Set Interfaces

LazySet{N}

Abstract type for the set types in LazySets.

Notes

LazySet types should be parameterized with a type N, typically N<:Real, for using different numeric types.

Every concrete LazySet must define the following method:

• dim(S::LazySet) – the ambient dimension of S

While not strictly required, it is useful to define the following method:

• σ(d::AbstractVector, S::LazySet) – the support vector of S in a given direction d

The method

• ρ(d::AbstractVector, S::LazySet) – the support function of S in a given direction d

is optional because there is a fallback implementation relying on σ. However, for potentially unbounded sets (which includes most lazy set types) this fallback cannot be used and an explicit method must be implemented.

The subtypes of LazySet (including abstract interfaces):

julia> subtypes(LazySet, false)
18-element Vector{Any}:
 AbstractAffineMap
 AbstractPolynomialZonotope
 Bloating
 CachedMinkowskiSumArray
 CartesianProduct
 CartesianProductArray
 Complement
 ConvexSet
 Intersection
 IntersectionArray
 LazySets.AbstractStar
 MinkowskiSum
 MinkowskiSumArray
 Polygon
 QuadraticMap
 Rectification
 UnionSet
 UnionSetArray

If we only consider concrete subtypes, then:

julia> concrete_subtypes = subtypes(LazySet, true);

julia> length(concrete_subtypes)
54

julia> println.(concrete_subtypes);
AffineMap
Ball1
Ball2
BallInf
Ballp
Bloating
CachedMinkowskiSumArray
CartesianProduct
CartesianProductArray
Complement
ConvexHull
ConvexHullArray
DensePolynomialZonotope
Ellipsoid
EmptySet
ExponentialMap
ExponentialProjectionMap
HParallelotope
HPolygon
HPolygonOpt
HPolyhedron
HPolytope
HalfSpace
Hyperplane
Hyperrectangle
Intersection
IntersectionArray
Interval
InverseLinearMap
LazySets.AbstractStar
Line
Line2D
LineSegment
LinearMap
MinkowskiSum
MinkowskiSumArray
Polygon
QuadraticMap
Rectification
ResetMap
RotatedHyperrectangle
SimpleSparsePolynomialZonotope
Singleton
SparsePolynomialZonotope
Star
SymmetricIntervalHull
Translation
UnionSet
UnionSetArray
Universe
VPolygon
VPolytope
ZeroSet
Zonotope

plot_lazyset(X::LazySet{N}, [ε]::Real=N(PLOT_PRECISION); ...) where {N}

Plot a set.

Input

• X – set
• ε – (optional, default: PLOT_PRECISION) approximation error bound

Notes

This recipe just defines the default plotting options and then calls the function plot_recipe, which then implements the set-specific plotting.

The argument ε is ignored by some set types, e.g., for polyhedra (subtypes of AbstractPolyhedron).

Examples

julia> B = Ball2(ones(2), 0.1);

julia> plot(B, 1e-3)  # default accuracy value (explicitly given for clarity here)

julia> plot(B, 1e-2)  # faster but less accurate than the previous call

plot_list(list::AbstractVector{VN}, [ε]::Real=N(PLOT_PRECISION),
          [Nφ]::Int=PLOT_POLAR_DIRECTIONS; [same_recipe]=false; ...)
    where {N, VN<:LazySet{N}}

Plot a list of sets.

Input

• list – list of sets (1D or 2D)
• ε – (optional, default: PLOT_PRECISION) approximation error bound
• Nφ – (optional, default: PLOT_POLAR_DIRECTIONS) number of polar directions (used to plot lazy intersections)
• same_recipe – (optional, default: false) switch for faster plotting but without individual plot recipes (see notes below)

Notes

For each set in the list we apply an individual plot recipe.

The option same_recipe provides access to a faster plotting scheme where all sets in the list are first converted to polytopes and then plotted in one single run. This, however, is not suitable when plotting flat sets (line segments, singletons) because then the polytope plot recipe does not deliver good results. Hence by default we do not use this option. For plotting a large number of (non-flat) polytopes, we highly advise activating this option.

Examples

julia> B1 = BallInf(zeros(2), 0.4);

julia> B2 = BallInf(ones(2), 0.4);

julia> plot([B1, B2])

Some of the sets in the list may not be plotted precisely but rather overapproximated first. The second argument ε controls the accuracy of this overapproximation.

julia> Bs = [BallInf(zeros(2), 0.4), Ball2(ones(2), 0.4)];

julia> plot(Bs, 1e-3)  # default accuracy value (explicitly given for clarity)

julia> plot(Bs, 1e-2)  # faster but less accurate than the previous call

plot_vlist(X::S, ε::Real) where {S<:LazySet}

Return a list of vertices used for plotting a two-dimensional set.

Input

• X – two-dimensional set
• ε – precision parameter

Output

A list of vertices of a polygon P. For convex X, P usually satisfies that the Hausdorff distance to X is less than ε.

plot3d(S::LazySet; [backend]=default_polyhedra_backend(S), [alpha]=1.0,
       [color]=:blue, [colormap]=:viridis, [colorrange]=nothing,
       [interpolate]=false, [linewidth]=1, [overdraw]=false, [shading]=true,
       [transparency]=true, [visible]=true)

Plot a three-dimensional set using Makie.

Input

• S – set
• backend – (optional, default: default_polyhedra_backend(S)) backend for polyhedral computations
• alpha – (optional, default: 1.0) float in [0,1]; the alpha or transparency value
• color – (optional, default: :blue) Symbol or Colorant; the color of the main plot element (markers, lines, etc.), which can be a color symbol/string like :red
• colormap – (optional, default: :viridis) the color map of the main plot; use available_gradients() to see which gradients are available, which can also be used as [:red, :black]
• colorrange – (optional, default: nothing, which falls back to Makie.Automatic()) a tuple (min, max) where min and max specify the data range to be used for indexing the colormap
• interpolate – (optional, default: false) a boolean for heatmap and images; toggles color interpolation between nearby pixels
• linewidth – (optional, default: 1) a number that specifies the width of the line in line and linesegments plots
• overdraw – (optional, default: false)
• shading – (optional, default: true) a boolean that toggles shading (for meshes)
• transparency – (optional, default: true) if true, the set is transparent, otherwise it is displayed as a solid object
• visible – (optional, default: true) a boolean that toggles visibility of the plot

For a complete list of attributes and usage see Makie's documentation.

Notes

This plot recipe works by computing the list of constraints of S and converting to a polytope in H-representation. Then, this polytope is transformed with Polyhedra.Mesh and plotted using the mesh function.

If the function constraints_list is not applicable to your set S, try overapproximation first; e.g. via

julia> Sapprox = overapproximate(S, SphericalDirections(10))

julia> using Polyhedra, GLMakie

julia> plot3d(Sapprox)

The number 10 above corresponds to the number of directions considered; for better resolution use higher values (but it will take longer).

For efficiency consider using the CDDLib backend, as in

julia> using CDDLib

julia> plot3d(Sapprox, backend=CDDLib.Library())

Examples

The functionality requires both Polyhedra and a Makie backend. After loading LazySets, do using Polyhedra, GLMakie (or another Makie backend).

julia> using LazySets, Polyhedra, GLMakie

julia> plot3d(10 * rand(Hyperrectangle, dim=3))

julia> plot3d!(10 * rand(Hyperrectangle, dim=3), color=:red)

plot3d!(S::LazySet; backend=default_polyhedra_backend(S), [alpha]=1.0,
       [color]=:blue, [colormap]=:viridis, [colorrange]=nothing,
       [interpolate]=false, [linewidth]=1, [overdraw]=false, [shading]=true,
       [transparency]=true, [visible]=true)

Plot a three-dimensional set using Makie.

Input

See plot3d for the description of the inputs. For a complete list of attributes and usage see Makie's documentation.

Notes

See the documentation of plot3d for examples.

isconvextype(X::Type{<:LazySet})

Check whether the given LazySet type is convex.

Input

• X – subtype of LazySet

Output

true if the given set type is guaranteed to be convex by using only type information, and false otherwise.

Notes

Since this operation only acts on types (not on values), it can return false negatives, i.e., there may be instances where the set is convex, even though the answer of this function is false. The examples below illustrate this point.

Examples

A ball in the infinity norm is always convex, hence we get:

julia> isconvextype(BallInf)
true

For instance, the union (UnionSet) of two sets may in general be either convex or not. Since convexity cannot be decided by just using type information, isconvextype returns false.

julia> isconvextype(UnionSet)
false

However, the type parameters of set operations allow to decide convexity in some cases by falling back to the convexity information of the type of its arguments. Consider for instance the lazy intersection. The intersection of two convex sets is always convex, hence we get:

julia> isconvextype(Intersection{Float64, BallInf{Float64}, BallInf{Float64}})
true

low(X::LazySet)

Return a vector with the lowest coordinates of the set in each canonical direction.

Input

• X – set

Output

A vector with the lower coordinate of the set in each dimension.

Notes

See also low(X::LazySet, i::Int).

The result is the lowermost corner of the box approximation, so it is not necessarily contained in X.

high(X::LazySet)

Return a vector with the highest coordinate of the set in each canonical direction.

Input

• X – set

Output

A vector with the highest coordinate of the set in each dimension.

Notes

See also high(X::LazySet, i::Int).

The result is the uppermost corner of the box approximation, so it is not necessarily contained in X.

extrema(X::LazySet, i::Int)

Return the lower and higher coordinate of a set in a given dimension.

Input

• X – set
• i – dimension of interest

Output

The lower and higher coordinate of the set in the given dimension.

Notes

The result is equivalent to (low(X, i), high(X, i)), but sometimes it can be computed more efficiently.

Algorithm

The bounds are computed with low and high.

extrema(X::LazySet)

Return two vectors with the lowest and highest coordinate of X in each canonical direction.

Input

• X – set

Output

Two vectors with the lowest and highest coordinates of X in each dimension.

Notes

See also extrema(X::LazySet, i::Int).

The result is equivalent to (low(X), high(X)), but sometimes it can be computed more efficiently.

The resulting points are the lowermost and uppermost corners of the box approximation, so they are not necessarily contained in X.

Algorithm

The bounds are computed with low and high by default.

convex_hull(X::LazySet; kwargs...)

Compute the convex hull of a polytopic set.

Input

• X – polytopic set

Output

The set X itself if its type indicates that it is convex, or a new set with the list of the vertices describing the convex hull.

Algorithm

For non-convex sets, this method relies on the vertices_list method.

triangulate(X::LazySet)

Triangulate a three-dimensional polyhedral set.

Input

• X – three-dimensional polyhedral set

Output

A tuple (p, c) where p is a matrix, with each column containing a point, and c is a list of 3-tuples containing the indices of the points in each triangle.

basetype(T::Type{<:LazySet})

Return the base type of the given set type (i.e., without type parameters).

Input

• T – set type

Output

The base type of T.

basetype(S::LazySet)

Return the base type of the given set (i.e., without type parameters).

Input

• S – set

Output

The base type of S.

Examples

julia> Z = rand(Zonotope);

julia> basetype(Z)
Zonotope

julia> basetype(Z + Z)
MinkowskiSum

julia> basetype(LinearMap(rand(2, 2), Z + Z))
LinearMap

isboundedtype(T::Type{<:LazySet})

Check whether a set type only represents bounded sets.

Input

• T – set type

Output

true if the set type only represents bounded sets. Note that some sets may still represent an unbounded set even though their type actually does not (example: HPolytope, because the construction with non-bounding linear constraints is allowed).

Notes

By default this function returns false. All set types that can determine boundedness should override this behavior.

isbounded(S::LazySet)

Check whether a set is bounded.

Input

• S – set
• algorithm – (optional, default: "support_function") algorithm choice, possible options are "support_function" and "stiemke"

Output

true iff the set is bounded.

Algorithm

See the documentation of _isbounded_unit_dimensions or _isbounded_stiemke for details.

_isbounded_unit_dimensions(S::LazySet)

Check whether a set is bounded in each unit dimension.

Input

• S – set

Output

true iff the set is bounded in each unit dimension.

Algorithm

This function asks for upper and lower bounds in each ambient dimension.

is_polyhedral(S::LazySet)

Trait for polyhedral sets.

Input

• S – set

Output

true only if the set behaves like an AbstractPolyhedron.

