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Affine maps (AbstractAffineMap)

An affine map consists of a linear map and a translation.

LazySets.AbstractAffineMap — Type
AbstractAffineMap{N, S<:LazySet{N}} <: LazySet{N}

Abstract type for affine maps.

Notes

See AffineMap for a standard implementation of this interface.

Every concrete AbstractAffineMap must define the following methods:

  • matrix(::AbstractAffineMap) – return the linear map
  • vector(::AbstractAffineMap) – return the affine translation vector
  • set(::AbstractAffineMap) – return the set that the map is applied to

The subtypes of AbstractAffineMap:

julia> subtypes(AbstractAffineMap)
7-element Vector{Any}:
 AffineMap
 ExponentialMap
 ExponentialProjectionMap
 InverseLinearMap
 LinearMap
 ResetMap
 Translation
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This interface requires to implement the following functions:

LazySets.matrix — Method
matrix(X::AbstractAffineMap)

Return the matrix of an affine map.

Input

  • X – affine map

Output

The matrix of X.

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LazySets.vector — Method
vector(X::AbstractAffineMap)

Return the vector of an affine map.

Input

  • X – affine map

Output

The vector of X.

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LazySets.set — Method
set(X::AbstractAffineMap)

Return the set of an affine map.

Input

  • X – affine map

Output

The set of X before applying the map.

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This interface defines the following functions:

LazySets.API.an_element — Method
an_element(am::AbstractAffineMap)

Return some element of an affine map.

Input

  • am – affine map

Output

An element of the affine map.

Algorithm

The implementation relies on the an_element method of the wrapped set.

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LazySets.API.center — Method
center(am::AbstractAffineMap)

Return the center of an affine map of a centrally-symmetric set.

Input

  • cp – affine map of a centrally-symmetric set

Output

The center of the affine map.

Algorithm

The implementation relies on the center method of the wrapped set.

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LazySets.API.constraints_list — Method
constraints_list(am::AbstractAffineMap)

Return the list of constraints of a (polyhedral) affine map.

Input

  • am – affine map of a polyhedral set

Output

The list of constraints of the affine map.

Notes

We assume that the underlying set X is polyhedral, i.e., offers a method constraints_list(X).

Algorithm

This implementation uses the method to compute the list of constraints of the translation of a lazy linear map.

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LazySets.API.dim — Method
dim(am::AbstractAffineMap)

Return the dimension of an affine map.

Input

  • am – affine map

Output

The ambient dimension of an affine map.

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LazySets.API.isbounded — Method
isbounded(am::AbstractAffineMap; cond_tol::Number=DEFAULT_COND_TOL)

Check whether an affine map is bounded.

Input

  • am – affine map
  • cond_tol – (optional) tolerance of matrix condition (used to check whether the matrix is invertible)

Output

true iff the affine map is bounded.

Algorithm

We first check if the matrix is zero or the wrapped set is bounded. If not, we perform a sufficient check whether the matrix is invertible. If the matrix is invertible, then the map being bounded is equivalent to the wrapped set being bounded, and hence the map is unbounded. Otherwise, we check boundedness via _isbounded_unit_dimensions.

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Base.isempty — Method
isempty(am::AbstractAffineMap)

Check whether an affine map is empty.

Input

  • am – affine map

Output

true iff the wrapped set is empty.

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Base.:∈ — Method
∈(x::AbstractVector, am::AbstractAffineMap)

Check whether a given point is contained in the affine map of a convex set.

Input

  • x – point/vector
  • am – affine map of a convex set

Output

true iff $x ∈ am$.

Algorithm

Observe that $x ∈ M⋅S ⊕ v$ iff $M^{-1}⋅(x - v) ∈ S$. This implementation does not explicitly invert the matrix, which is why it also works for non-square matrices.

Examples

julia> am = AffineMap([2.0 0.0; 0.0 1.0], BallInf([1., 1.], 1.), [-1.0, -1.0]);

julia> [5.0, 1.0] ∈ am
false

julia> [3.0, 1.0] ∈ am
true

An example with a non-square matrix:

julia> B = BallInf(zeros(4), 1.);

julia> M = [1. 0 0 0; 0 1 0 0]/2;

julia> [0.5, 0.5] ∈ M*B
true
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LazySets.API.linear_map — Method
linear_map(M::AbstractMatrix, am::AbstractAffineMap)

Return the linear map of a lazy affine map.

Input

  • M – matrix
  • am – affine map

Output

A set corresponding to the linear map of the lazy affine map of a set.

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LazySets.API.ρ — Method
ρ(d::AbstractVector, am::AbstractAffineMap)

Evaluate the support function of an affine map.

Input

  • d – direction
  • am – affine map

Output

The evaluation of the support function in the given direction.

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LazySets.API.σ — Method
σ(d::AbstractVector, am::AbstractAffineMap)

Return a support vector of an affine map.

Input

  • d – direction
  • am – affine map

Output

A support vector in the given direction.

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LazySets.API.vertices_list — Method
vertices_list(am::AbstractAffineMap; [apply_convex_hull]::Bool)

Return the list of vertices of a (polytopic) affine map.

Input

  • am – affine map of a polytopic set
  • apply_convex_hull – (optional, default: true) if true, apply the convex hull operation to the list of vertices transformed by the affine map

Output

A list of vertices.

Algorithm

This implementation computes all vertices of X, then transforms them through the affine map, i.e., x ↦ M*x + v for each vertex x of X. By default, the convex-hull operation is taken before returning this list. For dimensions three or higher, this operation relies on the functionality through the concrete polyhedra library Polyhedra.jl.

If you are not interested in taking the convex hull of the resulting vertices under the affine map, pass apply_convex_hull=false as a keyword argument.

Note that we assume that the underlying set X is polytopic, either concretely or lazily, i.e., the function vertices_list should be applicable.

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Implementations