Common Set Operations

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.

Constructors:

• CartesianProduct{N<:Real, S1<:LazySet{N}, S2<:LazySet{N}}(X1::S1, X2::S2) – default constructor

• CartesianProduct(Xarr::Vector{S}) where {S<:LazySet} – constructor from an array of convex sets

    *(X::LazySet, Y::LazySet)

Alias for the binary Cartesian product.

    *(a::N, X::LazySet) where {N}

Return a linear map of a convex set by a scalar value.

Input

• a – scalar

• X – convex set

Output

The linear map of the convex set.

dim(cp::CartesianProduct)::Int

Return the dimension of a Cartesian product.

Input

• cp – Cartesian product

Output

The ambient dimension of the Cartesian product.

σ(d::V, cp::CartesianProduct) where {N<:Real, V<:AbstractVector{N}}

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 product sets.

Algorithm

∈(x::AbstractVector{<:Real}, cp::CartesianProduct)::Bool

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

Input

• x – point/vector

• cp – Cartesian product

Output

true iff $x ∈ cp$.

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

array(cpa::CartesianProductArray{N, S}
     )::Vector{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.

array(cha::ConvexHullArray{N, S})::Vector{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.

array(ia::IntersectionArray{N, S})::Vector{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.

array(msa::MinkowskiSumArray{N, S})::Vector{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.

array(cms::CacheMinkowskiSum{N, S})::Vector{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.

dim(cpa::CartesianProductArray)::Int

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.

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

Support vector of a Cartesian product.

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.

∈(x::AbstractVector{N}, cpa::CartesianProductArray{N, <:LazySet{N}}
 )::Bool  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 $x ∈ \text{cpa}$.

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

Type that represents the convex hull of the union of two convex sets.

Fields

• X – convex set

• Y – convex set

Notes

The EmptySet is the neutral element for ConvexHull.

Examples

Convex hull of two 100-dimensional Euclidean balls:

julia> b1, b2 = Ball2(zeros(100), 0.1), Ball2(4*ones(100), 0.2);

julia> c = ConvexHull(b1, b2);

julia> typeof(c)
LazySets.ConvexHull{Float64,LazySets.Ball2{Float64},LazySets.Ball2{Float64}}

CH

Alias for ConvexHull.

dim(ch::ConvexHull)::Int

Return the dimension of a convex hull of two convex sets.

Input

• ch – convex hull of two convex sets

Output

The ambient dimension of the convex hull of two convex sets.

σ(d::V, ch::ConvexHull) where {N<:Real, V<:AbstractVector{N}}

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

Input

• d – direction

• ch – convex hull of two convex sets

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

Type that represents the symbolic convex hull of a finite number of convex sets.

Fields

• array – array of sets

Notes

The EmptySet is the neutral element for ConvexHullArray.

Constructors:

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

• ConvexHullArray([n]::Int=0, [N]::Type=Float64) – constructor for an empty hull with optional size hint and numeric type

Examples

Convex hull of 100 two-dimensional balls whose centers follows a sinusoidal:

julia> b = [Ball2([2*pi*i/100, sin(2*pi*i/100)], 0.05) for i in 1:100];

julia> c = ConvexHullArray(b);

CHArray

Alias for ConvexHullArray.

array(cpa::CartesianProductArray{N, S}
     )::Vector{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.

array(cha::ConvexHullArray{N, S})::Vector{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.

array(ia::IntersectionArray{N, S})::Vector{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.

array(msa::MinkowskiSumArray{N, S})::Vector{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.

array(cms::CacheMinkowskiSum{N, S})::Vector{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.

dim(cha::ConvexHullArray)::Int

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

Input

• cha – convex hull array

Output

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

σ(d::V, cha::ConvexHullArray) where {N<:Real, V<:AbstractVector{N}}

Return the support vector of a convex hull array in a given direction.

Input

• d – direction

• cha – convex hull array

convex_hull(points::Vector{S}; [algorithm]::String="monotone_chain"
           )::Vector{S} where {S<:AbstractVector{N}} where {N<:Real}

Compute the convex hull of points in the plane.