Notes

The answer is conservative, i.e., may sometimes be false even if the set is polyhedral.

norm(S::LazySet, [p]::Real=Inf)

Return the norm of a set. It is the norm of the enclosing ball (of the given $p$-norm) of minimal volume that is centered in the origin.

Input

• S – set
• p – (optional, default: Inf) norm

Output

A real number representing the norm.

radius(S::LazySet, [p]::Real=Inf)

Return the radius of a set. It is the radius of the enclosing ball (of the given $p$-norm) of minimal volume with the same center.

Input

• S – set
• p – (optional, default: Inf) norm

Output

A real number representing the radius.

diameter(S::LazySet, [p]::Real=Inf)

Return the diameter of a set. It is the maximum distance between any two elements of the set, or, equivalently, the diameter of the enclosing ball (of the given $p$-norm) of minimal volume with the same center.

Input

• S – set
• p – (optional, default: Inf) norm

Output

A real number representing the diameter.

isempty(P::LazySet{N}, witness::Bool=false;
        [use_polyhedra_interface]::Bool=false, [solver]=nothing,
        [backend]=nothing) where {N}

Check whether a polyhedral set is empty.

Input

• P – polyhedral set
• witness – (optional, default: false) compute a witness if activated
• use_polyhedra_interface – (optional, default: false) if true, we use the Polyhedra interface for the emptiness test
• solver – (optional, default: nothing) LP-solver backend; uses default_lp_solver(N) if not provided
• backend – (optional, default: nothing) backend for polyhedral computations in Polyhedra; uses default_polyhedra_backend(P) if not provided

Output

• If witness option is deactivated: true iff $P = ∅$
• If witness option is activated:
  • (true, []) iff $P = ∅$
  • (false, v) iff $P ≠ ∅$ and $v ∈ P$

Notes

The default value of the backend is set internally and depends on whether the use_polyhedra_interface option is set or not. If the option is set, we use default_polyhedra_backend(P).

Witness production is not supported if use_polyhedra_interface is true.

Algorithm

The algorithm sets up a feasibility LP for the constraints of P. If use_polyhedra_interface is true, we call Polyhedra.isempty. Otherwise, we set up the LP internally.

affine_map(M::AbstractMatrix, X::LazySet, v::AbstractVector; kwargs...)

Compute the concrete affine map $M·X + v$.

Input

• M – linear map
• X – set
• v – translation vector

Output

A set representing the affine map $M·X + v$.

Algorithm

The implementation applies the functions linear_map and translate.

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

Compute the concrete exponential map of M and X, i.e., exp(M) * X.

Input

• M – matrix
• X – set

Output

A set representing the exponential map of M and X.

Algorithm

The implementation applies the functions exp and linear_map.

an_element(S::LazySet)

Return some element of a set.

Input

• S – set

Output

An element of a set.

Algorithm

An element of the set is obtained by evaluating its support vector along direction $[1, 0, …, 0]$. This may fail for unbounded sets.

tosimplehrep(S::LazySet)

Return the simple constraint representation $Ax ≤ b$ of a polyhedral set from its list of linear constraints.

Input

• S – polyhedral set

Output

The tuple (A, b) where A is the matrix of normal directions and b is the vector of offsets.

Algorithm

This fallback implementation relies on constraints_list(S).

reflect(P::LazySet)

Concrete reflection of a set P, resulting in the reflected set -P.

Algorithm

This function requires that the list of constraints of the set P is available, i.e., that it can be written as $P = \{z ∈ ℝⁿ: ⋂ sᵢᵀz ≤ rᵢ, i = 1, ..., N\}.$

This function can be used to implement the alternative definition of the Minkowski Difference

\[A ⊖ B = \{a − b | a ∈ A, b ∈ B\} = A ⊕ (-B)\]

by calling minkowski_sum(A, reflect(B)).

is_interior_point(d::AbstractVector{N}, X::LazySet{N};
                  p=N(Inf), ε=_rtol(N)) where {N}

Check whether the point d is contained in the interior of the set X.

Input

• d – point
• X – set
• p – (optional; default: N(Inf)) norm of the ball used to apply the error tolerance
• ε – (optional; default: _rtol(N)) error tolerance of check

Output

Boolean which indicates if the point d is contained in X.

Algorithm

The implementation checks if a Ballp of norm p with center d and radius ε is contained in the set X. This is a numerical check for d ∈ interior(X) with error tolerance ε.

isoperationtype(X::Type{<:LazySet})

Check whether the given set type is an operation or not.

Input

• X – set type

Output

true if the given set type is a set operation and false otherwise.

Notes

This fallback implementation returns an error that isoperationtype is not implemented. Subtypes of LazySet should dispatch on this function as required.

See also isoperation(X<:LazySet).

Examples

julia> isoperationtype(BallInf)
false

julia> isoperationtype(LinearMap)
true

isoperation(X::LazySet)

Check whether a set is an instance of a set operation or not.

Input

• X – set

Output

true if X is an instance of a set-based operation and false otherwise.

Notes

This fallback implementation checks whether the set type of the input is an operation type using isoperationtype(::Type{<:LazySet}).

Examples

julia> B = BallInf([0.0, 0.0], 1.0);

julia> isoperation(B)
false

julia> isoperation(B ⊕ B)
true

isequivalent(X::LazySet, Y::LazySet)

Check whether two sets are equal in the mathematical sense, i.e., equivalent.

Input

• X – set
• Y – set

Output

true iff X is equivalent to Y (up to some precision).

Algorithm

First we check X ≈ Y, which returns true if and only if X and Y have the same type and approximately the same values (checked with LazySets._isapprox). If that fails, we check the double inclusion X ⊆ Y && Y ⊆ X.

Examples

julia> X = BallInf([0.1, 0.2], 0.3);

julia> Y = convert(HPolytope, X);

julia> X == Y
false

julia> isequivalent(X, Y)
true

surface(X::LazySet)

Compute the surface area of a set.

Input

• X – set

Output

A real number representing the surface area of X.

area(X::LazySet{N}) where {N}

Compute the area of a two-dimensional polytopic set using the Shoelace formula.

Input

• X – two-dimensional polytopic set

Output

A number representing the area of X.

Notes

This algorithm is applicable to any polytopic set X whose list of vertices can be computed via vertices_list.

Algorithm

Let m be the number of vertices of X. We consider the following instances:

• m = 0, 1, 2: the output is zero.
• m = 3: the triangle case is solved using the Shoelace formula with 3 points.
• m = 4: the quadrilateral case is solved by the factored version of the Shoelace formula with 4 points.

Otherwise, the general Shoelace formula is used; for details see the Wikipedia page.

concretize(X::LazySet)

Construct a concrete representation of a (possibly lazy) set.

Input

• X – set

Output

A concrete representation of X (as far as possible).

Notes

Since not every lazy set has a concrete set representation in this library, the result may be partially lazy.

complement(X::LazySet)

Return the complement of a polyhedral set.

Input

• X – polyhedral set

Output

A UnionSetArray of half-spaces, i.e., the output is the union of the linear constraints which are obtained by complementing each constraint of X.

Algorithm

The principle used in this implementation is that for any pair of sets $(X, Y)$ we have that $(X ∩ Y)^C = X^C ∪ Y^C$. In particular, we can apply this rule for each constraint that defines a polyhedral set. Hence the concrete complement can be represented as the set union of the complement of each constraint.

polyhedron(P::LazySet; [backend]=default_polyhedra_backend(P))

Compute a set representation from Polyhedra.jl.

Input

• P – polyhedral set
• backend – (optional, default: call default_polyhedra_backend(P)) the polyhedral computations backend

Output

A set representation in the Polyhedra library.

Notes

For further information on the supported backends see Polyhedra's documentation.

Algorithm

This default implementation uses tosimplehrep, which computes the constraint representation of P. Set types preferring the vertex representation should implement their own method.

project(S::LazySet, block::AbstractVector{Int}, [::Nothing=nothing],
        [n]::Int=dim(S); [kwargs...])

Project a set to a given block by using a concrete linear map.

Input

• S – set
• block – block structure - a vector with the dimensions of interest
• nothing – (default: nothing)
• n – (optional, default: dim(S)) ambient dimension of the set S

Output

A set representing the projection of the set S to block block.

Algorithm

We apply the function linear_map.

project(S::LazySet, block::AbstractVector{Int}, set_type::Type{TS},
        [n]::Int=dim(S); [kwargs...]) where {TS<:LazySet}

Project a set to a given block and set type, possibly involving an overapproximation.

Input

• S – set
• block – block structure - a vector with the dimensions of interest
• set_type – target set type
• n – (optional, default: dim(S)) ambient dimension of the set S

Output

A set of type set_type representing an overapproximation of the projection of S.

Algorithm

1. Project the set S with M⋅S, where M is the identity matrix in the block

coordinates and zero otherwise.

1. Overapproximate the projected set using overapproximate and set_type.

project(S::LazySet, block::AbstractVector{Int},
        set_type_and_precision::Pair{T, N}, [n]::Int=dim(S);
        [kwargs...]) where {T<:UnionAll, N<:Real}

Project a set to a given block and set type with a certified error bound.

Input

• S – set
• block – block structure - a vector with the dimensions of interest
• set_type_and_precision – pair (T, ε) of a target set type T and an error bound ε for approximation
• n – (optional, default: dim(S)) ambient dimension of the set S

Output

A set representing the epsilon-close approximation of the projection of S.

Notes

Currently we only support HPolygon as set type, which implies that the set must be two-dimensional.

Algorithm

1. Project the set S with M⋅S, where M is the identity matrix in the block

coordinates and zero otherwise.

1. Overapproximate the projected set with the given error bound ε.

project(S::LazySet, block::AbstractVector{Int}, ε::Real, [n]::Int=dim(S);
        [kwargs...])

Project a set to a given block and set type with a certified error bound.

Input

• S – set
• block – block structure - a vector with the dimensions of interest
• ε – error bound for approximation
• n – (optional, default: dim(S)) ambient dimension of the set S

Output

A set representing the epsilon-close approximation of the projection of S.

Algorithm

1. Project the set S with M⋅S, where M is the identity matrix in the block

coordinates and zero otherwise.

1. Overapproximate the projected set with the given error bound ε.

The target set type is chosen automatically.

rectify(X::LazySet, [concrete_intersection]::Bool=false)

Concrete rectification of a set.

Input

• X – set
• concrete_intersection – (optional, default: false) flag to compute concrete intersections for intermediate results

Output

A set corresponding to the rectification of X, which is in general a union of linear maps of intersections.

Algorithm

For each dimension in which X is both positive and negative, we split X into these two parts. Additionally we project the negative part to zero.

permute(X::LazySet, p::AbstractVector{Int})

Permute the dimensions of a set according to a given permutation vector.

Input

• X – set
• p – permutation vector

Output

A new set corresponding to X where the dimensions have been permuted according to p.

rationalize(::Type{T}, X::LazySet{<:AbstractFloat}, tol::Real)
    where {T<:Integer}

Approximate a set of floating-point numbers as a set whose entries are rationals of the given integer type.

Input

• T – (optional, default: Int) integer type to represent the rationals
• X – set which has floating-point components
• tol – (optional, default: eps(N)) tolerance of the result; each rationalized component will differ by no more than tol with respect to the floating-point value

Output

A set of the same base type of X where each numerical component is of type Rational{T}.

singleton_list(P::LazySet)

Return the vertices of a polytopic set as a list of singletons.

Input

• P – polytopic set

Output

A list of the vertices of P as Singletons.

Notes

This function relies on vertices_list, which raises an error if the set is not polytopic (e.g., unbounded).

constraints(X::LazySet)

Construct an iterator over the constraints of a polyhedral set.

Input

• X – polyhedral set

Output

An iterator over the constraints of X.

vertices(X::LazySet)

Construct an iterator over the vertices of a polytopic set.

Input

• X – polytopic set

Output

An iterator over the vertices of X.

delaunay(X::LazySet)

Compute the Delaunay triangulation of the given polytopic set.

Input

• X – polytopic set

Output

A union of polytopes in vertex representation.

Notes

This function requires that you have properly installed the package MiniQhull.jl, including the library Qhull.

The method works in arbitrary dimension and the requirement is that the list of vertices of X can be obtained.

chebyshev_center_radius(P::LazySet{N};
                        [backend]=default_polyhedra_backend(P),
                        [solver]=default_lp_solver_polyhedra(N; presolve=true)
                       ) where {N}

Compute a Chebyshev center and the corresponding radius of a polytopic set.

