Symmetric interval hull (SymmetricIntervalHull)

SymmetricIntervalHull{N, S<:LazySet{N}} <: AbstractHyperrectangle{N}

Type that represents the symmetric interval hull of a compact set.

Fields

• X – compact set
• cache – partial storage of already computed bounds, organized as mapping from the dimension to the bound value

Notes

The symmetric interval hull can be computed with $2n$ support-function queries (of unit vectors), where $n$ is the dimension of the wrapped set (i.e., two queries per dimension). When asking for the support vector (or support function) in a direction $d$, one needs $2k$ such queries, where $k$ is the number of non-zero entries in $d$.

However, if one asks for many support vectors (or support-function evaluations) in a loop, the number of computations may exceed $2n$. To be most efficient in such cases, this type stores the intermediately computed bounds in the cache field.

The set X must be bounded. The flag check_boundedness (which defaults to true) can be used to elide the boundedness check in the inner constructor. Misuse of this flag can result in incorrect behavior.

The symmetric interval hull of a set is a hyperrectangle centered in the origin, which in particular is convex.

The convenience alias ⊡ can be typed by \boxdot<tab>.

dim(sih::SymmetricIntervalHull)

Return the dimension of the symmetric interval hull of a set.

Input

• sih – symmetric interval hull of a set

Output

The ambient dimension of the symmetric interval hull of a set.

σ(d::AbstractVector, sih::SymmetricIntervalHull)

Return a support vector of the symmetric interval hull of a set in a given direction.

Input

• d – direction
• sih – symmetric interval hull of a set

Output

A support vector of the symmetric interval hull of a set in the given direction. If the direction has norm zero, the origin is returned.

Algorithm

For each non-zero entry in d we need to either look up the bound (if it has been computed before) or compute it, in which case we store it for future queries.

center(sih::SymmetricIntervalHull, i::Int)

Return the center along a given dimension of the symmetric interval hull of a set.

Input

• sih – symmetric interval hull of a set
• i – dimension of interest

Output

The center along a given dimension of the symmetric interval hull of a set.

center(sih::SymmetricIntervalHull)

Return the center of the symmetric interval hull of a set.

Input

• sih – symmetric interval hull of a set

Output

The origin.

radius_hyperrectangle(sih::SymmetricIntervalHull)

Return the box radius of the symmetric interval hull of a set in every dimension.

Input

• sih – symmetric interval hull of a set

Output

The box radius of the symmetric interval hull of a set.

Notes

This function computes the symmetric interval hull explicitly.

radius_hyperrectangle(sih::SymmetricIntervalHull, i::Int)

Return the box radius of the symmetric interval hull of a set in a given dimension.

Input

• sih – symmetric interval hull of a set
• i – dimension of interest

Output

The radius in the given dimension.

Notes

If the radius was computed before, this is just a look-up. Otherwise it is computed.