Input

• points – list of 2D vectors

• algorithm – (optional, default: "monotone_chain") the convex hull algorithm, valid options are:

  • "monotone_chain"

  • "monotone_chain_sorted"

Output

The convex hull as a list of 2D vectors with the coordinates of the points.

Examples

Compute the convex hull of a random set of points:

julia> points = [randn(2) for i in 1:30]; # 30 random points in 2D

julia> hull = convex_hull(points);

julia> typeof(hull)
Array{Array{Float64,1},1}

Plot both the random points and the computed convex hull polygon:

julia> using Plots;

julia> plot([Tuple(pi) for pi in points], seriestype=:scatter);

julia> plot!(VPolygon(hull), alpha=0.2);

convex_hull!(points::Vector{S}; [algorithm]::String="monotone_chain"
            )::Vector{S} where {S<:AbstractVector{N}} where {N<:Real}

Compute the convex hull of points in the plane, in-place.

Input

• points – list of 2D vectors (is modified)

• algorithm – (optional, default: "monotone_chain") the convex hull algorithm; valid options are:

  • "monotone_chain"

  • "monotone_chain_sorted"

Output

The convex hull as a list of 2D vectors with the coordinates of the points.

Notes

See the non-modifying version convex_hull for more details.

right_turn(O::AbstractVector{N}, A::AbstractVector{N}, B::AbstractVector{N}
          )::N where {N<:Real}

Determine if the acute angle defined by the three points O, A, B in the plane is a right turn (counter-clockwise) with respect to the center O.

Input

• O – 2D center point

• A – 2D one point

• B – 2D another point

Output

Scalar representing the rotation.

Algorithm

The cross product is used to determine the sense of rotation. If the result is 0, the points are collinear; if it is positive, the three points constitute a positive angle of rotation around O from A to B; otherwise they constitute a negative angle.

monotone_chain!(points::Vector{S}; sort::Bool=true
               )::Vector{S} where {S<:AbstractVector{N}} where {N<:Real}

Compute the convex hull of points in the plane using Andrew's monotone chain method.

Input

• points – list of 2D vectors; is sorted in-place inside this function

• sort – (optional, default: true) flag for sorting the vertices lexicographically; sortedness is required for correctness

Output

List of vectors containing the 2D coordinates of the corner points of the convex hull.

Notes

For large sets of points, it is convenient to use static vectors to get maximum performance. For information on how to convert usual vectors into static vectors, see the type SVector provided by the StaticArrays package.

Algorithm

This function implements Andrew's monotone chain convex hull algorithm to construct the convex hull of a set of $n$ points in the plane in $O(n \log n)$ time. For further details see Monotone chain

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

Type that represents the intersection of two convex sets.

Fields

• X – convex set

• Y – convex set

∩

Alias for Intersection.

dim(cap::Intersection)::Int

Return the dimension of an intersection of two convex sets.

Input

• cap – intersection of two convex sets

Output

The ambient dimension of the intersection of two convex sets.

σ(d::V, cap::Intersection) where {N<:Real, V<:AbstractVector{N}}

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

Input

• d – direction

• cap – intersection of two convex sets

Output

The support vector in the given direction.

∈(x::AbstractVector{N}, cap::Intersection{N})::Bool where {N<:Real}

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

Input

• x – point/vector

• cap – intersection of two convex sets

Output

true iff $x ∈ cap$.

isempty(cap::Intersection)::Bool

Return if the intersection is empty or not.

Input

• cap – intersection of two convex sets

Output

true iff the intersection is empty.

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

Type that represents the intersection of a finite number of convex sets.

Fields

• array – array of convex sets

Notes

This type assumes that the dimensions of all elements match.

The EmptySet is the absorbing element for IntersectionArray.

