# Intersection

## Binary intersection (Intersection)

`LazySets.Intersection`

— Type`Intersection{N, S1<:LazySet{N}, S2<:LazySet{N}} <: LazySet{N}`

Type that represents the intersection of two sets.

**Fields**

`X`

– set`Y`

– set`cache`

– internal cache for avoiding recomputation; see`IntersectionCache`

**Notes**

If the arguments of the lazy intersection are half-spaces, the set is simplified to a polyhedron in constraint representation (`HPolyhedron`

).

The intersection preserves convexity: if the set arguments are convex, then their intersection is convex as well.

**Examples**

Create an expression $Z$ that lazily represents the intersection of two squares $X$ and $Y$:

```
julia> X, Y = BallInf([0.0, 0.0], 0.5), BallInf([1.0, 0.0], 0.75);
julia> Z = X ∩ Y;
julia> typeof(Z)
Intersection{Float64, BallInf{Float64, Vector{Float64}}, BallInf{Float64, Vector{Float64}}}
julia> dim(Z)
2
```

We can check if the intersection is empty with `isempty`

:

```
julia> isempty(Z)
false
```

Do not confuse `Intersection`

with the concrete operation, which is computed with the lowercase `intersection`

function:

```
julia> W = intersection(X, Y)
Hyperrectangle{Float64, Vector{Float64}, Vector{Float64}}([0.375, 0.0], [0.125, 0.5])
```

`Base.:∩`

— Method`∩(X::LazySet, Y::LazySet)`

Alias for the lazy intersection.

**Notes**

The function symbol can be typed via `\cap<tab>`

.

`LazySets.API.dim`

— Method`dim(cap::Intersection)`

Return the dimension of an intersection of two sets.

**Input**

`cap`

– intersection of two sets

**Output**

The ambient dimension of the intersection of two sets.

`LazySets.API.ρ`

— Method`ρ(d::AbstractVector, cap::Intersection)`

Return an upper bound on the support function of the intersection of two sets in a given direction.

**Input**

`d`

– direction`cap`

– intersection of two sets

**Output**

An upper bound on the support function in the given direction.

**Algorithm**

The support function of an intersection of $X$ and $Y$ is upper-bounded by the minimum of the support-function evaluations for $X$ and $Y$.

`LazySets.API.ρ`

— Method```
ρ(d::AbstractVector, cap::Intersection{N, S1, S2};
algorithm::String="line_search", kwargs...
) where {N, S1<:LazySet{N},
S2<:Union{HalfSpace{N}, Hyperplane{N}, Line2D{N}}}
```

Evaluate the support function of the intersection of a compact set and a half-space/hyperplane/line in a given direction.

**Input**

`d`

– direction`cap`

– lazy intersection of a compact set and a half-space/hyperplane/ line`algorithm`

– (optional, default:`"line_search"`

): the algorithm to calculate the support function; valid options are:`"line_search"`

– solve the associated univariate optimization problem using a line-search method (either Brent or the Golden Section method)`"projection"`

– only valid for intersection with a hyperplane/line; evaluate the support function by reducing the problem to the 2D intersection of a rank-2 linear transformation of the given compact set in the plane generated by the given direction`d`

and the hyperplane's normal vector`n`

`"simple"`

– take the $\min$ of the support-function evaluation of each operand

**Output**

The scalar value of the support function of the set `cap`

in the given direction.

**Notes**

It is assumed that the first set of the intersection (`cap.X`

) is compact.

Any additional number of arguments to the algorithm backend can be passed as keyword arguments.

**Algorithm**

The algorithms are based on solving the associated optimization problem

\[\min_{λ ∈ D_h} ρ(ℓ - λa, X) + λb.\]

where $D_h = \{ λ : λ ≥ 0 \}$ if $H$ is a half-space or $D_h = \{ λ : λ ∈ ℝ \}$ if $H$ is a hyperplane.

For additional information we refer to:

[1] G. Frehse, R. Ray. *Flowpipe-Guard Intersection for Reachability Computations with Support Functions.* [2] C. Le Guernic. *Reachability Analysis of Hybrid Systems with Linear Continuous Dynamics*, PhD thesis [3] T. Rockafellar, R. Wets. *Variational Analysis*.

`LazySets.API.ρ`

— Method```
ρ(d::AbstractVector, cap::Intersection{N, S1, S2};
kwargs...) where {N, S1<:LazySet{N}, S2<:AbstractPolyhedron{N}}
```

Return an upper bound on the support function of the intersection between a compact set and a polyhedron along a given direction.

**Input**

`d`

– direction`cap`

– intersection of a compact set and a polyhedron`kwargs`

– additional arguments that are passed to the support-function algorithm

**Output**

An upper bound of the support function of the given intersection.