Input

• P – polytopic set
• backend – (optional; default: default_polyhedra_backend(P)) the backend for polyhedral computations
• solver – (optional; default: default_lp_solver_polyhedra(N; presolve=true)) the LP solver passed to Polyhedra

Output

The pair (c, r) where c is a Chebyshev center of P and r is the radius of the largest ball with center c enclosed by P.

Notes

The Chebyshev center is the center of a largest Euclidean ball enclosed by P. In general, the center of such a ball is not unique, but the radius is.

Algorithm

We call Polyhedra.chebyshevcenter.

plot_recipe(X::LazySet, [ε])

Convert a compact convex set to a pair (x, y) of points for plotting.

Input

• X – compact convex set
• ε – approximation-error bound

Output

A pair (x, y) of points that can be plotted.

Notes

We do not support three-dimensional or higher-dimensional sets at the moment.

Algorithm

One-dimensional sets are converted to an Interval.

For two-dimensional sets, we first compute a polygonal overapproximation. The second argument, ε, corresponds to the error in Hausdorff distance between the overapproximating set and X. On the other hand, if you only want to produce a fast box-overapproximation of X, pass ε=Inf.

Finally, we use the plot recipe for the constructed set (interval or polygon).

low(X::ConvexSet{N}, i::Int) where {N}

Return the lower coordinate of a convex set in a given dimension.

Input

• X – convex set
• i – dimension of interest

Output

The lower coordinate of the set in the given dimension.

high(X::ConvexSet{N}, i::Int) where {N}

Return the higher coordinate of a convex set in a given dimension.

Input

• X – convex set
• i – dimension of interest

Output

The higher coordinate of the set in the given dimension.

an_element(P::AbstractPolyhedron{N};
           [solver]=default_lp_solver(N)) where {N}

Return some element of a polyhedron.

Input

• P – polyhedron
• solver – (optional, default: default_lp_solver(N)) LP solver

Output

An element of the polyhedron, or an error if the polyhedron is empty.

Algorithm

An element is obtained by solving a feasibility linear program.

an_element(U::Universe{N}) where {N}

Return some element of a universe.

Input

• U – universe

Output

The origin.

σ

Function to compute the support vector σ.

support_vector

Alias for the support vector σ.

ρ(d::AbstractVector, S::LazySet)

Evaluate the support function of a set in a given direction.

Input

• d – direction
• S – set

Output

The evaluation of the support function of the set S for the direction d.

support_function

Alias for the support function ρ.

==(X::LazySet, Y::LazySet)

Check whether two sets use exactly the same set representation.

Input

• X – set
• Y – set

Output

• true iff X is equal to Y.

Notes

The check is purely syntactic and the sets need to have the same base type. For instance, X::VPolytope == Y::HPolytope returns false even if X and Y represent the same polytope. However X::HPolytope{Int64} == Y::HPolytope{Float64} is a valid comparison.

Algorithm

We recursively compare the fields of X and Y until a mismatch is found.

Examples

julia> HalfSpace([1], 1) == HalfSpace([1], 1)
true

julia> HalfSpace([1], 1) == HalfSpace([1.0], 1.0)
true

julia> Ball1([0.0], 1.0) == Ball2([0.0], 1.0)
false

≈(X::LazySet, Y::LazySet)

Check whether two sets of the same type are approximately equal.

Input

• X – set
• Y – set of the same base type as X

Output

• true iff X is equal to Y.

Notes

The check is purely syntactic and the sets need to have the same base type. For instance, X::VPolytope ≈ Y::HPolytope returns false even if X and Y represent the same polytope. However X::HPolytope{Int64} ≈ Y::HPolytope{Float64} is a valid comparison.

Algorithm

We recursively compare the fields of X and Y until a mismatch is found.

Examples

julia> HalfSpace([1], 1) ≈ HalfSpace([1], 1)
true

julia> HalfSpace([1], 1) ≈ HalfSpace([1.00000001], 0.99999999)
true

julia> Ball1([0.0], 1.0) ≈ Ball2([0.0], 1.0)
false

copy(S::LazySet)

Return a copy of a set by copying its values recursively.

Input

• S – set

Output

A copy of S.

Notes

This function computes a copy of each field in S. See the documentation of ?copy for further details.

eltype(::Type{<:LazySet{N}}) where {N}

Return the numeric type (N) of the given set type.

Input

• T – set type

Output

The numeric type of T.

eltype(::LazySet{N}) where {N}

Return the numeric type (N) of the given set.

Input

• X – set

Output

The numeric type of X.

CompactSet

An alias for compact set types.

Notes

Most lazy operations are not captured by this alias because whether their result is compact or not depends on the argument(s).

NonCompactSet

An alias for non-compact set types.

Notes

Most lazy operations are not captured by this alias because whether their result is non-compact or not depends on the argument(s).

ConvexSet{N} <: LazySet{N}

Abstract type for convex sets, i.e., sets characterized by a (possibly infinite) intersection of halfspaces, or equivalently, sets $S$ such that for any two elements $x, y ∈ S$ and $0 ≤ λ ≤ 1$ it holds that $λ·x + (1-λ)·y ∈ S$.

AbstractCentrallySymmetric{N} <: ConvexSet{N}

Abstract type for centrally symmetric compact convex sets.

Notes

Every concrete AbstractCentrallySymmetric must define the following functions:

• center(::AbstractCentrallySymmetric) – return the center point
• center(::AbstractCentrallySymmetric, i::Int) – return the center point at index i

The subtypes of AbstractCentrallySymmetric:

julia> subtypes(AbstractCentrallySymmetric)
3-element Vector{Any}:
 Ball2
 Ballp
 Ellipsoid

dim(S::AbstractCentrallySymmetric)

Return the ambient dimension of a centrally symmetric set.

Input

• S – centrally symmetric set

Output

The ambient dimension of the set.

isbounded(S::AbstractCentrallySymmetric)

Check whether a centrally symmetric set is bounded.

Input

• S – centrally symmetric set

Output

true (since a set with a unique center must be bounded).

isuniversal(S::AbstractCentrallySymmetric{N},
            [witness]::Bool=false) where {N}

Check whether a centrally symmetric set is universal.

Input

• S – centrally symmetric set
• witness – (optional, default: false) compute a witness if activated

Output

• If witness option is deactivated: false
• If witness option is activated: (false, v) where $v ∉ S$

Algorithm

Centrally symmetric sets are bounded. A witness is obtained by computing the support vector in direction d = [1, 0, …, 0] and adding d on top.

an_element(S::AbstractCentrallySymmetric)

Return some element of a centrally symmetric set.

Input

• S – centrally symmetric set

Output

The center of the centrally symmetric set.

isempty(S::AbstractCentrallySymmetric)

Check whether a centrally symmetric set is empty.

Input

• S – centrally symmetric set

Output

false.

center(H::AbstractCentrallySymmetric, i::Int)

Return the center of a centrally symmetric set along a given dimension.

Input

• S – centrally symmetric set
• i – dimension of interest

Output

The center along the given dimension.

extrema(S::AbstractCentrallySymmetric, i::Int)

Return the lower and higher coordinate of a centrally symmetric set in a given dimension.

Input

• S – centrally symmetric set
• i – dimension of interest

Output

The lower and higher coordinate of the centrally symmetric set in the given dimension.

Notes

The result is equivalent to (low(S, i), high(S, i)).

Algorithm

We compute high(S, i) and then compute the lowest coordinates with the help of center(S, i) (which is assumed to be cheaper to obtain).

extrema(S::AbstractCentrallySymmetric)

Return two vectors with the lowest and highest coordinate of a centrally symmetric set.

Input

• S – centrally symmetric set

Output

Two vectors with the lowest and highest coordinates of S.

Notes

The result is equivalent to (low(S), high(S)).

Algorithm

We compute high(S) and then compute the lowest coordinates with the help of center(S) (which is assumed to be cheaper to obtain).

AbstractPolyhedron{N} <: ConvexSet{N}

Abstract type for compact convex polyhedral sets.

Notes

Every concrete AbstractPolyhedron must define the following functions:

• constraints_list(::AbstractPolyhedron{N}) – return a list of all facet constraints

Polyhedra are defined as the intersection of a finite number of closed half-spaces. As such, polyhedra are closed and convex but not necessarily bounded. Bounded polyhedra are called polytopes (see AbstractPolytope).

The subtypes of AbstractPolyhedron (including abstract interfaces):

julia> subtypes(AbstractPolyhedron)
8-element Vector{Any}:
 AbstractPolytope
 HPolyhedron
 HalfSpace
 Hyperplane
 Line
 Line2D
 Star
 Universe

∈(x::AbstractVector, P::AbstractPolyhedron)

Check whether a given point is contained in a polyhedron.

Input

• x – point/vector
• P – polyhedron

Output

true iff $x ∈ P$.

Algorithm

This implementation checks if the point lies inside each defining half-space.

isuniversal(P::AbstractPolyhedron{N}, [witness]::Bool=false) where {N}

Check whether a polyhedron is universal.

Input

• P – polyhedron
• witness – (optional, default: false) compute a witness if activated

Output

• If witness option is deactivated: true iff $P$ is universal
• If witness option is activated:
  • (true, []) iff $P$ is universal
  • (false, v) iff $P$ is not universal and $v ∉ P$

Algorithm

P is universal iff it has no constraints.

A witness is produced using isuniversal(H) where H is the first linear constraint of P.

constrained_dimensions(P::AbstractPolyhedron)

Return the indices in which a polyhedron is constrained.

Input

• P – polyhedron

Output

A vector of ascending indices i such that the polyhedron is constrained in dimension i.

Examples

A 2D polyhedron with constraint $x1 ≥ 0$ is constrained in dimension 1 only.

linear_map(M::AbstractMatrix{NM},
           P::AbstractPolyhedron{NP};
           [algorithm]::Union{String, Nothing}=nothing,
           [check_invertibility]::Bool=true,
           [cond_tol]::Number=DEFAULT_COND_TOL,
           [inverse]::Union{AbstractMatrix{N}, Nothing}=nothing,
           [backend]=nothing,
           [elimination_method]=nothing) where {NM, NP}

Concrete linear map of a polyhedral set.

Input

• M – matrix

• P – polyhedral set

• algorithm – (optional; default: nothing) algorithm to be used; for the description see the Algorithm section below; possible choices are:

  • "inverse", alias: "inv"
  • "inverse_right", alias: "inv_right"
  • "elimination", alias: "elim"
  • "lift"
  • "vrep"
  • "vrep_chull"

• check_invertibility – (optional, default: true) if true check whether the given matrix M is invertible; set to false only if you know that M is invertible

• cond_tol – (optional; default: DEFAULT_COND_TOL) tolerance of matrix condition (used to check whether the matrix is invertible)

• inverse – (optional; default: nothing) matrix inverse M⁻¹; use this option if you have already computed the inverse matrix of M

• backend – (optional: default: nothing) polyhedra backend

• elimination_method – (optional: default: nothing) elimination method for the "elimination" algorithm

Output

The type of the result is "as close as possible" to the the type of P. Let (m, n) be the size of M, where m ≠ n is allowed for rectangular maps.

To fix the type of the output to something different than the default value, consider post-processing the result of this function with a call to a suitable convert method.

In particular, the output depends on the type of P, on m, and the algorithm that was used:

• If the vertex-based approach was used:

  • If P is a VPolygon and m = 2 then the output is a VPolygon.
  • If P is a VPolytope then the output is a VPolytope.
  • Otherwise the output is an Interval if m = 1, a VPolygon if m = 2, and a VPolytope in other cases.

• If the invertibility criterion was used:

  • The types of HalfSpace, Hyperplane, Line2D, and subtypes of AbstractHPolygon are preserved.
  • If P is an AbstractPolytope, then the output is an Interval if m = 1, an HPolygon if m = 2, and an HPolytope in other cases.
  • Otherwise the output is an HPolyhedron.

Notes

Since the different linear-map algorithms work at the level of constraints, this method uses dispatch on two stages: once the algorithm has been defined, first the helper methods _linear_map_hrep_helper (resp. _linear_map_vrep) are invoked, which dispatch on the set type. Then, each helper method calls the concrete implementation of _linear_map_hrep, which dispatches on the algorithm, and returns a list of constraints.

To simplify working with different algorithms and options, the types <: AbstractLinearMapAlgorithm are used. These types are singleton type or types that carry only the key data for the given algorithm, such as the matrix inverse or the polyhedra backend.