Constructors:

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

• IntersectionArray([n]::Int=0, [N]::Type=Float64) – constructor for an empty sum with optional size hint and numeric type

array(cpa::CartesianProductArray{N, S}
     )::Vector{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.

array(cha::ConvexHullArray{N, S})::Vector{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.

array(ia::IntersectionArray{N, S})::Vector{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.

array(msa::MinkowskiSumArray{N, S})::Vector{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.

array(cms::CacheMinkowskiSum{N, S})::Vector{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.

dim(ia::IntersectionArray)::Int

Return the dimension of an intersection of a finite number of sets.

Input

• ia – intersection of a finite number of convex sets

Output

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

σ(d::AbstractVector{<:Real}, ia::IntersectionArray)::Vector{<:Real}

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

Input

• d – direction

• ia – intersection of a finite number of convex sets

Output

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

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

Type that represents the Minkowski sum of two convex sets.

Fields

• X – first convex set

• Y – second convex set

Notes

The ZeroSet is the neutral element and the EmptySet is the absorbing element for MinkowskiSum.

X + Y

Convenience constructor for Minkowski sum.

Input

• X – a convex set

• Y – another convex set

Output

The symbolic Minkowski sum of $X$ and $Y$.

⊕(X::LazySet, Y::LazySet)

Unicode alias constructor ⊕ (oplus) for the lazy Minkowski sum operator.

dim(ms::MinkowskiSum)::Int

Return the dimension of a Minkowski sum.

Input

• ms – Minkowski sum

Output

The ambient dimension of the Minkowski sum.

σ(d::V, ms::MinkowskiSum) where {N<:Real, V<:AbstractVector{N}}

Return the support vector of a Minkowski sum.

Input

• d – direction

• ms – Minkowski sum

Output

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

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

Type that represents the Minkowski sum of a finite number of convex sets.

Fields

• array – array of convex sets

Notes

This type assumes that the dimensions of all elements match.

The ZeroSet is the neutral element and the EmptySet is the absorbing element for MinkowskiSumArray.

Constructors:

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

• MinkowskiSumArray([n]::Int=0, [N]::Type=Float64) – constructor for an empty sum with optional size hint and numeric type

array(cpa::CartesianProductArray{N, S}
     )::Vector{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.

array(cha::ConvexHullArray{N, S})::Vector{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.

array(ia::IntersectionArray{N, S})::Vector{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.

array(msa::MinkowskiSumArray{N, S})::Vector{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.

array(cms::CacheMinkowskiSum{N, S})::Vector{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.

dim(msa::MinkowskiSumArray)::Int

Return the dimension of a Minkowski sum of a finite number of sets.

Input

• msa – Minkowski sum array

Output

The ambient dimension of the Minkowski sum of a finite number of sets.

σ(d::AbstractVector{<:Real}, msa::MinkowskiSumArray)::Vector{<:Real}

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

Input

• d – direction

• msa – Minkowski sum array

Output

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

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

Type that represents the Minkowski sum of a finite number of convex sets. Support vector queries are cached.

Fields

• array – array of convex sets

• cache – cache of support vector query results

Notes

This type assumes that the dimensions of all elements match.

The ZeroSet is the neutral element and the EmptySet is the absorbing element for CacheMinkowskiSum.

The cache (field cache) is implemented as dictionary whose keys are directions and whose values are pairs (k, s) where k is the number of elements in the array array when the support vector was evaluated last time, and s is the support vector that was obtained. Thus this type assumes that array is not modified except by adding new sets at the end.

Constructors:

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

• CacheMinkowskiSum([n]::Int=0, [N]::Type=Float64) – constructor for an empty sum with optional size hint and numeric type

array(cpa::CartesianProductArray{N, S}
     )::Vector{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.

array(cha::ConvexHullArray{N, S})::Vector{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.

array(ia::IntersectionArray{N, S})::Vector{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.

array(msa::MinkowskiSumArray{N, S})::Vector{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.

array(cms::CacheMinkowskiSum{N, S})::Vector{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.

dim(cms::CacheMinkowskiSum)::Int

Return the dimension of a caching Minkowski sum.

