# Polyhedron in constraint representation (HPolyhedron)

`LazySets.HPolyhedronModule.HPolyhedron`

— Type`HPolyhedron{N, VN<:AbstractVector{N}} <: AbstractPolyhedron{N}`

Type that represents a convex polyhedron in constraint representation, that is, a finite intersection of half-spaces,

\[P = ⋂_{i = 1}^m H_i,\]

where each $H_i = \{x ∈ ℝ^n : a_i^T x ≤ b_i \}$ is a half-space, $a_i ∈ ℝ^n$ is the normal vector of the $i$-th half-space and $b_i$ is the displacement. The set $P$ may or may not be bounded (see also `HPolytope`

, which assumes boundedness).

**Fields**

`constraints`

– vector of linear constraints

## Conversion

`Base.convert`

— Method`convert(::Type{HPolyhedron}, X::LazySet)`

Convert a polyhedral set to a polyhedron in constraint representation.

**Input**

`HPolyhedron`

– target type`X`

– polyhedral set

**Output**

The given set represented as a polyhedron in constraint representation.

**Algorithm**

This method uses `constraints_list`

.

`Base.convert`

— Method` convert(::Type{HPolyhedron}, P::HRep)`

Convert an `HRep`

polyhedron from `Polyhedra.jl`

to a polyhedron in constraint representation .

**Input**

`HPolyhedron`

– target type`P`

–`HRep`

polyhedron

**Output**

An `HPolyhedron`

.

## Operations

The following methods are shared between `HPolytope`

and `HPolyhedron`

.

`LazySets.API.constraints_list`

— Method`constraints_list(P::HPoly)`

Return the list of constraints defining a polyhedron in constraint representation.

**Input**

`P`

– polyhedron in constraint representation

**Output**

The list of constraints of the polyhedron.

`LazySets.API.dim`

— Method`dim(P::HPoly)`

Return the dimension of a polyhedron in constraint representation.

**Input**

`P`

– polyhedron in constraint representation

**Output**

The ambient dimension of the polyhedron in constraint representation. If it has no constraints, the result is $-1$.

`LinearAlgebra.normalize`

— Method`normalize(P::HPoly{N}, p::Real=N(2)) where {N}`

Normalize a polyhedron in constraint representation.

**Input**

`P`

– polyhedron in constraint representation`p`

– (optional, default:`2`

) norm

**Output**

A new polyhedron in constraint representation whose normal directions $a_i$ are normalized, i.e., such that $‖a_i‖_p = 1$ holds.

`LazySets.remove_redundant_constraints`

— Method`remove_redundant_constraints(P::HPoly{N}; [backend]=nothing) where {N}`

Remove the redundant constraints in a polyhedron in constraint representation.

**Input**

`P`

– polyhedron`backend`

– (optional, default:`nothing`

) the backend used to solve the linear program

**Output**

A polyhedron equivalent to `P`

but with no redundant constraints, or an empty set if `P`

is detected to be empty, which may happen if the constraints are infeasible.

**Notes**

If `backend`

is `nothing`

, it defaults to `default_lp_solver(N)`

.

**Algorithm**

See `remove_redundant_constraints!(::AbstractVector{<:HalfSpace})`

for details.

`LazySets.remove_redundant_constraints!`

— Method`remove_redundant_constraints!(P::HPoly{N}; [backend]=nothing) where {N}`

Remove the redundant constraints of a polyhedron in constraint representation; the polyhedron is updated in-place.

**Input**

`P`

– polyhedron`backend`

– (optional, default:`nothing`

) the backend used to solve the linear program

**Output**

`true`

if the method was successful and the polyhedron `P`

is modified by removing its redundant constraints, and `false`

if `P`

is detected to be empty, which may happen if the constraints are infeasible.

**Notes**

If `backend`

is `nothing`

, it defaults to `default_lp_solver(N)`

.

**Algorithm**

See `remove_redundant_constraints!(::AbstractVector{<:HalfSpace})`

for details.