New subtypes of the AbstractPolyhedron interface may define their own helper methods _linear_map_vrep (respectively _linear_map_hrep_helper) for special handling of the constraints returned by the implementations of _linear_map_hrep; otherwise the fallback implementation for AbstractPolyhedron is used, which instantiates an HPolyhedron.

Algorithm

This method mainly implements several approaches for the linear map: inverse, right inverse, transformation to vertex representation, variable elimination, and variable lifting. Depending on the properties of M and P, one algorithm may be preferable over the other. Details on the algorithms are given in the following subsections.

Otherwise, if the algorithm argument is not specified, a default option is chosen based on heuristics on the types and values of M and P:

• If the "inverse" algorithm applies, it is used.
• Otherwise, if the "inverse_right" algorithm applies, it is used.
• Otherwise, if the "lift" algorithm applies, it is used.
• Otherwise, the "elimination" algorithm is used.

Note that the algorithms "inverse" and "inverse_right" do not require the external library Polyhedra. However, the fallback method "elimination" requires Polyhedra as well as the library CDDLib.

The optional keyword arguments inverse and check_invertibility modify the default behavior:

• If an inverse matrix is passed in inverse, the given algorithm is applied, and if none is given, either "inverse" or "inverse_right" is applied (in that order of preference).
• If check_invertibility is set to false, the given algorithm is applied, and if none is given, either "inverse" or "inverse_right" is applied (in that order of preference).

Inverse

This algorithm is invoked with the keyword argument algorithm="inverse" (or algorithm="inv"). The algorithm requires that M is invertible, square, and dense. If you know a priori that M is invertible, set the flag check_invertibility=false, such that no extra checks are done. Otherwise, we check the sufficient condition that the condition number of M is not too high. The threshold for the condition number can be modified from its default value, DEFAULT_COND_TOL, by passing a custom cond_tol.

The algorithm is described next. Assuming that the matrix $M$ is invertible (which we check via a sufficient condition,), $y = M x$ implies $x = \text{inv}(M) y$ and we can transform the polyhedron $A x ≤ b$ to the polyhedron $A \text{inv}(M) y ≤ b$.

If the dense condition on M is not fullfilled, there are two suggested workarounds: either transform to a dense matrix, i.e., calling linear_map with Matrix(M), or use the "inverse_right" algorithm, which does not compute the inverse matrix explicitly, but uses a polyalgorithm; see the documentation of ? for details.

Inverse-right

This algorithm is invoked with the keyword argument algorithm="inverse_right" (or algorithm="inv_right"). This algorithm applies to square and invertible matrices M. The idea is essentially the same as for the "inverse" algorithm; the difference is that in "inverse" the full matrix inverse is computed, and in "inverse_right" only the left division on the normal vectors is used. In particular, "inverse_right" is good as a workaround when M is sparse (since the inv function is not available for sparse matrices).

Elimination

This algorithm is invoked with the keyword argument algorithm = "elimination" (or algorithm = "elim"). The algorithm applies to any matrix M (invertible or not), and any polyhedron P (bounded or not).

The idea is described next. If P : Ax <= b and y = Mx denote the polyhedron and the linear map, respectively, we consider the vector z = [y, x], write the given equalities and the inequalities, and then eliminate the last x variables (there are length(x) in total) using a call to Polyhedra.eliminate to a backend library that can do variable elimination (typically CDDLib with the BlockElimination() algorithm). In this way we have eliminated the "old" variables x and kept the "new" or transformed variables "y".

The default elimination method is block elimination. For possible options we refer to the documentation of Polyhedra, projection/elimination.

Lift

This algorithm is invoked with the keyword argument algorithm="lift". The algorithm applies if M is rectangular of size m × n with m > n and full rank (i.e., of rank n).

The idea is to embed the polyhedron into the m-dimensional space by appending zeros, i.e. extending all constraints of P to m dimensions, and constraining the last m - n dimensions to 0. The resulting matrix is extended to an invertible m × m matrix, and the algorithm using the inverse of the linear map is applied. For technical details of extending M to a higher-dimensional invertible matrix, see ReachabilityBase.Arrays.extend.

Vertex representation

This algorithm is invoked with the keyword argument algorithm="vrep" (or algorithm="vrep_chull"). If the polyhedron is bounded, the idea is to convert it to its vertex representation and apply the linear map to each vertex.

The returned set is a polytope in vertex representation. Note that conversion of the result back to half-space representation is not computed by default, since this may be costly. If you use this algorithm and still want to convert back to half-space representation, apply tohrep to the result of this method.

an_element(P::AbstractPolyhedron{N};
           [solver]=default_lp_solver(N)) where {N}

Return some element of a polyhedron.

Input

• P – polyhedron
• solver – (optional, default: default_lp_solver(N)) LP solver

Output

An element of the polyhedron, or an error if the polyhedron is empty.

Algorithm

An element is obtained by solving a feasibility linear program.

an_element(U::Universe{N}) where {N}

Return some element of a universe.

Input

• U – universe

Output

The origin.

isbounded(P::AbstractPolyhedron{N}; [solver]=default_lp_solver(N)) where {N}

Check whether a polyhedron is bounded.

Input

• P – polyhedron
• solver – (optional, default: default_lp_solver(N)) the backend used to solve the linear program

Output

true iff the polyhedron is bounded

Algorithm

We first check if the polyhedron has more than dim(P) constraints, which is a necessary condition for boundedness.

If so, we check boundedness via _isbounded_stiemke.

vertices_list(P::AbstractPolyhedron; check_boundedness::Bool=true)

Return the list of vertices of a polyhedron in constraint representation.

Input

• P – polyhedron in constraint representation
• check_boundedness – (optional, default: true) if true, check whether the polyhedron is bounded

Output

The list of vertices of P, or an error if P is unbounded.

Notes

This function throws an error if the polyhedron is unbounded. Otherwise, the polyhedron is converted to an HPolytope and its list of vertices is computed.

Examples

julia> P = HPolyhedron([HalfSpace([1.0, 0.0], 1.0),
                        HalfSpace([0.0, 1.0], 1.0),
                        HalfSpace([-1.0, 0.0], 1.0),
                        HalfSpace([0.0, -1.0], 1.0)]);

julia> length(vertices_list(P))
4

project(P::AbstractPolyhedron{N}, block::AbstractVector{Int};
        [kwargs...]) where {N}

Concrete projection of a polyhedral set.

Input

• P – set
• block – block structure, a vector with the dimensions of interest

Output

A polyhedron representing the projection of P on the dimensions specified by block. If P was bounded, the result is an HPolytope; otherwise the result is an HPolyhedron. Note that there are more specific methods for specific input types, which give a different output type; e.g., projecting a Ball1 results in a Ball1.

Algorithm

• We first try to exploit the special case where each of the constraints of P and block are compatible, which is one of the two cases described below. Let c be a constraint of P and let $D_c$ and $D_b$ be the set of dimensions in which c resp. block are constrained.
  • If $D_c ⊆ D_b$, then one can project the normal vector of c.
  • If $D_c ∩ D_b = ∅$, then the constraint becomes redundant.
• In the general case, we compute the concrete linear map of the projection matrix associated to the given block structure.

Examples

Consider the four-dimensional cross-polytope (unit ball in the 1-norm):

julia> P = convert(HPolytope, Ball1(zeros(4), 1.0));

All dimensions are constrained, and computing the (trivial) projection on the whole space behaves as expected:

julia> constrained_dimensions(P)
4-element Vector{Int64}:
 1
 2
 3
 4

julia> project(P, [1, 2, 3, 4]) == P
true

Each constraint of the cross polytope is constrained in all dimensions.

Now let us take a ball in the infinity norm and remove some constraints:

julia> B = BallInf(zeros(4), 1.0);

julia> c = constraints_list(B)[1:2]
2-element Vector{HalfSpace{Float64, ReachabilityBase.Arrays.SingleEntryVector{Float64}}}:
 HalfSpace{Float64, ReachabilityBase.Arrays.SingleEntryVector{Float64}}([1.0, 0.0, 0.0, 0.0], 1.0)
 HalfSpace{Float64, ReachabilityBase.Arrays.SingleEntryVector{Float64}}([0.0, 1.0, 0.0, 0.0], 1.0)

julia> P = HPolyhedron(c);

julia> constrained_dimensions(P)
2-element Vector{Int64}:
 1
 2

Finally, we take the concrete projection onto variables 1 and 2:

julia> project(P, [1, 2]) |> constraints_list
2-element Vector{HalfSpace{Float64, Vector{Float64}}}:
 HalfSpace{Float64, Vector{Float64}}([1.0, 0.0], 1.0)
 HalfSpace{Float64, Vector{Float64}}([0.0, 1.0], 1.0)

_isbounded_stiemke(constraints::AbstractVector{<:HalfSpace{N}};
                   solver=LazySets.default_lp_solver(N),
                   check_nonempty::Bool=true) where {N}

Check whether a list of constraints is bounded using Stiemke's theorem of alternatives.

Input

• constraints – list of constraints
• backend – (optional, default: default_lp_solver(N)) the backend used to solve the linear program
• check_nonempty – (optional, default: true) if true, check the precondition to this algorithm that P is non-empty

Output

true iff the list of constraints is bounded.

Notes

The list of constraints represents a polyhedron.

The algorithm calls isempty to check whether the polyhedron is empty. This computation can be avoided using the check_nonempty flag.

Algorithm

The algorithm is based on Stiemke's theorem of alternatives, see, e.g., [1].

Let the polyhedron $P$ be given in constraint form $Ax ≤ b$. We assume that the polyhedron is non-empty.

Proposition 1. If $\ker(A)≠\{0\}$, then $P$ is unbounded.

Proposition 2. Assume that $ker(A)={0}$ and $P$ is non-empty. Then $P$ is bounded if and only if the following linear program admits a feasible solution: $\min∥y∥_1$ subject to $A^Ty=0$ and $y≥1$.

[1] Mangasarian, Olvi L. Nonlinear programming. Society for Industrial and Applied Mathematics, 1994.

constraints_list(A::AbstractMatrix{N}, b::AbstractVector)

Convert a matrix-vector representation to a linear-constraint representation.

Input

• A – matrix
• b – vector

Output

A list of linear constraints.

tosimplehrep(constraints::AbstractVector{LC};
             [n]::Int=0) where {N, LC<:HalfSpace{N}}

Return the simple H-representation $Ax ≤ b$ from a list of linear constraints.

Input

• constraints – a list of linear constraints
• n – (optional; default: 0) dimension of the constraints

Output

The tuple (A, b) where A is the matrix of normal directions and b is the vector of offsets.

Notes

The parameter n can be used to create a matrix with no constraints but a non-zero dimension.

remove_redundant_constraints(constraints::AbstractVector{S};
                             backend=nothing) where {S<:HalfSpace}

Remove the redundant constraints of a given list of linear constraints.

Input

• constraints – list of constraints
• backend – (optional, default: nothing) the backend used to solve the linear program

Output

The list of constraints with the redundant ones removed, or an empty set if the constraints are infeasible.

Notes

If backend is nothing, it defaults to default_lp_solver(N).

Algorithm

See remove_redundant_constraints!(::AbstractVector{<:HalfSpace}) for details.

remove_redundant_constraints!(constraints::AbstractVector{S};
                              [backend]=nothing) where {S<:HalfSpace}

Remove the redundant constraints of a given list of linear constraints; the list is updated in-place.

Input

• constraints – list of constraints
• backend – (optional, default: nothing) the backend used to solve the linear program

Output

true if the removal was successful and the list of constraints constraints is modified by removing the redundant constraints, and false only if the constraints are infeasible.

Notes

Note that the result may be true even if the constraints are infeasible. For example, $x ≤ 0 && x ≥ 1$ will return true without removing any constraint. To check if the constraints are infeasible, use isempty(HPolyhedron(constraints)).

If backend is nothing, it defaults to default_lp_solver(N).

Algorithm

If there are m constraints in n dimensions, this function checks one by one if each of the m constraints is implied by the remaining ones.

To check if the k-th constraint is redundant, an LP is formulated using the constraints that have not yet been removed. If, at an intermediate step, it is detected that a subgroup of the constraints is infeasible, this function returns false. If the calculation finished successfully, this function returns true.