Input

• cms – caching Minkowski sum

Output

The ambient dimension of the caching Minkowski sum.

σ(d::AbstractVector{<:Real}, cms::CacheMinkowskiSum)::Vector{<:Real}

Return the support vector of a caching Minkowski sum in a given direction.

Input

• d – direction

• cms – caching Minkowski sum

Output

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

Notes

The result is cached, i.e., any further query with the same direction runs in constant time. When sets are added to the caching Minkowski sum, the query is only performed for the new sets.

forget_sets!(cms::CacheMinkowskiSum)::Int

Tell a caching Minkowski sum to forget the stored sets (but not the support vectors). Only those sets are forgotten such that for each cached direction the support vector has been computed before.

Input

• cms – caching Minkowski sum

Output

The number of sets that have been forgotten.

Notes

This function should only be used under the assertion that no new directions are queried in the future; otherwise such support vector results will be incorrect.

This implementation is optimistic and first tries to remove all sets. However, it also checks that for all cached directions the support vector has been computed before. If it finds that this is not the case, the implementation identifies the biggest index $k$ such that the above holds for the $k$ oldest sets, and then it only removes these. See the example below.

Examples

julia> x1 = BallInf(ones(3), 3.); x2 = Ball1(ones(3), 5.);

julia> cms1 = CacheMinkowskiSum(2); cms2 = CacheMinkowskiSum(2);

julia> d = ones(3);

julia> a1 = array(cms1); a2 = array(cms2);

julia> push!(a1, x1); push!(a2, x1);

julia> σ(d, cms1); σ(d, cms2);

julia> push!(a1, x2); push!(a2, x2);

julia> σ(d, cms1);

julia> idx1 = forget_sets!(cms1) # support vector was computed for both sets
2

julia> idx1 = forget_sets!(cms2) # support vector was only computed for first set
1

LinearMap{NM, N} <: LazySet{N}

Type that represents a linear transformation $M⋅S$ of a convex set $S$.

Fields

• M – matrix/linear map

• X – convex set

Notes

This type is parametric in the elements of the linear map, NM, and independently on the type of elements of the target set (N). Typically NM = N but it is not necessarily always the case e.g. if NM is an interval that holds numbers of type N, where N is a floating point number type such as Float64.

dim(lm::LinearMap)::Int

Return the dimension of a linear map.

Input

• lm – linear map

Output

The ambient dimension of the linear map.

σ(d::AbstractVector{<:Real}, lm::LinearMap)::AbstractVector{<:Real}

Return the support vector of the linear map.

Input

• d – direction

• lm – linear map

Output

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

Notes

If $L = M⋅S$, where $M$ is a matrix and $S$ is a convex set, it follows that $σ(d, L) = M⋅σ(M^T d, S)$ for any direction $d$.

    *(M::AbstractMatrix, X::LazySet)

Return the linear map of a convex set.

Input

• M – matrix/linear map

• X – convex set

Output

A lazy linear map, i.e. a LinearMap instance.

    *(a::N, X::LazySet) where {N}

Return a linear map of a convex set by a scalar value.

Input

• a – scalar

• X – convex set

Output

The linear map of the convex set.

∈(x::AbstractVector{N}, lm::LinearMap{NM, N})::Bool where {NM, N<:Real}

Check whether a given point is contained in a linear map of a convex set.

Input

• x – point/vector

• lm – linear map of a convex set

Output

true iff $x ∈ lm$.

Algorithm

Note that $x ∈ M⋅S$ iff $M^{-1}⋅x ∈ S$. This implementation does not explicitly invert the matrix, which is why it also works for non-square matrices.

Examples

julia> lm = LinearMap([2.0 0.0; 0.0 1.0], BallInf([1., 1.], 1.));

julia> ∈([5.0, 1.0], lm)
false
julia> ∈([3.0, 1.0], lm)
true

An example with non-square matrix:

julia> B = BallInf(zeros(4), 1.);

julia> M = [1. 0 0 0; 0 1 0 0]/2;

julia> ∈([0.5, 0.5], M*B)
true

an_element(lm::LinearMap)

Return some element of a linear map.

