CartesianProduct

Cartesian product

Binary Cartesian product (CartesianProduct)

LazySets.CartesianProduct — Type.
CartesianProduct{N<:Real, S1<:LazySet{N}, S2<:LazySet{N}} <: LazySet{N}

Type that represents a Cartesian product of two convex sets.

Fields

  • X – first convex set
  • Y – second convex set

Notes

The Cartesian product of three elements is obtained recursively. See also CartesianProductArray for an implementation of a Cartesian product of many sets without recursion, instead using an array.

The EmptySet is the absorbing element for CartesianProduct.

Examples

The Cartesian product between two sets X and Y can be constructed either using CartesianProduct(X, Y) or the short-cut notation X × Y:

julia> I1 = Interval(0, 1);

julia> I2 = Interval(2, 4);

julia> I12 = I1 × I2;

julia> typeof(I12)
CartesianProduct{Float64,Interval{Float64,IntervalArithmetic.Interval{Float64}},Interval{Float64,IntervalArithmetic.Interval{Float64}}}

A hyperrectangle is the cartesian product of intervals, so we can convert I12 exactly to a Hyperrectangle type:

julia> convert(Hyperrectangle, I12)
Hyperrectangle{Float64,Array{Float64,1},Array{Float64,1}}([0.5, 3.0], [0.5, 1.0])
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LinearAlgebra.:× — Method.
×

Alias for the binary Cartesian product.

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Base.:* — Method.
    *(X::LazySet, Y::LazySet)

Alias for the binary Cartesian product.

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LazySets.swap — Method.
swap(cp::CartesianProduct)

Return a new CartesianProduct object with the arguments swapped.

Input

  • cp – Cartesian product of two convex sets

Output

A new CartesianProduct object with the arguments swapped.

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LazySets.dim — Method.
dim(cp::CartesianProduct)

Return the dimension of a Cartesian product.

Input

  • cp – Cartesian product

Output

The ambient dimension of the Cartesian product.

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LazySets.ρ — Method.
ρ(d::AbstractVector{N}, cp::CartesianProduct{N}) where {N<:Real}

Return the support function of a Cartesian product.

Input

  • d – direction
  • cp – Cartesian product

Output

The support function in the given direction. If the direction has norm zero, the result depends on the wrapped sets.

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LazySets.σ — Method.
σ(d::AbstractVector{N}, cp::CartesianProduct{N}) where {N<:Real}

Return the support vector of a Cartesian product.

Input

  • d – direction
  • cp – Cartesian product

Output

The support vector in the given direction. If the direction has norm zero, the result depends on the wrapped sets.

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LazySets.isbounded — Method.
isbounded(cp::CartesianProduct)

Determine whether a Cartesian product is bounded.

Input

  • cp – Cartesian product

Output

true iff both wrapped sets are bounded.

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Base.:∈ — Method.
∈(x::AbstractVector{N}, cp::CartesianProduct{N}) where {N<:Real}

Check whether a given point is contained in a Cartesian product.

Input

  • x – point/vector
  • cp – Cartesian product

Output

true iff $x ∈ cp$.

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Base.isempty — Method.
isempty(cp::CartesianProduct)

Return if a Cartesian product is empty or not.

Input

  • cp – Cartesian product

Output

true iff any of the sub-blocks is empty.

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LazySets.constraints_list — Method.
constraints_list(cp::CartesianProduct{N}) where {N<:Real}

Return the list of constraints of a (polyhedral) Cartesian product.

Input

  • cp – Cartesian product

Output

A list of constraints.

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LazySets.vertices_list — Method.
vertices_list(cp::CartesianProduct{N}) where {N<:Real}

Return the list of vertices of a (polytopic) Cartesian product.

Input

  • cp – Cartesian product

Output

A list of vertices.

Algorithm

We assume that the underlying sets are polytopic. Then the high-dimensional set of vertices is just the Cartesian product of the low-dimensional sets of vertices.

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LazySets.linear_map — Method.
linear_map(M::AbstractMatrix{N}, cp::CartesianProduct{N}) where {N<:Real}

Concrete linear map of a (polyhedral) Cartesian product.

Input

  • M – matrix
  • cp – Cartesian product of two convex sets

Output

A polytope if cp is bounded and a polyhedron otherwise.

Algorithm

We convert the Cartesian product to constraint representation and then call linear_map on the corresponding polyhedron.

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Inherited from LazySet:

$n$-ary Cartesian product (CartesianProductArray)

LazySets.CartesianProductArray — Type.

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

Type that represents the Cartesian product of a finite number of convex sets.

Fields

  • array – array of sets

Notes

The EmptySet is the absorbing element for CartesianProductArray.

Constructors:

  • CartesianProductArray(array::Vector{<:LazySet}) – default constructor

  • CartesianProductArray([n]::Int=0, [N]::Type=Float64)

– constructor for an empty product with optional size hint and numeric type

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LazySets.dim — Method.

dim(cpa::CartesianProductArray)

Return the dimension of a Cartesian product of a finite number of convex sets.

Input

  • cpa – Cartesian product array

Output

The ambient dimension of the Cartesian product of a finite number of convex sets.

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LazySets.ρ — Method.

ρ(d::AbstractVector{N}, cp::CartesianProductArray{N}) where {N<:Real}

Return the support function of a Cartesian product array.

Input

  • d – direction
  • cpa – Cartesian product array

Output

The support function in the given direction. If the direction has norm zero, the result depends on the wrapped sets.

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LazySets.σ — Method.

σ(d::AbstractVector{N}, cpa::CartesianProductArray{N}) where {N<:Real}

Support vector of a Cartesian product array.

Input

  • d – direction
  • cpa – Cartesian product array

Output

The support vector in the given direction. If the direction has norm zero, the result depends on the product sets.