`LazySets.tohrep`

— Method`tohrep(P::HPoly)`

Return a constraint representation of the given polyhedron in constraint representation (no-op).

**Input**

`P`

– polyhedron in constraint representation

**Output**

The same polyhedron instance.

`LazySets.tovrep`

— Method`tovrep(P::HPoly; [backend]=default_polyhedra_backend(P))`

Transform a polytope in constraint representation to a polytope in vertex representation.

**Input**

`P`

– polytope in constraint representation`backend`

– (optional, default:`default_polyhedra_backend(P)`

) the backend for polyhedral computations

**Output**

A `VPolytope`

which is a vertex representation of the given polytope in constraint representation.

**Notes**

The conversion may not preserve the numeric type (e.g., with `N == Float32`

) depending on the backend. For further information on the supported backends see Polyhedra's documentation.

`LazySets.addconstraint!`

— Method`addconstraint!(P::HPoly, constraint::HalfSpace)`

Add a linear constraint to a polyhedron in constraint representation.

**Input**

`P`

– polyhedron in constraint representation`constraint`

– linear constraint to add

**Notes**

It is left to the user to guarantee that the dimension of all linear constraints is the same.

`LazySets.API.ρ`

— Method```
ρ(d::AbstractVector{M}, P::HPoly{N};
solver=default_lp_solver(M, N)) where {M, N}
```

Evaluate the support function of a polyhedron in constraint representation in a given direction.

**Input**

`d`

– direction`P`

– polyhedron in constraint representation`solver`

– (optional, default:`default_lp_solver(M, N)`

) the backend used to solve the linear program

**Output**

The evaluation of the support function for the polyhedron. If a polytope is unbounded in the given direction, we throw an error. If a polyhedron is unbounded in the given direction, the result is `Inf`

.

`LazySets.API.σ`

— Method```
σ(d::AbstractVector{M}, P::HPoly{N};
solver=default_lp_solver(M, N) where {M, N}
```

Return a support vector of a polyhedron in constraint representation in a given direction.

**Input**

`d`

– direction`P`

– polyhedron in constraint representation`solver`

– (optional, default:`default_lp_solver(M, N)`

) the backend used to solve the linear program

**Output**

The support vector in the given direction. If a polytope is unbounded in the given direction, we throw an error. If a polyhedron is unbounded in the given direction, the result contains `±Inf`

entries.

`LazySets.API.translate`

— Method`translate(P::HPoly, v::AbstractVector; [share]::Bool=false)`

Translate (i.e., shift) a polyhedron in constraint representation by a given vector.

**Input**

`P`

– polyhedron in constraint representation`v`

– translation vector`share`

– (optional, default:`false`

) flag for sharing unmodified parts of the original set representation

**Output**

A translated polyhedron in constraint representation.

**Notes**

The normal vectors of the constraints (vector `a`

in `a⋅x ≤ b`

) are shared with the original constraints if `share == true`

.

**Algorithm**

We translate every constraint.

The following methods are specific to `HPolyhedron`

.

`Base.rand`

— Method```
rand(::Type{HPolyhedron}; [N]::Type{<:Real}=Float64, [dim]::Int=2,
[rng]::AbstractRNG=GLOBAL_RNG, [seed]::Union{Int, Nothing}=nothing)
```

Create a random polyhedron.

**Input**

`HPolyhedron`

– type for dispatch`N`

– (optional, default:`Float64`

) numeric type`dim`

– (optional, default: 2) dimension (is ignored)`rng`

– (optional, default:`GLOBAL_RNG`

) random number generator`seed`

– (optional, default:`nothing`

) seed for reseeding

**Output**

A random polyhedron.

**Algorithm**

We first create a random polytope and then for each constraint randomly (50%) decide whether to include it.

Inherited from `LazySet`

:

`diameter`

`high`

`isempty`

`low`

`norm`

`radius`

`reflect`

`singleton_list`

- [
`linear_map`

](@ref linear_map(::AbstractMatrix, ::LazySet)

Inherited from `AbstractPolyhedron`

: