Lazy Intersections

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

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

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

pell = plot(E₁, aspectratio=1)
pell = plot!(pell, E₂)

The accuracy to which this set is plotted can be controlled by passing a numerical argument as in plot(E₁, 1e-3, aspectratio=1). Here, 1e-3 stands for an upper-bound on the error, measured in terms of the Hausdorff distance between the ellipsoid and the polygonal overapproximation which is actually computed for display.

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

Z = E₁ ∩ E₂
Intersection{Float64, Ellipsoid{Float64, Vector{Float64}, Matrix{Float64}}, Ellipsoid{Float64, Vector{Float64}, Matrix{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)})

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}})
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)])...);
Hint (generic function with 1 method)
pell = plot(E₁, aspectratio=1)
pell = plot!(pell, E₂)
pεsmaller = plot!(pell, convert(HPolygon, Hint(0.5)), alpha=.4)

pell = plot(E₁, aspectratio=1)
pell = plot!(pell, E₂)
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.

using 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, aspectratio=1)
pell = plot!(pell, E₂)
pbox = plot!(pell, Xbox, alpha=.4)

pell = plot(E₁, aspectratio=1)
pell = plot!(pell, E₂)
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!

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)

for n in [2, 5, 10]
    println("\nn = $n\n")
    global E₁, E₂ = rand_ellipsoid(n), rand_ellipsoid(n)

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

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

n = 2

  0.000769 seconds (5.26 k allocations: 205.344 KiB)
  0.001370 seconds (11.45 k allocations: 467.062 KiB)

n = 5

  0.001505 seconds (15.36 k allocations: 637.859 KiB)
  0.011614 seconds (131.70 k allocations: 7.047 MiB)

n = 10

  0.003619 seconds (38.34 k allocations: 1.741 MiB)
  0.160052 seconds (1.36 M allocations: 89.490 MiB, 33.36% gc time)