**Algorithm**

The idea is to solve the univariate optimization problem `ρ(di, X ∩ Hi)`

for each half-space in the polyhedron and then take the minimum. This gives an overapproximation of the exact support value.

This algorithm is inspired from [1].

[1] G. Frehse, R. Ray. *Flowpipe-Guard Intersection for Reachability Computations with Support Functions*.

**Notes**

This method relies on the `constraints_list`

of the polyhedron.

`LazySets.API.ρ`

— Method```
ρ(d::AbstractVector, cap::Intersection{N, S1, S2}; kwargs...
) where {N, S1<:AbstractPolyhedron{N}, S2<:AbstractPolyhedron{N}}
```

Evaluate the support function of the intersection between two polyhedral sets.

**Input**

`d`

– direction`cap`

– intersection of two polyhedral sets`kwargs`

– additional arguments that are passed to the support-function algorithm

**Output**

The evaluation of the support function in the given direction.

**Algorithm**

We combine the constraints of the two polyhedra to a new `HPolyhedron`

, for which we then evaluate the support function.

`LazySets.API.σ`

— Method`σ(d::AbstractVector, cap::Intersection)`

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

**Input**

`d`

– direction`cap`

– intersection of two sets

**Output**

A support vector in the given direction.

**Algorithm**

We compute the concrete intersection, which may be expensive.

`LazySets.API.isbounded`

— Method`isbounded(cap::Intersection)`

Check whether an intersection of two sets is bounded.

**Input**

`cap`

– intersection of two sets

**Output**

`true`

iff the intersection is bounded.

**Algorithm**

We first check if any of the wrapped sets is bounded. Otherwise we check boundedness via `LazySets._isbounded_unit_dimensions`

.

`Base.isempty`

— Method`isempty(cap::Intersection)`

Check whether the intersection of two sets is empty.

**Input**

`cap`

– intersection of two sets

**Output**

`true`

iff the intersection is empty.

**Notes**

The result will be cached, so a second query will be fast.

`Base.:∈`

— Method`∈(x::AbstractVector, cap::Intersection)`

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

**Input**

`x`

– point/vector`cap`

– intersection of two sets

**Output**

`true`

iff $x ∈ cap$.

**Algorithm**

A point $x$ is in the intersection iff it is in each set.

`LazySets.API.constraints_list`

— Method`constraints_list(cap::Intersection)`

Return a list of constraints of an intersection of two (polyhedral) sets.

**Input**

`cap`

– intersection of two (polyhedral) sets

**Output**

A list of constraints of the intersection.

**Notes**

We assume that the underlying sets are polyhedral, i.e., offer a method `constraints_list`

.

**Algorithm**

We create the polyhedron by taking the intersection of the `constraints_list`

s of the sets and remove redundant constraints.

This function ignores the boolean output from the in-place `remove_redundant_constraints!`

, which may inform the user that the constraints are infeasible. In that case, the list of constraints at the moment when the infeasibility was detected is returned.

`LazySets.API.vertices_list`

— Method`vertices_list(cap::Intersection)`

Return a list of vertices of a lazy intersection of two (polyhedral) sets.

**Input**

`cap`

– intersection of two (polyhedral) sets

**Output**

A list containing the vertices of the lazy intersection of two sets.

**Notes**

We assume that the underlying sets are polyhedral and that the intersection is bounded.

**Algorithm**

We compute the concrete intersection using `intersection`

and then take the vertices of that representation.

`LazySets.isempty_known`

— Method`isempty_known(cap::Intersection)`

Ask whether the status of emptiness is known.

**Input**

`cap`

– intersection of two sets

**Output**

`true`

iff the emptiness status is known. In this case, `isempty(cap)`

can be used to obtain the status in constant time.

`LazySets.set_isempty!`

— Method`set_isempty!(cap::Intersection, isempty::Bool)`

Set the status of emptiness in the cache.

**Input**

`cap`

– intersection of two sets`isempty`

– new status of emptiness

`LazySets.swap`

— Method`swap(cap::Intersection)`

Return a new `Intersection`

object with the arguments swapped.

**Input**

`cap`

– intersection of two sets

**Output**

A new `Intersection`

object with the arguments swapped. The old cache is shared between the old and new objects.

**Notes**

The advantage of using this function instead of manually swapping the arguments is that the cache is shared.

`LazySets.use_precise_ρ`

— Function`use_precise_ρ(cap::Intersection)`

Check whether a precise algorithm for computing $ρ$ shall be applied.

**Input**

`cap`

– intersection of two sets

**Output**

`true`

if a precise algorithm shall be applied.

**Notes**

The default implementation always returns `true`

.

If the result is `false`

, a coarse approximation of the support function is returned.

This function can be overwritten by the user to control the policy.

`LazySets._line_search`

— Function`_line_search(ℓ, X, H::Union{<:HalfSpace, <:Hyperplane, <:Line2D}; [kwargs...])`

Given a convex set $X$ and a half-space $H = \{x: a^T x ≤ b \}$ or a hyperplane/line $H = \{x: a^T x = b \}$, calculate:

\[\min_{λ ∈ D_h} ρ(ℓ - λa, X) + λb.\]

where $D_h = \{ λ : λ ≥ 0 \}$ if $H$ is a half-space or $D_h = \{ λ : λ ∈ ℝ \}$ if $H$ is a hyperplane.

**Input**

`ℓ`

– direction`X`

– convex set`H`

– half-space or hyperplane or line

**Output**

The tuple `(fmin, λmin)`

, where `fmin`

is the minimum value of the function $f(λ) = ρ(ℓ - λa) + λb$ over the feasible set $λ ≥ 0$, and $λmin$ is the minimizer.

**Notes**

This function requires the `Optim`

package, and relies on the univariate optimization interface `Optim.optimize(...)`

.

Additional arguments to the `optimize`

backend can be passed as keyword arguments. The default method is `Optim.Brent()`

.

**Examples**

