated set

Output

true if the set is polyhedral.

Algorithm

We check the sufficient condition that the base set is polyhedral and that the norm for bloating is either 1-norm or the infinity norm.