Lazy Intersections

In this section we illustrate the use of lazy intersection in LazySets. We will use the ellipsoid set type.

Lazy Intersections

An ellipsoid $E$ can be created by giving its center $c$ and its shape matrix $Q$, which should be positive definite, i.e. its eigenvalues must be positive. Mathematically, it is the set

\[ E = \{ x ∈ \mathbb{R}^n : (x-c)Q^{-1}(x-c) ≤ 1\}.\]

Let's make two rotated ellipsoids and plot them in the same pair of axes.

using Plots, LazySets, LazySets.Approximations

E₁ = Ellipsoid(zeros(2), [1 0; 0 2.])
E₂ = Ellipsoid(ones(2), [2 0; 0 1.])

pell = plot(E₁, 1e-3, aspectratio=1, alpha=.5)
pell = plot!(pell, E₂, 1e-3, alpha=.5)

Note

If you are wondering about the paremeter 1e-3 passed to plot, this parameter controls the accuracy to which the set is plotted (because the set that we actually plot is a polygonal overapproximation of the ellipses!).

Now let's take the lazy intersection of the ellipses:

Z = E₁ ∩ E₂
typeof(Z)

Intersection{Float64,Ellipsoid{Float64},Ellipsoid{Float64}}

On the other hand, the concrete intersection of sets, called intersection in LazySets, is not yet available for ellipsoids:

hasmethod(intersection, Tuple{typeof(E₁), typeof(E₂)})

false

So how can we work with the intersection of the ellipsoids?

One way is to overapproximate them by polygons (or polytopes in higher dims) and then take their intersection, because this function is defined, whose return type is again a HPolytope:

hasmethod(intersection, Tuple{HPolytope{Float64}, HPolytope{Float64}})

true

import LazySets.Approximations.overapproximate

# the parameter epsilon controls the accuracy of the iterative refinement,
# with respect to the Hausdorff distance
H₁(ε) = overapproximate(E₁, HPolygon, ε)
H₂(ε) = overapproximate(E₂, HPolygon, ε)

# using the concrete hpolytope-hpolytope intersection here
Hint(ε) = intersection(convert.(HPolytope, [H₁(ε), H₂(ε)])...);

pell = plot(E₁, 1e-3, aspectratio=1, alpha=.5)
pell = plot!(pell, E₂, 1e-3, alpha=.5)
pεsmaller = plot!(pell, convert(HPolygon, Hint(0.5)), alpha=.4)

pell = plot(E₁, 1e-3, aspectratio=1, alpha=.5)
pell = plot!(pell, E₂, 1e-3, alpha=.5)
pεbigger = plot!(pell, convert(HPolygon, Hint(0.05)), alpha=.4)

plot(pεsmaller, pεbigger, layout=(1, 2))

Note how dividing the $\varepsilon$ threshold by 10 makes the polygonal overapproximation of the intersection tighter.

Yet another approach is to directly query the directions of the lazy intersection E₁ ∩ E₂. We can overapproximate using template directions, such as a box, an octagon, or other.

The idea behind the template overapproximation method is to use the property that the support function of the intersection of two convex sets is upper bounded by the min of the support function of each set. We can see in the following experiments that the resulting set is quite tight.

import LazySets.Approximations.overapproximate
using LazySets.Approximations, Polyhedra

# overapproximate the lazy intersection using a box
Xbox = overapproximate(E₁ ∩ E₂, BoxDirections(2))

# overapproximate the lazy intersection using octagonal directions
Xoct = overapproximate(E₁ ∩ E₂, OctDirections(2))

pell = plot(E₁, 1e-3, aspectratio=1, alpha=.5)
pell = plot!(pell, E₂, 1e-3, alpha=.5)
pbox = plot!(pell, Xbox, alpha=.4)

pell = plot(E₁, 1e-3, aspectratio=1, alpha=.5)
pell = plot!(pell, E₂, 1e-3, alpha=.5)
poct = plot!(pell, Xoct, alpha=.4)

plot(pbox, poct, layout=(1, 2))

Using support function evaluations over a set of fixed directions is in general more efficient than iterative refinement, but the drawback is that one does not have control on the overapproximation error. Moreover, iterative refinement is currently only available in two dimensions, but overapproximation with template directions can be used in any dimension.

Let's time it!

using BenchmarkTools

@btime overapproximate($E₁ ∩ $E₂, BoxDirections(2))
@btime overapproximate($E₁ ∩ $E₂, OctDirections(2));

  1.786 μs (69 allocations: 1.63 KiB)
  5.059 μs (146 allocations: 4.63 KiB)

We can work with higher dimensional ellipsoids as well:

using LinearAlgebra

# a random ellipsoid in n-dimensions
function rand_ellipsoid(n)
    A = rand(n,n)
    Q = (A+transpose(A))/2 + n * I
    Ellipsoid(rand(n), Q)
end;

for n in [2, 5, 50, 100]
    println("\nn = $n\n")
    global E₁, E₂ = rand_ellipsoid(n), rand_ellipsoid(n)

    # overapproximate the lazy intersection using an n-dimensional box
    @btime overapproximate($E₁ ∩ $E₂, BoxDirections($n))

    # overapproximate the lazy intersection using octagonal directions in R^n
    @btime overapproximate($E₁ ∩ $E₂, OctDirections($n))
end;


n = 2

  1.768 μs (69 allocations: 1.63 KiB)
  5.156 μs (146 allocations: 4.63 KiB)

n = 5

  4.227 μs (165 allocations: 3.83 KiB)
  36.114 μs (862 allocations: 35.05 KiB)

n = 50

  51.085 μs (1605 allocations: 36.95 KiB)
  7.190 ms (85018 allocations: 11.43 MiB)

n = 100

  131.879 μs (3205 allocations: 73.61 KiB)
  104.384 ms (340020 allocations: 72.76 MiB)