For details, see Fukuda's Polyhedra FAQ.

plot_recipe(P::AbstractPolyhedron{N}, [ε]=zero(N)) where {N}

Convert a (bounded) polyhedron to a pair (x, y) of points for plotting.

Input

• P – bounded polyhedron
• ε – (optional, default: 0) ignored, used for dispatch

Output

A pair (x, y) of points that can be plotted, where x is the vector of x-coordinates and y is the vector of y-coordinates.

Algorithm

We first assert that P is bounded (i.e., that P is a polytope).

One-dimensional polytopes are converted to an Interval. Three-dimensional or higher-dimensional polytopes are not supported.

For two-dimensional polytopes (i.e., polygons) we compute their set of vertices using vertices_list and then plot the convex hull of these vertices.

AbstractPolytope{N} <: AbstractPolyhedron{N}

Abstract type for compact convex polytopic sets.

Notes

Every concrete AbstractPolytope must define the following method:

• vertices_list(::AbstractPolytope) – return a list of all vertices

julia> subtypes(AbstractPolytope)
4-element Vector{Any}:
 AbstractCentrallySymmetricPolytope
 AbstractPolygon
 HPolytope
 VPolytope

A polytope is a bounded polyhedron (see AbstractPolyhedron). Polytopes are compact convex sets with either of the following equivalent properties:

1. They are the intersection of a finite number of closed half-spaces.
2. They are the convex hull of finitely many vertices.

isbounded(P::AbstractPolytope)

Check whether a polytopic set is bounded.

Input

• P – polytopic set

Output

true (since a polytopic set must be bounded).

isuniversal(P::AbstractPolytope{N}, [witness]::Bool=false) where {N}

Check whether a polytopic set is universal.

Input

• P – polytopic set
• witness – (optional, default: false) compute a witness if activated

Output

• If witness option is deactivated: false
• If witness option is activated: (false, v) where $v ∉ P$ unless the list of constraints is empty (which should not happen for a normal polytope)

Algorithm

A witness is produced using isuniversal(H) where H is the first linear constraint of P.

isempty(P::AbstractPolytope)

Check whether a polytopic set is empty.

Input

• P – polytopic set

Output

true if the given polytopic set contains no vertices, and false otherwise.

Algorithm

This algorithm checks whether the vertices_list of P is empty.

volume(P::AbstractPolytope; backend=default_polyhedra_backend(P))

Compute the volume of a polytopic set.

Input

• P – polytopic set
• backend – (optional, default: default_polyhedra_backend(P)) the backend for polyhedral computations; see Polyhedra's documentation for further information

Output

The volume of P.

Algorithm

The volume is computed by the Polyhedra library.

AbstractPolygon{N} <: AbstractPolytope{N}

Abstract type for convex polygons (i.e., two-dimensional polytopes).

Notes

Every concrete AbstractPolygon must define the following functions:

• tovrep(::AbstractPolygon{N}) – transform into vertex representation
• tohrep(::AbstractPolygon{N}) – transform into constraint representation

The subtypes of AbstractPolygon (including abstract interfaces):

julia> subtypes(AbstractPolygon)
2-element Vector{Any}:
 AbstractHPolygon
 VPolygon

dim(P::AbstractPolygon)

Return the ambient dimension of a convex polygon.

Input

• P – convex polygon

Output

The ambient dimension of the polygon, which is 2.

jump2pi(x::N) where {N<:AbstractFloat}

Return $x + 2π$ if $x$ is negative, otherwise return $x$.

Input

• x – real scalar

Output

$x + 2π$ if $x$ is negative, $x$ otherwise.

Examples

julia> using LazySets: jump2pi

julia> jump2pi(0.0)
0.0

julia> jump2pi(-0.5)
5.783185307179586

julia> jump2pi(0.5)
0.5

<=(u::AbstractVector, v::AbstractVector)

Compare two 2D vectors by their direction.

Input

• u – first 2D direction
• v – second 2D direction

Output

true iff $\arg(u) [2π] ≤ \arg(v) [2π]$.

Notes

The argument is measured in counter-clockwise fashion, with the 0 being the direction (1, 0).

Algorithm

The implementation checks the quadrant of each direction, and compares directions using the right-hand rule. In particular, this method does not use the arctangent.

_leq_trig(u::AbstractVector{N}, v::AbstractVector{N}) where {N<:AbstractFloat}

Compare two 2D vectors by their direction.

Input

• u – first 2D direction
• v – second 2D direction

Output

true iff $\arg(u) [2π] ≤ \arg(v) [2π]$.

Notes

The argument is measured in counter-clockwise fashion, with the 0 being the direction (1, 0).

Algorithm

The implementation uses the arctangent function with sign, atan, which for two arguments implements the atan2 function.

quadrant(w::AbstractVector{N}) where {N}

Compute the quadrant where the direction w belongs.

Input

• w – direction

Output

An integer from 0 to 3, with the following convention:

     ^
   1 | 0
  ---+-->
   2 | 3

Algorithm

The idea is to encode the following logic function: $11 ↦ 0, 01 ↦ 1, 00 ↦ 2, 10 ↦ 3$, according to the convention above.

This function is inspired from AGPX's answer in: Sort points in clockwise order?

AbstractHPolygon{N} <: AbstractPolygon{N}

Abstract type for polygons in constraint representation.

Notes

All subtypes must satisfy the invariant that constraints are sorted counter-clockwise.

Every concrete AbstractHPolygon must have the following fields:

• constraints::Vector{HalfSpace{N, AbstractVector{N}}} – the constraints

The subtypes of AbstractHPolygon:

julia> subtypes(AbstractHPolygon)
2-element Vector{Any}:
 HPolygon
 HPolygonOpt

an_element(P::AbstractHPolygon)

Return some element of a polygon in constraint representation.

Input

• P – polygon in constraint representation

Output

A vertex of the polygon in constraint representation (the first one in the order of the constraints).

∈(x::AbstractVector, P::AbstractHPolygon)

Check whether a given two-dimensional point is contained in a polygon in constraint representation.

Input

• x – two-dimensional point/vector
• P – polygon in constraint representation

Output

true iff $x ∈ P$.

Algorithm

This implementation checks if the point lies inside each constraint.

rand(::Type{HPOLYGON}; [N]::Type=Float64, [dim]::Int=2,
     [rng]::AbstractRNG=GLOBAL_RNG, [seed]::Union{Int, Nothing}=nothing,
     [num_constraints]::Int=-1) where {HPOLYGON<:AbstractHPolygon}

Create a random polygon in constraint representation.

Input

• HPOLYGON – type for dispatch
• N – (optional, default: Float64) numeric type
• dim – (optional, default: 2) dimension
• rng – (optional, default: GLOBAL_RNG) random number generator
• seed – (optional, default: nothing) seed for reseeding
• num_constraints – (optional, default: -1) number of constraints of the polygon (must be 3 or bigger; see comment below)

Output

A random polygon in constraint representation.

Algorithm

We create a random polygon in vertex representation and convert it to constraint representation. See rand(::Type{VPolygon}). For non-flat polygons the number of vertices and the number of constraints are identical.

tohrep(P::HPOLYGON) where {HPOLYGON<:AbstractHPolygon}

Build a constraint representation of the given polygon.

Input

• P – polygon in constraint representation

Output

The identity, i.e., the same polygon instance.

tovrep(P::AbstractHPolygon)

Build a vertex representation of a polygon in constraint representation.

Input

• P – polygon in constraint representation

Output

The same polygon but in vertex representation, a VPolygon.

addconstraint!(P::AbstractHPolygon, constraint::HalfSpace;
               [linear_search]::Bool=length(P.constraints) < 10,
               [prune]::Bool=true)

Add a linear constraint to a polygon in constraint representation, keeping the constraints sorted by their normal directions.

Input

• P – polygon in constraint representation
• constraint – linear constraint to add
• linear_search – (optional, default: length(constraints) < 10) flag to choose between linear and binary search
• prune – (optional, default: true) flag for removing redundant constraints in the end

addconstraint!(constraints::Vector{LC}, new_constraint::HalfSpace;
               [linear_search]::Bool=length(P.constraints) < 10,
               [prune]::Bool=true) where {LC<:HalfSpace}

Add a linear constraint to a sorted vector of constrains, keeping the constraints sorted by their normal directions.

Input

• constraints – vector of linear constraints
• new_constraint – linear constraint to add
• linear_search – (optional, default: length(constraints) < 10) flag to choose between linear and binary search
• prune – (optional, default: true) flag for removing redundant constraints in the end

Algorithm

If prune is active, we check if the new constraint is redundant. If the constraint is not redundant, we perform the same check to the left and to the right until we find the first constraint that is not redundant.

normalize(P::AbstractHPolygon{N}, p=N(2)) where {N}

Normalize a polygon in constraint representation.

Input

• P – polygon in constraint representation
• p – (optional, default: 2) norm

Output

A new polygon in constraint representation whose normal directions $a_i$ are normalized, i.e., such that $‖a_i‖_p = 1$ holds.

isredundant(cmid::HalfSpace, cright::HalfSpace, cleft::HalfSpace)

Check whether a linear constraint is redundant wrt. two surrounding constraints.

Input

• cmid – linear constraint of concern
• cright – linear constraint to the right (clockwise turn)
• cleft – linear constraint to the left (counter-clockwise turn)

Output

true iff the constraint is redundant.

Algorithm

We first check whether the angle between the surrounding constraints is < 180°, which is a necessary condition (unless the direction is identical to one of the other two constraints). If so, we next check if the angle is 0°, in which case the constraint cmid is redundant unless it is strictly tighter than the other two constraints. If the angle is strictly between 0° and 180°, the constraint cmid is redundant if and only if the vertex defined by the other two constraints lies inside the set defined by cmid.

remove_redundant_constraints!(P::AbstractHPolygon)

Remove all redundant constraints of a polygon in constraint representation.

Input

• P – polygon in constraint representation

Output

The same polygon with all redundant constraints removed.

Notes

Since we only consider bounded polygons and a polygon needs at least three constraints to be bounded, we stop removing redundant constraints if there are three or fewer constraints left. Hence for unbounded polygons the result may be unexpected.

Algorithm

We go through all consecutive triples of constraints and check if the one in the middle is redundant. For this we assume that the constraints are sorted.

constraints_list(P::AbstractHPolygon)

Return the list of constraints defining a polygon in constraint representation.

Input

• P – polygon in constraint representation

Output

The list of constraints of the polygon. The implementation guarantees that the constraints are sorted counter-clockwise.

vertices_list(P::AbstractHPolygon{N};
              apply_convex_hull::Bool=true,
              check_feasibility::Bool=true) where {N}

Return the list of vertices of a polygon in constraint representation.

Input

• P – polygon in constraint representation
• apply_convex_hull – (optional, default: true) flag to post-process the intersection of constraints with a convex hull
• check_feasibility – (optional, default: true) flag to check whether the polygon was empty (required for correctness in case of empty polygons)

Output

List of vertices.

Notes

By construction an AbstractHPolygon should not contain any redundant vertices. Still the apply_convex_hull argument is activated by default to remove potential duplicate vertices. They can exist due to numeric instability.

julia> p = HPolygon([HalfSpace([1.0, 0.0], 1.0),
                     HalfSpace([0.0, 1.0], 1.0),
                     HalfSpace([-1.0, 0.0], -1.0),
                     HalfSpace([0.0, -1.0], -1.0)]);

julia> vertices_list(p, apply_convex_hull=false)
4-element Vector{Vector{Float64}}:
 [1.0, 1.0]
 [1.0, 1.0]
 [1.0, 1.0]
 [1.0, 1.0]

If it is known that each constraint has a "proper" distance to the next vertex, this step can be skipped.

Algorithm

We compute each vertex as the intersection of consecutive lines defined by the half-spaces. If check_feasibility is active, we then check if the constraints of the polygon were actually feasible (i.e., they pointed in the right direction). For this we compute the average of all vertices and check membership in each constraint.

isbounded(P::AbstractHPolygon, [use_type_assumption]::Bool=true)

Determine whether a polygon in constraint representation is bounded.

Input

• P – polygon in constraint representation
• use_type_assumption – (optional, default: true) flag for ignoring the type assumption that polygons are bounded

Output

true if use_type_assumption is activated. Otherwise, true iff P is bounded.

Algorithm

If !use_type_assumption, we use _isbounded_unit_dimensions.