```
julia> X = Ball1(zeros(2), 1.0);
julia> H = HalfSpace([-1.0, 0.0], -1.0); # x >= 1
julia> using Optim
julia> using LazySets: _line_search
julia> v = _line_search([1.0, 0.0], X, H); # uses Brent's method by default
julia> v[1]
1.0
```

We can specify the upper bound in Brent's method:

```
julia> v = _line_search([1.0, 0.0], X, H, upper=1e3);
julia> v[1]
1.0
```

Instead of Brent's method we can use the Golden Section method:

```
julia> v = _line_search([1.0, 0.0], X, H, upper=1e3, method=GoldenSection());
julia> v[1]
1.0
```

`LazySets._projection`

— Function```
_projection(ℓ, X, H::Union{Hyperplane{N}, Line2D{N}};
[lazy_linear_map]=false,
[lazy_2d_intersection]=true,
[algorithm_2d_intersection]=nothing,
[kwargs...]) where {N}
```

Given a convex set $X$ and a hyperplane $H = \{x: n ⋅ x = γ \}$, calculate the support function of the intersection between the rank-2 projection $Π_{nℓ} X$ and the line $Lγ = \{(x, y): x = γ \}$.

**Input**

`ℓ`

– direction`X`

– convex set`H`

– hyperplane`lazy_linear_map`

– (optional, default:`false`

) flag to perform the projection lazily or concretely`lazy_2d_intersection`

– (optional, default:`true`

) flag to perform the 2D intersection between the projected set and the line lazily or concretely`algorithm_2d_intersection`

– (optional, default:`nothing`

) if given, fixes the support-function algorithm used for the intersection in 2D; otherwise the default is used

**Output**

The evaluation of the support function of $X ∩ H$ along direction $ℓ$.

**Algorithm**

This projection method is based on Prop. 8.2, [1, page 103].

In the original algorithm, Section 8.2 of [1], the linear map is performed concretely and the intersection is performed lazily (these are the default options in this algorithm, but here the four combinations are available). If the set $X$ is a zonotope, its concrete projection is again a zonotope (sometimes called "zonogon"). The intersection between this zonogon and the line can be taken efficiently in a lazy way (see Section 8.2.2 of [1]), if one uses dispatch on `ρ(y_dir, Sℓ⋂Lγ; kwargs...)`

given that `Sℓ`

is itself a zonotope.

**Notes**

This function depends on the calculation of the support function of another set in two dimensions. Obviously one does not want to use `algorithm="projection"`

again for this second calculation. The option `algorithm_2d_intersection`

is used for that: if not given, the default support-function algorithm is used (e.g., `"line_search"`

). You can still pass additional arguments to the `"line_search"`

backend through the `kwargs`

arguments.

[1] C. Le Guernic. *Reachability Analysis of Hybrid Systems with Linear Continuous Dynamics*, PhD thesis.

`LazySets.API.linear_map`

— Method`linear_map(M::AbstractMatrix, cap::Intersection)`

Return the concrete linear map of an intersection of two sets.

**Input**

`M`

– matrix`cap`

– intersection of two sets

**Output**

The set obtained by applying the given linear map to the intersection.

**Algorithm**

This method computes the concrete intersection.

`LazySets.plot_recipe`

— Method```
plot_recipe(cap::Intersection{N}, [ε]::N=-one(N),
[Nφ]::Int=PLOT_POLAR_DIRECTIONS) where {N}
```

Convert an intersection of two sets to a pair `(x, y)`

of points for plotting.

**Input**

`cap`

– intersection of two sets`ε`

– (optional, default`0`

) ignored, used for dispatch`Nφ`

– (optional, default:`PLOT_POLAR_DIRECTIONS`

) number of polar directions used in the template overapproximation

**Output**

A pair `(x, y)`

of points that can be plotted.

`RecipesBase.apply_recipe`

— Method```
plot_intersection(cap::Intersection{N}, [ε]::Real=zero(N),
[Nφ]::Int=PLOT_POLAR_DIRECTIONS) where {N}
```

Plot a lazy intersection.