Input

• lmap – linear map

Output

An element in the linear map. It relies on the an_element function of the wrapped set.

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

Type that represents the action of an exponential map on a convex set.

Fields

• spmexp – sparse matrix exponential

• X – convex set

Examples

The ExponentialMap type is overloaded to the usual times * operator when the linear map is a lazy matrix exponential. For instance,

julia> A = sprandn(100, 100, 0.1);

julia> E = SparseMatrixExp(A);

julia> B = BallInf(zeros(100), 1.);

julia> M = E * B; # represents the image set: exp(A) * B

julia> M isa ExponentialMap
true

julia> dim(M)
100

dim(em::ExponentialMap)::Int

Return the dimension of an exponential map.

Input

• em – an ExponentialMap

Output

The ambient dimension of the exponential map.

σ(d::V, em::ExponentialMap) where {N<:Real, V<:AbstractVector{N}}

Return the support vector of the exponential map.

Input

• d – direction

• em – exponential map

Output

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

Notes

If $E = \exp(M)⋅S$, where $M$ is a matrix and $S$ is a convex set, it follows that $σ(d, E) = \exp(M)⋅σ(\exp(M)^T d, S)$ for any direction $d$.

We allow sparse direction vectors, but will convert them to dense vectors to be able to use expmv.

∈(x::AbstractVector{N}, em::ExponentialMap{<:LazySet{N}})::Bool where {N<:Real}

Check whether a given point is contained in an exponential map of a convex set.

Input

• x – point/vector

• em – exponential map of a convex set

Output

true iff $x ∈ em$.

Algorithm

This implementation exploits that $x ∈ \exp(M)⋅S$ iff $\exp(-M)⋅x ∈ S$. This follows from $\exp(-M)⋅\exp(M) = I$ for any $M$.

Examples

julia> em = ExponentialMap(SparseMatrixExp(SparseMatrixCSC([2.0 0.0; 0.0 1.0])),
                           BallInf([1., 1.], 1.));

julia> ∈([5.0, 1.0], em)
false
julia> ∈([1.0, 1.0], em)
true

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

Type that represents the application of a projection of a sparse matrix exponential to a convex set.

Fields

• spmexp – projection of a sparse matrix exponential

• X – convex set

dim(eprojmap::ExponentialProjectionMap)::Int

Return the dimension of a projection of an exponential map.

Input

• eprojmap – projection of an exponential map

Output

The ambient dimension of the projection of an exponential map.

σ(d::V, eprojmap::ExponentialProjectionMap) where {N<:Real, V<:AbstractVector{N}}

Return the support vector of a projection of an exponential map.

Input

• d – direction

• eprojmap – projection of an exponential map

Output

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

Notes

If $S = (L⋅M⋅R)⋅X$, where $L$ and $R$ are matrices, $M$ is a matrix exponential, and $X$ is a set, it follows that $σ(d, S) = L⋅M⋅R⋅σ(R^T⋅M^T⋅L^T⋅d, X)$ for any direction $d$.

We allow sparse direction vectors, but will convert them to dense vectors to be able to use expmv.

SparseMatrixExp{N}

Type that represents the matrix exponential, $\exp(M)$, of a sparse matrix.

Fields

• M – sparse matrix

Examples

Take for exammple a random sparse matrix:

julia> A = sprandn(100, 100, 0.1);

julia> E = SparseMatrixExp(A);

julia> size(E)
(100, 100)

Now, E is a lazy representation of $\exp(A)$. To compute with E, use get_row and get_column (or get_rows and get_columns; they return row and column vectors (or matrices). For example:

julia> get_row(E, 10); # compute E[10, :]

julia> get_column(E, 10); # compute E[:, 10]

julia> get_rows(E, [10]); # same as get_row(E, 10) but a 1x100 matrix is returned

julia> get_columns(E, [10]); # same as get_column(E, 10) but a 100x1 matrix is returned

Notes

This type is provided for use with very large and very sparse matrices. The evaluation of the exponential matrix action over vectors relies on the Expokit package.