AbstractCentrallySymmetricPolytope{N} <: AbstractPolytope{N}

Abstract type for centrally symmetric, polytopic sets. It combines the AbstractCentrallySymmetric and AbstractPolytope interfaces. Such a type combination is necessary as long as Julia does not support multiple inheritance.

Notes

Every concrete AbstractCentrallySymmetricPolytope must define the following methods:

• from AbstractCentrallySymmetric:
  • center(::AbstractCentrallySymmetricPolytope) – return the center point
  • center(::AbstractCentrallySymmetricPolytope, i::Int) – return the center point at index i
• from AbstractPolytope:
  • vertices_list(::AbstractCentrallySymmetricPolytope) – return a list of all vertices

The subtypes of AbstractCentrallySymmetricPolytope (including abstract interfaces):

julia> subtypes(AbstractCentrallySymmetricPolytope)
2-element Vector{Any}:
 AbstractZonotope
 Ball1

dim(P::AbstractCentrallySymmetricPolytope)

Return the ambient dimension of a centrally symmetric, polytopic set.

Input

• P – centrally symmetric, polytopic set

Output

The ambient dimension of the polytopic set.

an_element(P::AbstractCentrallySymmetricPolytope)

Return some element of a centrally symmetric, polytopic set.

Input

• P – centrally symmetric, polytopic set

Output

The center of the centrally symmetric, polytopic set.

isempty(P::AbstractCentrallySymmetricPolytope)

Check whether a centrally symmetric, polytopic set is empty.

Input

• P – centrally symmetric, polytopic set

Output

false.

isuniversal(S::AbstractCentrallySymmetricPolytope{N},
            [witness]::Bool=false) where {N}

Check whether a centrally symmetric, polytopic set is universal.

Input

• S – centrally symmetric, polytopic set
• witness – (optional, default: false) compute a witness if activated

Output

• If witness option is deactivated: false
• If witness option is activated: (false, v) where $v ∉ S$

Algorithm

Centrally symmetric, polytopic sets are bounded. A witness is obtained by computing the support vector in direction d = [1, 0, …, 0] and adding d on top.

center(S::AbstractCentrallySymmetricPolytope, i::Int)

Return the center of a centrally symmetric, polytopic set along a given dimension.

Input

• S – centrally symmetric, polytopic set
• i – dimension of interest

Output

The center along the given dimension.

extrema(S::AbstractCentrallySymmetricPolytope, i::Int)

Return the lower and higher coordinate of a centrally symmetric, polytopic set in a given dimension.

Input

• S – centrally symmetric, polytopic set
• i – dimension of interest

Output

The lower and higher coordinate of the centrally symmetric, polytopic set in the given dimension.

Notes

The result is equivalent to (low(S, i), high(S, i)).

Algorithm

We compute high(S, i) and then compute the lowest coordinates with the help of center(S, i) (which is assumed to be cheaper to obtain).

extrema(S::AbstractCentrallySymmetricPolytope)

Return two vectors with the lowest and highest coordinate of a centrally symmetric, polytopic set.

Input

• S – centrally symmetric, polytopic set

Output

Two vectors with the lowest and highest coordinates of S.

Notes

The result is equivalent to (low(S), high(S)).

Algorithm

We compute high(S) and then compute the lowest coordinates with the help of center(S) (which is assumed to be cheaper to obtain).

AbstractZonotope{N} <: AbstractCentrallySymmetricPolytope{N}

Abstract type for zonotopic sets.

Notes

Mathematically, a zonotope is defined as the set

\[Z = \left\{ c + ∑_{i=1}^p ξ_i g_i,~~ ξ_i \in [-1, 1]~~ ∀ i = 1,…, p \right\},\]

where $c \in \mathbb{R}^n$ is its center and $\{g_i\}_{i=1}^p$, $g_i \in \mathbb{R}^n$, is the set of generators. This characterization defines a zonotope as the finite Minkowski sum of line segments. Zonotopes can be equivalently described as the image of a unit infinity-norm ball in $\mathbb{R}^n$ by an affine transformation.

See Zonotope for a standard implementation of this interface.

Every concrete AbstractZonotope must define the following functions:

• genmat(::AbstractZonotope) – return the generator matrix

• generators(::AbstractZonotope) – return an iterator over the generators

Since the functions genmat and generators can be defined in terms of each other, it is sufficient to only genuinely implement one of them and let the implementation of the other function call the fallback implementation genmat_fallback resp. generators_fallback.

The subtypes of AbstractZonotope (including abstract interfaces):

julia> subtypes(AbstractZonotope)
5-element Vector{Any}:
 AbstractHyperrectangle
 HParallelotope
 LineSegment
 RotatedHyperrectangle
 Zonotope

ngens(Z::AbstractZonotope)

Return the number of generators of a zonotopic set.

Input

• Z – zonotopic set

Output

An integer representing the number of generators.

genmat_fallback(Z::AbstractZonotope{N};
                [gens]=generators(Z), [ngens]=nothing) where {N}

Fallback definition of genmat for zonotopic sets.

Input

• Z – zonotopic set
• gens – (optional; default: generators(Z)) iterator over generators
• ngens – (optional; default: nothing) number of generators or nothing if unknown

Output

A matrix where each column represents one generator of Z.

Notes

Passing the number of generators (ngens) is more efficient, since otherwise the generators have to be obtained from the iterator (gens) and stored in an intermediate vector until the final result matrix can be allocated.

generators_fallback(Z::AbstractZonotope)

Fallback definition of generators for zonotopic sets.

Input

• Z – zonotopic set

Output

An iterator over the generators of Z.

ρ(d::AbstractVector, Z::AbstractZonotope)

Evaluate the support function of a zonotopic set in a given direction.

Input

• d – direction
• Z – zonotopic set

Output

The evaluation of the support function in the given direction.

Algorithm

The support value is $cᵀ d + ‖Gᵀ d‖₁$, where $c$ is the center and $G$ is the generator matrix of Z.

σ(d::AbstractVector, Z::AbstractZonotope)

Return a support vector of a zonotopic set in a given direction.

Input

• d – direction
• Z – zonotopic set

Output

A support vector in the given direction. If the direction has norm zero, the vertex with $ξ_i = 1 \ \ ∀ i = 1,…, p$ is returned.

∈(x::AbstractVector, Z::AbstractZonotope; solver=nothing)

Check whether a given point is contained in a zonotopic set.

Input

• x – point/vector
• Z – zonotopic set
• solver – (optional, default: nothing) the backend used to solve the linear program

Output

true iff $x ∈ Z$.

Examples

julia> Z = Zonotope([1.0, 0.0], [0.1 0.0; 0.0 0.1]);

julia> [1.0, 0.2] ∈ Z
false
julia> [1.0, 0.1] ∈ Z
true

Notes

If solver == nothing, we fall back to default_lp_solver(N).

Algorithm

The membership problem is computed by stating and solving the following linear program. Let $p$ and $n$ be the number of generators and ambient dimension, respectively. We consider the minimization of $x_0$ in the $p+1$-dimensional space of elements $(x_0, ξ_1, …, ξ_p)$ constrained to $0 ≤ x_0 ≤ ∞$, $ξ_i ∈ [-1, 1]$ for all $i = 1, …, p$, and such that $x-c = Gξ$ holds. If a feasible solution exists, the optimal value $x_0 = 0$ is achieved.

linear_map(M::AbstractMatrix, Z::AbstractZonotope)

Concrete linear map of a zonotopic set.

Input

• M – matrix
• Z – zonotopic set

Output

The zonotope obtained by applying the linear map.

Algorithm

We apply the linear map to the center and the generators.

translate(Z::AbstractZonotope, v::AbstractVector)

Translate (i.e., shift) a zonotopic set by a given vector.

Input

• Z – zonotopic set
• v – translation vector

Output

A translated zonotopic set.

Notes

See also translate!(Z::AbstractZonotope, v::AbstractVector) for the in-place version.

translate!(Z::AbstractZonotope, v::AbstractVector)

Translate (i.e., shift) a zonotopic set by a given vector in-place.

Input

• Z – zonotopic set
• v – translation vector

Output

A translated zonotopic set.

Notes

See also translate(Z::AbstractZonotope, v::AbstractVector) for the out-of-place version.

Algorithm

We add the translation vector to the center of the zonotopic set.

split(Z::AbstractZonotope, j::Int)

Return two zonotopes obtained by splitting the given zonotopic set.

Input

• Z – zonotopic set
• j – index of the generator to be split

Output

The zonotope obtained by splitting Z into two zonotopes such that their union is Z and their intersection is possibly non-empty.

Algorithm

This function implements [Prop. 3, 1], which we state next. The zonotopic set $Z = ⟨c, g^{(1, …, p)}⟩$ is split into:

\[Z₁ = ⟨c - \frac{1}{2}g^{(j)}, (g^{(1, …,j-1)}, \frac{1}{2}g^{(j)}, g^{(j+1, …, p)})⟩ \\ Z₂ = ⟨c + \frac{1}{2}g^{(j)}, (g^{(1, …,j-1)}, \frac{1}{2}g^{(j)}, g^{(j+1, …, p)})⟩,\]

such that $Z₁ ∪ Z₂ = Z$ and $Z₁ ∩ Z₂ = Z^*$, where

\[Z^* = ⟨c, (g^{(1,…,j-1)}, g^{(j+1,…, p)})⟩.\]

[1] Althoff, M., Stursberg, O., & Buss, M. Reachability analysis of nonlinear systems with uncertain parameters using conservative linearization. CDC 2008.

split(Z::AbstractZonotope, gens::AbstractVector{Int},
      nparts::AbstractVector{Int})

Split a zonotopic set along the given generators into a vector of zonotopes.

Input

• Z – zonotopic set
• gens – vector of indices of the generators to be split
• n – vector of integers describing the number of partitions in the corresponding generator

Output

The zonotopes obtained by splitting Z into 2^{n_i} zonotopes for each generator i such that their union is Z and their intersection is possibly non-empty.

Examples

Splitting of a two-dimensional zonotopic set along its first generator:

julia> Z = Zonotope([1.0, 0.0], [0.1 0.0; 0.0 0.1])
Zonotope{Float64, Vector{Float64}, Matrix{Float64}}([1.0, 0.0], [0.1 0.0; 0.0 0.1])

julia> split(Z, [1], [1])
2-element Vector{Zonotope{Float64, Vector{Float64}, Matrix{Float64}}}:
 Zonotope{Float64, Vector{Float64}, Matrix{Float64}}([0.95, 0.0], [0.05 0.0; 0.0 0.1])
 Zonotope{Float64, Vector{Float64}, Matrix{Float64}}([1.05, 0.0], [0.05 0.0; 0.0 0.1])

Here, the first vector in the arguments corresponds to the zonotopic set's generator to be split, and the second vector corresponds to the exponent of 2^n parts that the set will be split into along the corresponding generator.

As an example, below we split a two-dimensional zonotope along both of its generators, each time into four parts.

julia> Z = Zonotope([1.0, 0.0], [0.1 0.0; 0.0 0.1])
Zonotope{Float64, Vector{Float64}, Matrix{Float64}}([1.0, 0.0], [0.1 0.0; 0.0 0.1])

julia> split(Z, [1, 2], [2, 2])
16-element Vector{Zonotope{Float64, Vector{Float64}, Matrix{Float64}}}:
Zonotope{Float64, Vector{Float64}, Matrix{Float64}}([0.925, -0.075], [0.025 0.0; 0.0 0.025])
Zonotope{Float64, Vector{Float64}, Matrix{Float64}}([0.925, -0.025], [0.025 0.0; 0.0 0.025])
Zonotope{Float64, Vector{Float64}, Matrix{Float64}}([0.925, 0.025], [0.025 0.0; 0.0 0.025])
Zonotope{Float64, Vector{Float64}, Matrix{Float64}}([0.925, 0.075], [0.025 0.0; 0.0 0.025])
Zonotope{Float64, Vector{Float64}, Matrix{Float64}}([0.975, -0.075], [0.025 0.0; 0.0 0.025])
Zonotope{Float64, Vector{Float64}, Matrix{Float64}}([0.975, -0.025], [0.025 0.0; 0.0 0.025])
Zonotope{Float64, Vector{Float64}, Matrix{Float64}}([0.975, 0.025], [0.025 0.0; 0.0 0.025])
Zonotope{Float64, Vector{Float64}, Matrix{Float64}}([0.975, 0.075], [0.025 0.0; 0.0 0.025])
Zonotope{Float64, Vector{Float64}, Matrix{Float64}}([1.025, -0.075], [0.025 0.0; 0.0 0.025])
Zonotope{Float64, Vector{Float64}, Matrix{Float64}}([1.025, -0.025], [0.025 0.0; 0.0 0.025])
Zonotope{Float64, Vector{Float64}, Matrix{Float64}}([1.025, 0.025], [0.025 0.0; 0.0 0.025])
Zonotope{Float64, Vector{Float64}, Matrix{Float64}}([1.025, 0.075], [0.025 0.0; 0.0 0.025])
Zonotope{Float64, Vector{Float64}, Matrix{Float64}}([1.075, -0.075], [0.025 0.0; 0.0 0.025])
Zonotope{Float64, Vector{Float64}, Matrix{Float64}}([1.075, -0.025], [0.025 0.0; 0.0 0.025])
Zonotope{Float64, Vector{Float64}, Matrix{Float64}}([1.075, 0.025], [0.025 0.0; 0.0 0.025])
Zonotope{Float64, Vector{Float64}, Matrix{Float64}}([1.075, 0.075], [0.025 0.0; 0.0 0.025])

constraints_list(P::AbstractZonotope)

Return a list of constraints defining a zonotopic set.