**Input**

`cap`

– lazy intersection`ε`

– (optional, default`0`

) ignored, used for dispatch`Nφ`

– (optional, default:`PLOT_POLAR_DIRECTIONS`

) number of polar directions used in the template overapproximation

**Notes**

This function is separated from the main `LazySet`

plot recipe because iterative refinement is not available for lazy intersections (since it uses the support vector (but see #1187)).

Also note that if the set is a *nested* intersection, you may have to manually overapproximate this set before plotting (see `overapproximate`

for details).

**Examples**

```
julia> X = Ball2(zeros(2), 1.) ∩ Ball2(ones(2), 1.5); # lazy intersection
julia> plot(X)
```

You can specify the accuracy of the overapproximation of the lazy intersection by passing an explicit value for `Nφ`

, which stands for the number of polar directions used in the overapproximation. This number can also be passed to the `plot`

function directly.

```
julia> plot(overapproximate(X, PolarDirections(100)))
julia> plot(X, 0.0, 100) # equivalent to the above line
```

Inherited from `LazySet`

:

`norm`

`radius`

`diameter`

- [
`an_element`

](@ref an_element(::LazySet) `singleton_list`

`reflect`

### Intersection cache

`LazySets.IntersectionCache`

— Type`IntersectionCache`

Container for information cached by a lazy `Intersection`

object.

**Fields**

`isempty`

– is the intersection empty? There are three possible states, encoded as`Int8`

values -1, 0, 1:- $-1$ - it is currently unknown whether the intersection is empty
- $0$ - intersection is not empty
- $1$ - intersection is empty

## $n$-ary intersection (IntersectionArray)

`LazySets.IntersectionArray`

— Type`IntersectionArray{N, S<:LazySet{N}} <: LazySet{N}`

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

**Fields**

`array`

– array of sets

**Notes**

This type assumes that the dimensions of all elements match.

The `EmptySet`

is the absorbing element for `IntersectionArray`

.

The intersection preserves convexity: if the set arguments are convex, then their intersection is convex as well.

`Base.:∩`

— Method```
∩(X::LazySet, Xs::LazySet...)
∩(Xs::Vector{<:LazySet})
```

Alias for the n-ary lazy intersection.

`LazySets.API.dim`

— Method`dim(ia::IntersectionArray)`

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

**Input**

`ia`

– intersection of a finite number of sets

**Output**

The ambient dimension of the intersection of a finite number of sets, or `0`

if there is no set in the array.

`LazySets.API.σ`

— Method`σ(d::AbstractVector, ia::IntersectionArray)`

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

**Input**

`d`

– direction`ia`

– intersection of a finite number of sets

**Output**

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

**Algorithm**

This implementation computes the concrete intersection, which can be expensive.

`LazySets.API.isbounded`

— Method`isbounded(ia::IntersectionArray)`

Check whether an intersection of a finite number of sets is bounded.

**Input**

`ia`

– intersection of a finite number of sets

**Output**

`true`

iff the intersection is bounded.

**Algorithm**

We first check if any of the wrapped sets is bounded. Otherwise we check boundedness via `LazySets._isbounded_unit_dimensions`

.

`Base.:∈`

— Method`∈(x::AbstractVector, ia::IntersectionArray)`

Check whether a given point is contained in an intersection of a finite number of sets.

**Input**

`x`

– point/vector`ia`

– intersection of a finite number of sets

**Output**

`true`

iff $x ∈ ia$.

**Algorithm**

A point $x$ is in the intersection iff it is in each set.

`LazySets.array`

— Method`array(ia::IntersectionArray)`

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

**Input**

`ia`

– intersection of a finite number of sets

**Output**

The array of an intersection of a finite number of sets.

`LazySets.API.constraints_list`

— Method`constraints_list(ia::IntersectionArray)`

Return the list of constraints of an intersection of a finite number of (polyhedral) sets.

**Input**

`ia`

– intersection of a finite number of (polyhedral) sets

**Output**

The list of constraints of the intersection.

**Notes**

We assume that the underlying sets are polyhedral, i.e., offer a method `constraints_list`

.

**Algorithm**

We create the polyhedron from the `constraints_list`

s of the sets and remove redundant constraints.

Inherited from `LazySet`

:

`norm`

`radius`

`diameter`

- [
`an_element`

](@ref an_element(::LazySet) `singleton_list`

`reflect`