Union

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

Type that represents the set union of two convex sets.

Fields

• X – convex set
• Y – convex set

∪

Alias for UnionSet.

swap(cup::UnionSet)

Return a new UnionSet object with the arguments swapped.

Input

• cup – union of two convex sets

Output

A new UnionSet object with the arguments swapped.

dim(cup::UnionSet)

Return the dimension of the set union of two convex sets.

Input

• cup – union of two convex sets

Output

The ambient dimension of the union of two convex sets.

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

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

Input

• d – direction
• cup – union of two convex 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 convex 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 it happens that the support function can be more efficiently computed (without passing through the support vector), consider using the alternative algorithm="support_function" implementation, 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 convex sets in a given direction.

Input

• d – direction
• cup – union of two convex sets

Output

The support function in the given direction.

Algorithm

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

an_element(cup::UnionSet)

Return some element of a union of two convex sets.

Input

• cup – union of two convex sets

Output

An element in the union of two convex sets.

Algorithm

We use an_element on the first wrapped set.

∈(x::AbstractVector, cup::UnionSet)

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

Input

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

Output

true iff $x ∈ cup$.

isempty(cup::UnionSet)

Check whether a union of two convex sets is empty.

Input

• cup – union of two convex sets

Output

true iff the union is empty.

isbounded(cup::UnionSet)

Determine whether a union of two convex sets is bounded.

Input

• cup – union of two convex 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 a union of two convex sets.

Input

• cup – union of two convex 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 convex sets.

Fields

• array – array of convex sets

dim(cup::UnionSetArray)

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

Input

• cup – union of a finite number of convex sets

Output

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

array(cup::UnionSetArray)

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

Input

• cup – union of a finite number of convex sets

Output

The array that holds the union of a finite number of convex sets.

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

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

Input

• d – direction
• cup – union of a finite number of convex 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 convex 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 it happens that the support function can be more efficiently computed (without passing through the support vector), consider using the alternative algorithm="support_function" implementation, 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 convex sets in a given direction.

Input

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

Output

The support function in the given direction.

Algorithm

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

an_element(cup::UnionSetArray)

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

Input

• cup – union of a finite number of convex sets

Output

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

Algorithm

We use an_element on the first wrapped set.

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

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

Input

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

Output

true iff $x ∈ cup$.

isempty(cup::UnionSetArray)

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

Input

• cup – union of a finite number of convex sets

Output

true iff the union is empty.

isbounded(cup::UnionSetArray)

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

Input

• cup – union of a finite number of convex 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 a union of a finite number of convex sets.

Input

• cup – union of a finite number of convex 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.