Input

• Z – zonotopic set

Output

A list of constraints of the zonotopic set.

Algorithm

This is the (inefficient) fallback implementation for rational numbers. It first computes the vertices and then converts the corresponding polytope to constraint representation.

constraints_list(Z::AbstractZonotope{N}) where {N<:AbstractFloat}

Return a list of constraints defining a zonotopic set.

Input

• Z – zonotopic set

Output

A list of constraints of the zonotopic set.

Notes

The main algorithm assumes that the generator matrix is full rank. The result has $2 \binom{p}{n-1}$ (with $p$ being the number of generators and $n$ being the ambient dimension) constraints, which is optimal under this assumption. If this assumption is not given, the implementation tries to work around.

Algorithm

We follow the algorithm presented in [1]. Three cases are not covered by that algorithm, so we handle them separately. The first case is zonotopes in one dimension. The second case is that there are fewer generators than dimensions, $p < n$, or the generator matrix is not full rank, in which case we fall back to the (slower) computation based on the vertex representation. The third case is that the zonotope is flat in some dimensions, in which case we project the zonotope to the non-flat dimensions and extend the result later.

[1] Althoff, Stursberg, Buss. Computing Reachable Sets of Hybrid Systems Using a Combination of Zonotopes and Polytopes. 2009.

vertices_list(Z::AbstractZonotope; [apply_convex_hull]::Bool=true)

Return a list of the vertices of a zonotopic set.

Input

• Z – zonotopic set
• apply_convex_hull – (optional, default: true) if true, post-process the computation with the convex hull of the points

Output

A list of the vertices.

Algorithm

Two-dimensional case

We use a trick to speed up enumerating vertices of 2-dimensional zonotopic sets with all generators in the first quadrant or third quadrant (same sign). Namely, sort the generators by angle and add them clockwise in increasing order and counterclockwise in decreasing order. A more detailed explanation can be found here.

To avoid the cumulative sum from both directions separately, we build a 2D index matrix to sum generators for both directions in one matrix-vector product.

General case

If the zonotopic set has $p$ generators, each vertex is the result of summing the center with some linear combination of generators, where the combination factors are $ξ_i ∈ \{-1, 1\}$.

There are at most $2^p$ distinct vertices. Use the flag apply_convex_hull to control whether a convex-hull algorithm is applied to the vertices computed by this method; otherwise, redundant vertices may be present.

order(Z::AbstractZonotope)

Return the order of a zonotopic set.

Input

• Z – zonotopic set

Output

A rational number representing the order of the zonotopic set.

Notes

The order of a zonotopic set is defined as the quotient of its number of generators and its dimension.

togrep(Z::AbstractZonotope)

Return a generator representation of a zonotopic set.

Input

• Z – zonotopic set

Output

The same set in generator representation. This fallback implementation returns a Zonotope; however, more specific implementations may return other generator representations.

remove_redundant_generators(Z::AbstractZonotope)

Remove all redundant (pairwise linearly dependent) generators of a zonotopic set.

Input

• Z – zonotopic set

Output

A new zonotope with fewer generators, or the same zonotopic set if no generator could be removed.

Algorithm

By default this implementation returns the input zonotopic set. Subtypes of AbstractZonotope whose generators can be removed have to define a new method.

reduce_order(Z::AbstractZonotope, r::Real,
             [method]::AbstractReductionMethod=GIR05())

Reduce the order of a zonotopic set by overapproximating with a zonotope with fewer generators.

Input

• Z – zonotopic set
• r – desired order
• method – (optional, default: GIR05()) the reduction method used

Output

A new zonotope with fewer generators, if possible.

Algorithm

The available algorithms are:

julia> subtypes(AbstractReductionMethod)
3-element Vector{Any}:
 LazySets.ASB10
 LazySets.COMB03
 LazySets.GIR05

See the documentation of each algorithm for references. These methods split the given zonotopic set Z into two zonotopes, K and L, where K contains the most "representative" generators and L contains the generators that are reduced, Lred, using a box overapproximation. We follow the notation from [1]. See also [2].

• [1] Yang, X., & Scott, J. K. *A comparison of zonotope order reduction

techniques*. Automatica 2018.

• [2] Kopetzki, A. K., Schürmann, B., & Althoff, M. *Methods for order reduction

of zonotopes*. CDC 2017.

• [3] Althoff, M., Stursberg, O., & Buss, M. *Computing reachable sets of hybrid

systems using a combination of zonotopes and polytopes*. Nonlinear analysis: hybrid systems 2010.

AbstractReductionMethod

Abstract supertype for order-reduction methods of a zonotopic set.

ASB10 <: AbstractReductionMethod

Zonotope order-reduction method from [1].

• [1] Althoff, M., Stursberg, O., & Buss, M. *Computing reachable sets of hybrid

systems using a combination of zonotopes and polytopes*. Nonlinear analysis: hybrid systems 2010.

COMB03 <: AbstractReductionMethod

Zonotope order-reduction method from [1].

• [1] C. Combastel. A state bounding observer based on zonotopes. ECC 2003.

GIR05 <: AbstractReductionMethod

Zonotope order-reduction method from [1].

• [1] A. Girard. Reachability of Uncertain Linear Systems Using Zonotopes.

HSCC 2005.

AbstractHyperrectangle{N} <: AbstractZonotope{N}

Abstract type for hyperrectangular sets.

Notes

See Hyperrectangle for a standard implementation of this interface.

Every concrete AbstractHyperrectangle must define the following functions:

• radius_hyperrectangle(::AbstractHyperrectangle) – return the hyperrectangle's radius, which is a full-dimensional vector

• radius_hyperrectangle(::AbstractHyperrectangle, i::Int) – return the hyperrectangle's radius in the i-th dimension

• isflat(::AbstractHyperrectangle) – check whether the hyperrectangle's radius is zero in some dimension

Every hyperrectangular set is also a zonotopic set; see AbstractZonotope.

The subtypes of AbstractHyperrectangle (including abstract interfaces):

julia> subtypes(AbstractHyperrectangle)
5-element Vector{Any}:
 AbstractSingleton
 BallInf
 Hyperrectangle
 Interval
 SymmetricIntervalHull

norm(H::AbstractHyperrectangle, [p]::Real=Inf)

Return the norm of a hyperrectangular set.

The norm of a hyperrectangular set is defined as the norm of the enclosing ball of the given $p$-norm, of minimal volume, that is centered in the origin.

Input

• H – hyperrectangular set
• p – (optional, default: Inf) norm

Output

A real number representing the norm.

Algorithm

Recall that the norm is defined as

\[‖ X ‖ = \max_{x ∈ X} ‖ x ‖_p = max_{x ∈ \text{vertices}(X)} ‖ x ‖_p.\]

The last equality holds because the optimum of a convex function over a polytope is attained at one of its vertices.

This implementation uses the fact that the maximum is attained in the vertex $c + \text{diag}(\text{sign}(c)) r$ for any $p$-norm. Hence it suffices to take the $p$-norm of this particular vertex. This statement is proved below. Note that, in particular, there is no need to compute the $p$-norm for each vertex, which can be very expensive.

If $X$ is a hyperrectangle and the $n$-dimensional vectors center and radius of the hyperrectangle are denoted $c$ and $r$ respectively, then reasoning on the $2^n$ vertices we have that:

\[\max_{x ∈ \text{vertices}(X)} ‖ x ‖_p = \max_{α_1, …, α_n ∈ \{-1, 1\}} (|c_1 + α_1 r_1|^p + ... + |c_n + α_n r_n|^p)^{1/p}.\]

The function $x ↦ x^p$, $p > 0$, is monotonically increasing and thus the maximum of each term $|c_i + α_i r_i|^p$ is given by $|c_i + \text{sign}(c_i) r_i|^p$ for each $i$. Hence, $x^* := \text{argmax}_{x ∈ X} ‖ x ‖_p$ is the vertex $c + \text{diag}(\text{sign}(c)) r$.

radius(H::AbstractHyperrectangle, [p]::Real=Inf)

Return the radius of a hyperrectangular set.

Input

• H – hyperrectangular set
• p – (optional, default: Inf) norm

Output

A real number representing the radius.

Notes

The radius is defined as the radius of the enclosing ball of the given $p$-norm of minimal volume with the same center. It is the same for all corners of a hyperrectangular set.

σ(d::AbstractVector, H::AbstractHyperrectangle)

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

Input

• d – direction
• H – hyperrectangular set

Output

A support vector in the given direction.

If the direction vector is zero in dimension $i$, the result will have the center's coordinate in that dimension. For instance, for the two-dimensional infinity-norm ball, if the direction points to the right, the result is the vector [1, 0] in the middle of the right-hand facet.

If the direction has norm zero, the result can be any point in H. The default implementation returns the center of H.

ρ(d::AbstractVector, H::AbstractHyperrectangle)

Evaluate the support function of a hyperrectangular set in a given direction.

Input

• d – direction
• H – hyperrectangular set

Output

The evaluation of the support function in the given direction.

∈(x::AbstractVector, H::AbstractHyperrectangle)

Check whether a given point is contained in a hyperrectangular set.

Input

• x – point/vector
• H – hyperrectangular set

Output

true iff $x ∈ H$.

Algorithm

Let $H$ be an $n$-dimensional hyperrectangular set, $c_i$ and $r_i$ be the center and radius, and $x_i$ be the vector $x$ in dimension $i$, respectively. Then $x ∈ H$ iff $|c_i - x_i| ≤ r_i$ for all $i=1,…,n$.

vertices_list(H::AbstractHyperrectangle; kwargs...)

Return the list of vertices of a hyperrectangular set.

Input

• H – hyperrectangular set

Output

A list of vertices. Zeros in the radius are correctly handled, i.e., the result does not contain any duplicate vertices.

Algorithm

First we identify the dimensions where H is flat, i.e., its radius is zero. We also compute the number of vertices that we have to create.

Next we create the vertices. We do this by enumerating all vectors v of length n (the dimension of H) with entries -1/0/1 and construct the corresponding vertex as follows:

\[ \text{vertex}(v)(i) = \begin{cases} c(i) + r(i) & v(i) = 1 \\ c(i) & v(i) = 0 \\ c(i) - r(i) & v(i) = -1. \end{cases}\]

For enumerating the vectors v, we modify the current v from left to right by changing entries -1 to 1, skipping entries 0, and stopping at the first entry 1 (but changing it to -1). This way we only need to change the vertex in those dimensions where v has changed, which usually is a smaller number than n.

constraints_list(H::AbstractHyperrectangle{N}) where {N}

Return the list of constraints of a hyperrectangular set.

Input

• H – hyperrectangular set

Output

A list of $2n$ linear constraints, where $n$ is the dimension of H.

constraints_list(U::Universe{N}) where {N}

Return the list of constraints defining a universe.

Input

• U – universe

Output

The empty list of constraints, as the universe is unconstrained.

high(H::AbstractHyperrectangle)

Return the higher coordinates of a hyperrectangular set.

Input

• H – hyperrectangular set

Output

A vector with the higher coordinates of the hyperrectangular set.

high(H::AbstractHyperrectangle, i::Int)

Return the higher coordinate of a hyperrectangular set in a given dimension.