    *(spmexp::SparseMatrixExp, X::LazySet)::ExponentialMap

Return the exponential map of a convex set from a sparse matrix exponential.

Input

• spmexp – sparse matrix exponential

• X – convex set

Output

The exponential map of the convex set.

    *(M::AbstractMatrix, X::LazySet)

Return the linear map of a convex set.

Input

• M – matrix/linear map

• X – convex set

Output

A lazy linear map, i.e. a LinearMap instance.

    *(a::N, X::LazySet) where {N}

Return a linear map of a convex set by a scalar value.

Input

• a – scalar

• X – convex set

Output

The linear map of the convex set.

ProjectionSparseMatrixExp{N<:Real}

Type that represents the projection of a sparse matrix exponential, i.e., $L⋅\exp(M)⋅R$ for a given sparse matrix $M$.

Fields

• L – left multiplication matrix

• E – sparse matrix exponential

• R – right multiplication matrix

    *(projspmexp::ProjectionSparseMatrixExp,
      X::LazySet)::ExponentialProjectionMap

Return the application of a projection of a sparse matrix exponential to a convex set.

Input

• projspmexp – projection of a sparse matrix exponential

• X – convex set

Output

The application of the projection of a sparse matrix exponential to the convex set.

    *(a::N, X::LazySet) where {N}

Return a linear map of a convex set by a scalar value.

Input

• a – scalar

• X – convex set

Output

The linear map of the convex set.

SymmetricIntervalHull{N<:Real, S<:LazySet{N}} <: AbstractHyperrectangle{N}

Type that represents the symmetric interval hull of a convex set.

Fields

• X – convex set

• cache – partial storage of already computed bounds, organized as mapping from dimension to tuples (bound, valid), where valid is a flag indicating if the bound entry has been computed

Notes

The symmetric interval hull can be computed with $2n$ support vector queries of unit vectors, where $n$ is the dimension of the wrapped set (i.e., two queries per dimension). When asking for the support vector for a direction $d$, one needs $2k$ such queries, where $k$ is the number of non-zero entries in $d$.

However, if one asks for many support vectors in a loop, the number of computations may exceed $2n$. To be most efficient in such cases, this type stores the intermediately computed bounds in the cache field.

dim(P::AbstractPointSymmetricPolytope)::Int

Return the ambient dimension of a point symmetric set.

Input

• P – set

Output

The ambient dimension of the set.

dim(sih::SymmetricIntervalHull)::Int

Return the dimension of a symmetric interval hull of a convex set.

Input

• sih – symmetric interval hull of a convex set

Output

The ambient dimension of the symmetric interval hull of a convex set.

σ(d::V, H::AbstractHyperrectangle{N}) where {N<:Real, V<:AbstractVector{N}}

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

Input

• d – direction

• H – hyperrectangular set

Output

The support vector in the given direction. If the direction has norm zero, the vertex with biggest values is returned.

σ(d::V, sih::SymmetricIntervalHull{N}) where {N<:Real, V<:AbstractVector{N}}

Return the support vector of a symmetric interval hull of a convex set in a given direction.

Input

• d – direction

• sih – symmetric interval hull of a convex set

Output

The support vector of the symmetric interval hull of a convex set in the given direction. If the direction has norm zero, the origin is returned.

Algorithm

For each non-zero entry in d we need to either look up the bound (if it has been computed before) or compute it, in which case we store it for future queries. One such computation just asks for the support vector of the underlying set for both the positive and negative unit vector in the respective dimension.

an_element(S::LazySet{N}) where {N<:Real}

Return some element of a convex set.

Input

• S – convex set

Output

An element of a convex set.

an_element(P::AbstractPointSymmetricPolytope{N})::Vector{N} where {N<:Real}

Return some element of a point symmetric polytope.

Input

• P – point symmetric polytope

Output

The center of the point symmetric polytope.