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LazySets.isbounded — Method.

isbounded(cpa::CartesianProductArray)

Determine whether a Cartesian product of a finite number of convex sets is bounded.

Input

  • cpa – Cartesian product of a finite number of convex sets

Output

true iff all wrapped sets are bounded.

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Base.:∈ — Method.

∈(x::AbstractVector{N}, cpa::CartesianProductArray{N}) where {N<:Real}

Check whether a given point is contained in a Cartesian product of a finite number of sets.

Input

  • x – point/vector
  • cpa – Cartesian product array

Output

true iff $x ∈ \text{cpa}$.

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Base.isempty — Method.

isempty(cpa::CartesianProductArray)

Return if a Cartesian product is empty or not.

Input

  • cp – Cartesian product

Output

true iff any of the sub-blocks is empty.

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LazySets.constraints_list — Method.

constraints_list(cpa::CartesianProductArray{N}) where {N<:Real}

Return the list of constraints of a (polyhedral) Cartesian product of a finite number of sets.

Input

  • cpa – Cartesian product array

Output

A list of constraints.

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LazySets.vertices_list — Method.

vertices_list(cpa::CartesianProductArray{N}) where {N<:Real}

Return the list of vertices of a (polytopic) Cartesian product of a finite number of sets.

Input

  • cpa – Cartesian product array

Output

A list of vertices.

Algorithm

We assume that the underlying sets are polytopic. Then the high-dimensional set of vertices is just the Cartesian product of the low-dimensional sets of vertices.

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LazySets.linear_map — Method.

linear_map(M::AbstractMatrix{N}, cpa::CartesianProductArray{N} ) where {N<:Real}

Concrete linear map of a Cartesian product of a finite number of convex sets.

Input

  • M – matrix
  • cpa – Cartesian product of a finite number of convex sets

Output

A polytope.

Algorithm

We check if the matrix is invertible. If so, we convert the Cartesian product to constraint representation. Otherwise, we convert the Cartesian product to vertex representation. In both cases, we then call linear_map on the resulting polytope.

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LazySets.array — Method.

array(cpa::CartesianProductArray{N, S}) where {N<:Real, S<:LazySet{N}}

Return the array of a Cartesian product of a finite number of convex sets.

Input

  • cpa – Cartesian product array

Output

The array of a Cartesian product of a finite number of convex sets.

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array(cha::ConvexHullArray{N, S}) where {N<:Real, S<:LazySet{N}}

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

Input

  • cha – convex hull array

Output

The array of a convex hull of a finite number of convex sets.

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array(ia::IntersectionArray{N, S}) where {N<:Real, S<:LazySet{N}}

Return the array of an intersection of a finite number of convex sets.

Input

  • ia – intersection of a finite number of convex sets

Output

The array of an intersection of a finite number of convex sets.

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array(msa::MinkowskiSumArray{N, S}) where {N<:Real, S<:LazySet{N}}

Return the array of a Minkowski sum of a finite number of convex sets.

Input

  • msa – Minkowski sum array

Output

The array of a Minkowski sum of a finite number of convex sets.

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array(cms::CachedMinkowskiSumArray{N, S}) where {N<:Real, S<:LazySet{N}}

Return the array of a caching Minkowski sum.

Input

  • cms – caching Minkowski sum

Output

The array of a caching Minkowski sum.

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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.block_structure — Method.

block_structure(cpa::CartesianProductArray{N}) where {N}

Returns an array containing the dimension ranges of each block in a CartesianProductArray.

Input

  • cpa – Cartesian product array

Output

A vector of ranges

Example

julia> using LazySets: block_structure

julia> cpa = CartesianProductArray([BallInf(zeros(n), 1.0) for n in [3, 1, 2]]);

julia> block_structure(cpa)
3-element Array{UnitRange{Int64},1}:
 1:3
 4:4
 5:6
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LazySets.block_to_dimension_indices — Method.

blocktodimension_indices(cpa::CartesianProductArray{N}, vars::Vector{Int}) where {N}

Returns a vector mapping block index i to tuple (f, l) such that either f = l = -1 or f is the first dimension index and l is the last dimension index of the i-th block, depending on whether one of the block's dimension indices is specified in vars.

Input

  • cpa – Cartesian product array
  • vars – list containing the variables of interest, sorted in ascending order

Output

(i) A vector of tuples, where values in tuple relate to range of dimensions in the i-th block. (ii) Number of constrained blocks

Example

julia> using LazySets: block_to_dimension_indices

julia> cpa = CartesianProductArray([BallInf(zeros(n), 1.0) for n in [1, 3, 2, 3]]);

julia> m, k = block_to_dimension_indices(cpa, [2, 4, 8]);

julia> m
4-element Array{Tuple{Int64,Int64},1}:
 (-1, -1)
 (2, 4)
 (-1, -1)
 (7, 9)

julia> k
2

This vector represents the mapping "second block from dimension 2 to dimension 4, fourth block from dimension 7 to dimension 9." These blocks contain the dimensions specified in [2, 4, 8]. Number of constrained variables here is 2 (2nd and 4th blocks)

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LazySets.substitute_blocks — Method.

substituteblocks(lowdimcpa::CartesianProductArray{N}, origcpa::CartesianProductArray{N}, blocks::Vector{Tuple{Int,Int}}) where {N}

Return merged Cartesian Product Array between original CPA and some low-dimensional CPA, which represents updated subset of variables in specified blocks.

Input

  • low_dim_cpa – low-dimensional cartesian product array
  • orig_cpa – original high-dimensional Cartesian product array
  • blocks – index of the first variable in each block of orig_cpa

Output

Merged cartesian product array

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Inherited from LazySet: