Operations on sets

In this section we show which typical set operations this library supports.

• Operations on sets
• Unary operations
• dim
• ρ/σ
• an_element
• ∈
• isempty
• linear_map
• norm
• radius
• diameter
• Binary operations
• ⊆
• is_intersection_empty

We use the following four sets for illustration.

using LazySets, LazySets.Approximations, Plots
B1 = Ball1(-ones(2), 1.)
B2 = Ball2(ones(2), 1.)
BI = BallInf(zeros(2), 1.)
H = Hyperrectangle(ones(2), ones(2))
sets = [B1, B2, BI, H]

function plot_sets(sets)
    for S in sets
        println(S)
        plot!(S, 1e-2, fillalpha=0.1)
    end
end

function plot_points(points, prefix)
    for i in eachindex(points)
        p = points[i]
        num_occur = length(findfirst(x -> x == p, points[1:i]))
        x = p[1]
        y = p[2]
        if num_occur == 1
            x += 0.15
        elseif num_occur == 2
            y += 0.15
        elseif num_occur == 3
            x -= 0.15
        else
            y -= 0.15
        end
        plot!(Singleton(p))
        plot!(annotations=(x, y, text("$(prefix)$(i)")))
    end
end

plot1 = plot()
plot_sets(sets)
plot1

Unary operations

The following table lists all operations that take one convex set as argument in the columns. In the rows we list all set types, both the interfaces (where we abbreviate the Abstract prefix), the basic set types, and the lazy set operations, each sorted alphabetically. The table entries have the following meaning.

• "x" indicates that the operation is implemented for the respective set type.
• "i" indicates that the operation is inherited from a supertype.
• "(·)" indicates that the operation is partly implemented/inherited.

dim

This function returns the dimension of the set.

dim(B1), dim(B2), dim(BI), dim(H)

(2, 2, 2, 2)

ρ/σ

These functions return the support function resp. the support vector of the set.

an_element

This function returns some element in the set. Consecutive calls to this function typically return the same element.

an_element(B1), an_element(B2), an_element(BI), an_element(H)

([-1.0, -1.0], [1.0, 1.0], [0.0, 0.0], [1.0, 1.0])

∈

This function checks containment of a given vector in the set. The operator can be used in infix notation (v ∈ S) and in inverse operand order (S ∋ v). Alias: in

p1 = [1.5, 1.5]
p2 = [0.1, 0.1]
p3 = [-0.9, -0.8]
points = [p1, p2, p3]

for p in [p1, p2, p3]
    println("$p ∈ (B1, B2, BI, H)? ($(p ∈ B1), $(p ∈ B2), $(p ∈ BI), $(p ∈ H))")
end

[1.5, 1.5] ∈ (B1, B2, BI, H)? (false, true, false, true)
[0.1, 0.1] ∈ (B1, B2, BI, H)? (false, false, true, true)
[-0.9, -0.8] ∈ (B1, B2, BI, H)? (true, false, true, false)

plot1 = plot()
plot_sets(sets)
plot_points(points, "p")
plot1

isempty

This function checks if the set is empty.

linear_map

This function applies a concrete linear map to the set. The resulting set may be of a different type.

norm

This function returns the norm of a set. It is defined as the norm of the enclosing ball (of the given norm) of minimal volume centered in the origin.

# print 1-norm, 2-norm, and infinity norm (if available)
println(("-", "-", norm(B1, Inf)))
println(("-", "-", norm(B2, Inf)))
println((norm(BI, 1), norm(BI, 2), norm(BI, Inf)))
println((norm(H, 1), norm(H, 2), norm(H, Inf)))

("-", "-", 2.0)
("-", "-", 2.0)
(2.0, 1.4142135623730951, 1.0)
(4.0, 2.8284271247461903, 2.0)

radius

This function returns the radius of a set. It is defined as the radius of the enclosing ball (of the given norm) of minimal volume with the same center.

radius(B1), radius(B2), radius(BI), radius(H)

(1.0, 1.0, 1.0, 1.0)

diameter

This function returns the diameter of a set. It is defined as the diameter of the enclosing ball (of the given norm) of minimal volume with the same center. The implementation is inherited for all set types if the norm is the infinity norm, in which case the result is defined as twice the radius.

diameter(B1), diameter(B2), diameter(BI), diameter(H)

(2.0, 2.0, 2.0, 2.0)

Binary operations

The following table lists all operations that take two convex set as argument in the entries. In the rows we list all set types, both the interfaces (where we abbreviate the Abstract prefix), the basic set types, and the lazy set operations, each sorted alphabetically. In the columns we also list the operations, but abbreviated. The table entries consist of subsets of the following list of operations.

• "⊆" stands for the subset check issubset.
• "⊎" stands for the disjointness check isdisjoint.
• "∩" stands for the concrete intersection operation intersection.
• "C" stands for the conversion operation convert.
• "-" indicates that the two types' dimensionality constraints are incompatible.
• A suffix "i" indicates that the operation is inherited from a supertype.

⊆

This function checks whether a set is a subset of another set. It can optionally produce a witness if the subset relation does not hold. The operator can be used in infix notation (X ⊆ S). Alias: issubset

println(B1 ⊆ B2)
w1 = ⊆(B1, B2, true)[2]
println(B1 ⊆ BI)
w2 = ⊆(B1, BI, true)[2]
println(B1 ⊆ H)
w3 = ⊆(B1, H, true)[2]
# 'B2 ⊆ B1' is not supported yet
# w11 = ⊆(B2, B1, true)[2]
println(B2 ⊆ BI)
w4 = ⊆(B2, BI, true)[2]
println(B2 ⊆ H)
println(BI ⊆ B1)
w5 = ⊆(BI, B1, true)[2]
println(BI ⊆ B2)
w6 = ⊆(BI, B2, true)[2]
println(BI ⊆ H)
w7 = ⊆(BI, H, true)[2]
println(H ⊆ B1)
w8 = ⊆(H, B1, true)[2]
println(H ⊆ B2)
w9 = ⊆(H, B2, true)[2]
println(H ⊆ BI)
w10 = ⊆(H, BI, true)[2];

2-element Array{Float64,1}:
 2.0
 1.0

witnesses = [w1, w2, w3, w4, w5, w6, w7, w8, w9, w10]

plot1 = plot()
plot_sets(sets)
plot_points(witnesses, "w")
plot1

is_intersection_empty

This function checks whether the intersection of two sets is empty. It can optionally produce a witness if the intersection is nonempty.

println(is_intersection_empty(BI, H))
w1 = is_intersection_empty(BI, H, true)[2]
# none of the other combinations are supported yet
# is_intersection_empty(B1, B2)
# is_intersection_empty(B1, BI)
# is_intersection_empty(B1, H)
# w2 = is_intersection_empty(B1, H, true)[2]
# is_intersection_empty(B2, BI)
# is_intersection_empty(B2, H)

false

witnesses = [w1]

plot1 = plot()
plot_sets(sets)
plot_points(witnesses, "w")
plot1