Input

• H – hyperrectangular set
• i – dimension of interest

Output

The higher coordinate of the hyperrectangular set in the given dimension.

low(H::AbstractHyperrectangle)

Return the lower coordinates of a hyperrectangular set.

Input

• H – hyperrectangular set

Output

A vector with the lower coordinates of the hyperrectangular set.

low(H::AbstractHyperrectangle, i::Int)

Return the lower coordinate of a hyperrectangular set in a given dimension.

Input

• H – hyperrectangular set
• i – dimension of interest

Output

The lower coordinate of the hyperrectangular set in the given dimension.

extrema(H::AbstractHyperrectangle)

Return the lower and higher coordinates of a hyperrectangular set.

Input

• H – hyperrectangular set

Output

The lower and higher coordinates of the set.

Notes

The result is equivalent to (low(H), high(H)).

extrema(H::AbstractHyperrectangle, i::Int)

Return the lower and higher coordinate of a hyperrectangular set in a given dimension.

Input

• H – hyperrectangular set
• i – dimension of interest

Output

The lower and higher coordinate of the set in the given dimension.

Notes

The result is equivalent to (low(H, i), high(H, i)).

isflat(H::AbstractHyperrectangle)

Check whether a hyperrectangular set is flat, i.e., whether its radius is zero in some dimension.

Input

• H – hyperrectangular set

Output

true iff the hyperrectangular set is flat.

Notes

For robustness with respect to floating-point inputs, this function relies on the result of isapproxzero when applied to the radius in some dimension. Hence this function depends on the absolute zero tolerance ABSZTOL.

split(H::AbstractHyperrectangle{N},
      num_blocks::AbstractVector{Int}) where {N}

Partition a hyperrectangular set into uniform sub-hyperrectangles.

Input

• H – hyperrectangular set
• num_blocks – number of blocks in the partition for each dimension

Output

A list of Hyperrectangles.

generators(H::AbstractHyperrectangle)

Return an iterator over the generators of a hyperrectangular set.

Input

• H – hyperrectangular set

Output

An iterator over the generators of H.

genmat(H::AbstractHyperrectangle)

Return the generator matrix of a hyperrectangular set.

Input

• H – hyperrectangular set

Output

A matrix where each column represents one generator of H.

ngens(H::AbstractHyperrectangle{N}) where {N}

Return the number of generators of a hyperrectangular set.

Input

• H – hyperrectangular set

Output

The number of generators.

Algorithm

A hyperrectangular set has one generator for each non-flat dimension.

rectify(H::AbstractHyperrectangle)

Concrete rectification of a hyperrectangular set.

Input

• H – hyperrectangular set

Output

The Hyperrectangle that corresponds to the rectification of H.

volume(H::AbstractHyperrectangle)

Return the volume of a hyperrectangular set.

Input

• H – hyperrectangular set

Output

The volume of $H$.

Algorithm

The volume of the $n$-dimensional hyperrectangle $H$ with radius vector $r$ is $2ⁿ ∏ᵢ rᵢ$ where $rᵢ$ denotes the $i$-th component of $r$.

distance(x::AbstractVector, H::AbstractHyperrectangle{N};
         [p]::Real=N(2)) where {N}

Compute the distance between a point x and a hyperrectangular set H with respect to the given p-norm.

Input

• x – point/vector
• H – hyperrectangular set

Output

A scalar representing the distance between point x and hyperrectangle H.

AbstractSingleton{N} <: AbstractHyperrectangle{N}

Abstract type for sets with a single value.

Notes

Every concrete AbstractSingleton must define the following function:

• element(::AbstractSingleton) – return the single element

julia> subtypes(AbstractSingleton)
2-element Vector{Any}:
 Singleton
 ZeroSet

σ(d::AbstractVector, S::AbstractSingleton)

Return the support vector of a set with a single value.

Input

• d – direction
• S – set with a single value

Output

The support vector, which is the set's vector itself, irrespective of the given direction.

ρ(d::AbstractVector, S::AbstractSingleton)

Evaluate the support function of a set with a single value in a given direction.

Input

• d – direction
• S – set with a single value

Output

The support value in the given direction.

∈(x::AbstractVector, S::AbstractSingleton)

Check whether a given point is contained in a set with a single value.

Input

• x – point/vector
• S – set with a single value

Output

true iff $x ∈ S$.

Notes

This implementation performs an approximate comparison to account for imprecision in floating-point computations.

center(S::AbstractSingleton)

Return the center of a set with a single value.

Input

• S – set with a single value

Output

The center of the set.

center(S::AbstractSingleton, i::Int)

Return the center of a set with a single value in a given dimension.

Input

• S – set with a single value
• i – dimension of interest

Output

The i-th entry of the center of the set.

element(S::AbstractSingleton, i::Int)

Return the i-th entry of the element of a set with a single value.

Input

• S – set with a single value
• i – dimension of interest

Output

The i-th entry of the element.

vertices(S::AbstractSingleton{N}) where {N}

Construct an iterator over the vertices of a set with a single value.

Input

• S – set with a single value

Output

An iterator with a single value.

vertices_list(S::AbstractSingleton)

Return the list of vertices of a set with a single value.

Input

• S – set with a single value

Output

A list containing only a single vertex.

radius_hyperrectangle(S::AbstractSingleton{N}) where {N}

Return the box radius of a set with a single value in every dimension.

Input

• S – set with a single value

Output

The zero vector.

radius_hyperrectangle(S::AbstractSingleton{N}, i::Int) where {N}

Return the box radius of a set with a single value in a given dimension.

Input

• S – set with a single value
• i – dimension of interest

Output

Zero.

high(S::AbstractSingleton)

Return the higher coordinates of a set with a single value.

Input

• S – set with a single value

Output

A vector with the higher coordinates.

high(S::AbstractSingleton, i::Int)

Return the higher coordinate of a set with a single value in the given dimension.

Input

• S – set with a single value
• i – dimension of interest

Output

The higher coordinate in the given dimension.

low(S::AbstractSingleton)

Return the lower coordinates of a set with a single value.

Input

• S – set with a single value

Output

A vector with the lower coordinates.

low(S::AbstractSingleton, i::Int)

Return the lower coordinate of a set with a single value in the given dimension.

Input

• S – set with a single value
• i – dimension of interest

Output

The lower coordinate in the given dimension.

generators(S::AbstractSingleton{N}) where {N}

Return an (empty) iterator over the generators of a set with a single value.

Input

• S – set with a single value

Output

An empty iterator.

genmat(S::AbstractSingleton{N}) where {N}

Return the (empty) generator matrix of a set with a single value.

Input

• S – set with a single value

Output

A matrix with no columns representing the generators of S.

ngens(S::AbstractSingleton)

Return the number of generators of a set with a single value.

Input

• H – set with a single value

Output

The number of generators, which is $0$.

plot_recipe(S::AbstractSingleton{N}, [ε]=zero(N)) where {N}

Convert a singleton to a pair (x, y) of points for plotting.

Input

• S – singleton
• ε – (optional, default: 0) ignored, used for dispatch

Output

A pair (x, y) of one point that can be plotted.

plot_singleton(S::AbstractSingleton{N}, [ε]::Real=zero(N); ...) where {N}

Plot a singleton.

Input

• S – singleton
• ε – (optional, default: 0) ignored, used for dispatch

Examples

julia> plot(Singleton([0.5, 1.0]))

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

Abstract type for affine maps.

Notes

See AffineMap for a standard implementation of this interface.

Every concrete AbstractAffineMap must define the following methods:

• matrix(::AbstractAffineMap) – return the linear map
• vector(::AbstractAffineMap) – return the affine translation vector
• set(::AbstractAffineMap) – return the set that the map is applied to

The subtypes of AbstractAffineMap:

julia> subtypes(AbstractAffineMap)
7-element Vector{Any}:
 AffineMap
 ExponentialMap
 ExponentialProjectionMap
 InverseLinearMap
 LinearMap
 ResetMap
 Translation

dim(am::AbstractAffineMap)

Return the dimension of an affine map.

Input

• am – affine map

Output

The ambient dimension of an affine map.

σ(d::AbstractVector, am::AbstractAffineMap)

Return a support vector of an affine map.

Input

• d – direction
• am – affine map

Output

A support vector in the given direction.

ρ(d::AbstractVector, am::AbstractAffineMap)

Evaluate the support function of an affine map.

Input

• d – direction
• am – affine map

Output

The evaluation of the support function in the given direction.

an_element(am::AbstractAffineMap)

Return some element of an affine map.

Input

• am – affine map

Output

An element of the affine map.

Algorithm

The implementation relies on the an_element method of the wrapped set.

isempty(am::AbstractAffineMap)

Check whether an affine map is empty.

Input

• am – affine map

Output

true iff the wrapped set is empty.

isbounded(am::AbstractAffineMap; cond_tol::Number=DEFAULT_COND_TOL)

Check whether an affine map is bounded.

Input

• am – affine map
• cond_tol – (optional) tolerance of matrix condition (used to check whether the matrix is invertible)

Output

true iff the affine map is bounded.

Algorithm

We first check if the matrix is zero or the wrapped set is bounded. If not, we perform a sufficient check whether the matrix is invertible. If the matrix is invertible, then the map being bounded is equivalent to the wrapped set being bounded, and hence the map is unbounded. Otherwise, we check boundedness via _isbounded_unit_dimensions.

∈(x::AbstractVector, am::AbstractAffineMap)

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

Input

• x – point/vector
• am – affine map of a convex set

Output

true iff $x ∈ am$.

Algorithm

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

Examples

julia> am = AffineMap([2.0 0.0; 0.0 1.0], BallInf([1., 1.], 1.), [-1.0, -1.0]);

julia> [5.0, 1.0] ∈ am
false

julia> [3.0, 1.0] ∈ am
true

An example with a 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

center(am::AbstractAffineMap)

Return the center of an affine map of a centrally-symmetric set.

Input

• cp – affine map of a centrally-symmetric set

Output

The center of the affine map.

Algorithm

The implementation relies on the center method of the wrapped set.

vertices_list(am::AbstractAffineMap; [apply_convex_hull]::Bool)

Return the list of vertices of a (polytopic) affine map.

Input

• am – affine map of a polytopic set
• apply_convex_hull – (optional, default: true) if true, apply the convex hull operation to the list of vertices transformed by the affine map

Output

A list of vertices.

Algorithm

This implementation computes all vertices of X, then transforms them through the affine map, i.e., x ↦ M*x + v for each vertex x of X. By default, the convex-hull operation is taken before returning this list. For dimensions three or higher, this operation relies on the functionality through the concrete polyhedra library Polyhedra.jl.

If you are not interested in taking the convex hull of the resulting vertices under the affine map, pass apply_convex_hull=false as a keyword argument.

Note that we assume that the underlying set X is polytopic, either concretely or lazily, i.e., the function vertices_list should be applicable.

constraints_list(am::AbstractAffineMap)

Return the list of constraints of a (polyhedral) affine map.

Input

• am – affine map of a polyhedral set

Output

The list of constraints of the affine map.

Notes

We assume that the underlying set X is polyhedral, i.e., offers a method constraints_list(X).

Algorithm

This implementation uses the method to compute the list of constraints of the translation of a lazy linear map.

linear_map(M::AbstractMatrix, am::AbstractAffineMap)

Return the linear map of a lazy affine map.

Input

• M – matrix
• am – affine map

Output

A set corresponding to the linear map of the lazy affine map of a set.

AbstractStar{N} <: LazySet{N}

Abstract supertype for all star set types.

Notes

A set $X$ is star-like (also known as generalized star) if it can be represented by a center $x₀ ∈ \mathbb{R}^n$ and $m$ vectors $v₁, …, vₘ$ forming the basis, and a predicate $P : \mathbb{R}^n → \{⊤, ⊥\}$ such that

\[ X = \{x ∈ \mathbb{R}^n : x = x₀ + \sum_{i=1}^m α_i v_i,~~\textrm{s.t. } P(α) = ⊤ \}.\]

AbstractPolynomialZonotope{N} <: LazySet{N}

Abstract type for polynomial zonotope sets.

Notes

Polynomial zonotopes are in general non-convex. They are always bounded.

julia> subtypes(AbstractPolynomialZonotope)
3-element Vector{Any}:
 DensePolynomialZonotope
 SimpleSparsePolynomialZonotope
 SparsePolynomialZonotope