Union

Note that the union of convex sets is generally not convex. Hence these set types are not part of the convex-set family LazySet.

Binary set union (UnionSet)

LazySets.UnionSet — Type.

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

Type that represents the set union of two convex sets.

Fields

X – convex set
Y – convex set

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Base.:∪ — Method.

∪

Alias for UnionSet.

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LazySets.swap — Method.

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.

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LazySets.dim — Method.

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.

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LazySets.σ — Method.

σ(d::AbstractVector{N}, cup::UnionSet{N}; [algorithm]="support_vector") where {N<:Real}

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 $Y$.

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LazySets.ρ — Method.

ρ(d::AbstractVector{N}, cup::UnionSet{N}) where {N<:Real}

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 $Y$.

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LazySets.an_element — Method.

an_element(cup::UnionSet{N}) where {N<:Real}

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.

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Base.:∈ — Method.

∈(x::AbstractVector{N}, cup::UnionSet{N}) where {N<:Real}

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 $x ∈ cup$.

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Base.isempty — Method.

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.

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LazySets.isbounded — Method.

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.

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$n$-ary set union (UnionSetArray)

LazySets.UnionSetArray — Type.

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

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

Fields

array – array of convex sets

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LazySets.dim — Method.

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.

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LazySets.array — Method.

array(cup::UnionSetArray{N, S}) where {N<:Real, S<:LazySet{N}}

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.

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LazySets.σ — Method.

σ(d::AbstractVector{N}, cup::UnionSetArray{N}; [algorithm]="support_vector") where {N<:Real}

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 $Xᵢ$.

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LazySets.ρ — Method.

ρ(d::AbstractVector{N}, cup::UnionSetArray{N}) where {N<:Real}

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

Input