Union

UnionSet{N, S1<:LazySet{N}, S2<:LazySet{N}}

Type that represents the set union of two sets.

Fields

• X – set
• Y – set

Notes

The union of convex sets is typically not convex.

∪

Alias for UnionSet.

swap(cup::UnionSet)

Return a new UnionSet object with the arguments swapped.

Input

• cup – union of two sets

Output

A new UnionSet object with the arguments swapped.

dim(cup::UnionSet)

Return the dimension of the union of two sets.

Input

• cup – union of two sets

Output

The ambient dimension of the union of two sets.

σ(d::AbstractVector, cup::UnionSet; [algorithm]="support_vector")

Return the support vector of the union of two sets in a given direction.

Input

• d – direction
• cup – union of two sets
• algorithm – (optional, default: "supportvector"): the algorithm to compute the support vector; if "supportvector", use the support vector of each argument; if "support_function" use the support function of each argument and evaluate the support vector of only one of them

Output

The support vector in the given direction.

Algorithm

The support vector of the union of two sets $X$ and $Y$ can be obtained as the vector that maximizes the support function of either $X$ or $Y$, i.e., it is sufficient to find the $\argmax(ρ(d, X), ρ(d, Y)])$ and evaluate its support vector.

The default implementation, with option algorithm="support_vector", computes the support vector of $X$ and $Y$ and then compares the support function using a dot product.

If the support function can be computed more efficiently, the alternative implementation algorithm="support_function" can be used, which evaluates the support function of each set directly and then calls only the support vector of either $X$ or $Y$.

ρ(d::AbstractVector, cup::UnionSet)

Return the support function of the union of two sets in a given direction.

Input

• d – direction
• cup – union of two sets

Output

The support function in the given direction.

Algorithm

The support function of the union of two sets $X$ and $Y$ is the maximum of the support function of $X$ and $Y$.

an_element(cup::UnionSet)

Return some element of the union of two sets.

Input

• cup – union of two sets

Output

An element in the union of two sets.

Algorithm

We use an_element on the first wrapped set.

∈(x::AbstractVector, cup::UnionSet)

Check whether a given point is contained in the union of two sets.

Input

• x – point/vector
• cup – union of two sets

Output

true iff $x ∈ cup$.

isempty(cup::UnionSet)

Check whether the union of two sets is empty.

Input

• cup – union of two sets

Output

true iff the union is empty.

isbounded(cup::UnionSet)

Determine whether the union of two sets is bounded.

Input

• cup – union of two sets

Output

true iff the union is bounded.

vertices_list(cup::UnionSet; apply_convex_hull::Bool=false, backend=nothing)

Return the list of vertices of the union of two sets.

Input

• cup – union of two sets
• apply_convex_hull – (optional, default: false) if true, post-process the vertices using a convex-hull algorithm
• backend – (optional, default: nothing) backend for computing the convex hull (see argument apply_convex_hull)

Output

The list of vertices, possibly reduced to the list of vertices of the convex hull.

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

Type that represents the set union of a finite number of sets.

Fields

• array – array of sets

Notes

The union of convex sets is typically not convex.

dim(cup::UnionSetArray)

Return the dimension of the set union of a finite number of sets.

Input

• cup – union of a finite number of sets

Output

The ambient dimension of the union of a finite number of sets.

array(cup::UnionSetArray)

Return the array of the union of a finite number of sets.

Input

• cup – union of a finite number of sets

Output

The array that holds the sets.

σ(d::AbstractVector, cup::UnionSetArray; [algorithm]="support_vector")

Return the support vector of the union of a finite number of sets in a given direction.

Input

• d – direction
• cup – union of a finite number of sets
• algorithm – (optional, default: "supportvector"): the algorithm to compute the support vector; if "supportvector", use the support vector of each argument; if "support_function" use the support function of each argument and evaluate the support vector of only one of them

Output

The support vector in the given direction.

Algorithm

The support vector of the union of a finite number of sets $X₁, X₂, ...$ can be obtained as the vector that maximizes the support function, i.e., it is sufficient to find the $\argmax([ρ(d, X₂), ρ(d, X₂), ...])$ and evaluate its support vector.

The default implementation, with option algorithm="support_vector", computes the support vector of all $X₁, X₂, ...$ and then compares the support function using a dot product.

If the support function can be computed more efficiently, the alternative implementation algorithm="support_function" can be used, which evaluates the support function of each set directly and then calls only the support vector of one of the $Xᵢ$.

ρ(d::AbstractVector, cup::UnionSetArray)

Return the support function of the union of a finite number of sets in a given direction.

Input

• d – direction
• cup – union of a finite number of sets

Output

The support function in the given direction.

Algorithm

The support function of the union of a finite number of sets $X₁, X₂, ...$ can be obtained as the maximum of $ρ(d, X₂), ρ(d, X₂), ...$.

an_element(cup::UnionSetArray)

Return some element of the union of a finite number of sets.

Input

• cup – union of a finite number of sets

Output

An element in the union of a finite number of sets.

Algorithm

We use an_element on the first wrapped set.

∈(x::AbstractVector, cup::UnionSetArray)

Check whether a given point is contained in the union of a finite number of sets.

Input

• x – point/vector
• cup – union of a finite number of sets

Output

true iff $x ∈ cup$.

isempty(cup::UnionSetArray)

Check whether the union of a finite number of sets is empty.

Input

• cup – union of a finite number of sets

Output

true iff the union is empty.

isbounded(cup::UnionSetArray)

Determine whether the union of a finite number of sets is bounded.

Input

• cup – union of a finite number of sets

Output

true iff the union is bounded.

vertices_list(cup::UnionSetArray; apply_convex_hull::Bool=false,
              backend=nothing)

Return the list of vertices of the union of a finite number of sets.

Input

• cup – union of a finite number of sets
• apply_convex_hull – (optional, default: false) if true, post-process the vertices using a convex-hull algorithm
• backend – (optional, default: nothing) backend for computing the convex hull (see argument apply_convex_hull)

Output

The list of vertices, possibly reduced to the list of vertices of the convex hull.