Reset map (ResetMap)

ResetMap{N, S<:LazySet{N}} <: AbstractAffineMap{N, S}

Type that represents a lazy reset map. A reset map is a special case of an affine map $A x + b, x ∈ X$ where the linear map $A$ is the identity matrix with zero entries in all reset dimensions, and the translation vector $b$ is zero in all other dimensions.

Fields

• X – set
• resets – resets (a mapping from an index to a new value)

Notes

The reset map preserves convexity: if X is convex, then any reset map of X is convex as well.

Examples

julia> X = BallInf([2.0, 2.0, 2.0], 1.0);

julia> r = Dict(1 => 4.0, 3 => 0.0);

julia> rm = ResetMap(X, r);

Here rm modifies the set X such that x1 is reset to 4 and x3 is reset to 0, while x2 is not modified. Hence rm is equivalent to the set VPolytope([[4.0, 1.0, 0.0], [4.0, 3.0, 0.0]]), i.e., an axis-aligned line segment embedded in 3D.

The corresponding affine map $A x + b$ would be:

\[ egin{pmatrix} 0 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 0 nd{pmatrix} x + egin{pmatrix} 4 & 0 & 0 nd{pmatrix}\]

Use the function matrix (resp. vector) to create the matrix A (resp. vector b) corresponding to a given reset map.

julia> matrix(rm)
3×3 LinearAlgebra.Diagonal{Float64, Vector{Float64}}:
 0.0   ⋅    ⋅
  ⋅   1.0   ⋅
  ⋅    ⋅   0.0

julia> vector(rm)
3-element SparseArrays.SparseVector{Float64, Int64} with 1 stored entry:
  [1]  =  4.0

The application of a ResetMap to a ZeroSet or an EmptySet is simplified automatically.

julia> ResetMap(ZeroSet(3), r)
Singleton{Float64, SparseArrays.SparseVector{Float64, Int64}}(sparsevec([1], [4.0], 3))

julia> ResetMap(EmptySet(3), r)
∅(3)

The (in this case unique) support vector of rm in direction [1, 1, 1] is:

julia> σ(ones(3), rm)
3-element Vector{Float64}:
 4.0
 3.0
 0.0

dim(rm::ResetMap)

Return the dimension of a reset map.

Input

• rm – reset map

Output

The ambient dimension of a reset map.

ρ(d::AbstractVector, rm::ResetMap)

Evaluate the support function of a reset map.

Input

• d – direction
• rm – reset map

Output

The evaluation of the support function in the given direction.

Notes

We use the usual dot-product definition, but for unbounded sets we redefine the product between $0$ and $±∞$ as $0$; Julia returns NaN here.

julia> Inf * 0.0
NaN

See the discussion here.

σ(d::AbstractVector, rm::ResetMap)

Return a support vector of a reset map.

Input

• d – direction
• rm – reset map

Output

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

an_element(rm::ResetMap)

Return some element of a reset map.

Input

• rm – reset map

Output

An element in the reset map.

Algorithm

This method relies on the an_element implementation for the wrapped set.

matrix(rm::ResetMap{N}) where {N}

Return the $A$ matrix of the affine map $A x + b, x ∈ X$ represented by a reset map.

Input

• rm – reset map

Output

The (Diagonal) matrix for the affine map $A x + b, x ∈ X$ represented by the reset map.

Algorithm

We construct the identity matrix and set all entries in the reset dimensions to zero.

vector(rm::ResetMap)

Return the $b$ vector of the affine map $A x + b, x ∈ X$ represented by a reset map.

Input

• rm – reset map

Output

The (sparse) vector for the affine map $A x + b, x ∈ X$ represented by the reset map. The vector contains the reset value for all reset dimensions and is zero for all other dimensions.

set(rm::ResetMap)

Return the set wrapped by a reset map.

Input

• rm – reset map

Output

The wrapped set.

constraints_list(rm::ResetMap)

Return a list of constraints of a polyhedral reset map.

Input

• rm – reset map of a polyhedron

Output

A list of constraints of the reset map.

Notes

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

Algorithm

If the set rm.X is hyperrectangular, we iterate through all dimensions. For each reset we construct the corresponding (flat) constraints, and in the other dimensions we construct the corresponding constraints of the underlying set.

For more general sets, we fall back to constraints_list of a LinearMap of the A-matrix in the affine-map view of a reset map. Each reset dimension $i$ is projected to zero, expressed by two constraints for each reset dimension. Then it remains to shift these constraints to the new value.

For instance, if the dimension $5$ was reset to $4$, then there will be constraints $x₅ ≤ 0$ and $-x₅ ≤ 0$. We then modify the right-hand side of these constraints to $x₅ ≤ 4$ and $-x₅ ≤ -4$, respectively.