p-norm ball (Ballp)

Ballp{N<:AbstractFloat, VN<:AbstractVector{N}} <: AbstractBallp{N}

Type that represents a ball in the p-norm, for $1 ≤ p ≤ ∞$.

It is defined as the set

\[\mathcal{B}_p^n(c, r) = \{ x ∈ ℝ^n : ‖ x - c ‖_p ≤ r \},\]

where $c ∈ ℝ^n$ is its center and $r ∈ ℝ_+$ its radius. Here $‖ ⋅ ‖_p$ for $1 ≤ p ≤ ∞$ denotes the vector $p$-norm, defined as $‖ x ‖_p = \left( ∑\limits_{i=1}^n |x_i|^p \right)^{1/p}$ for any $x ∈ ℝ^n$.

Fields

• p – norm as a real scalar
• center – center of the ball as a real vector
• radius – radius of the ball as a scalar ($≥ 0$)

Notes

The special cases $p=1$, $p=2$ and $p=∞$ fall back to the specialized types Ball1, Ball2 and BallInf, respectively.

Examples

A five-dimensional ball in the $p=3/2$ norm centered at the origin of radius 0.5:

julia> B = Ballp(3/2, zeros(5), 0.5)
Ballp{Float64, Vector{Float64}}(1.5, [0.0, 0.0, 0.0, 0.0, 0.0], 0.5)

julia> dim(B)
5

We evaluate the support vector in direction $[1,2,…,5]$:

julia> σ([1.0, 2, 3, 4, 5], B)
5-element Vector{Float64}:
 0.013516004434607558
 0.05406401773843023
 0.12164403991146802
 0.21625607095372093
 0.33790011086518895

center(B::Ballp)

Return the center of a ball in the p-norm.

Input

• B – ball in the p-norm

Output

The center of the ball in the p-norm.

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

Create a random ball in the p-norm.

Input

• Ballp – type for dispatch
• N – (optional, default: Float64) numeric type
• dim – (optional, default: 2) dimension
• rng – (optional, default: GLOBAL_RNG) random number generator
• seed – (optional, default: nothing) seed for reseeding

Output

A random ball in the p-norm.

Algorithm

The center and radius are normally distributed with mean 0 and standard deviation 1. Additionally, the radius is nonnegative. The p-norm is a normally distributed number ≥ 1 with mean 1 and standard deviation 1.

reflect(B::Ballp)

Concrete reflection of a ball in the p-norm B, resulting in the reflected set -B.

Input

• B – ball in the p-norm

Output

The Ballp representing -B.

Algorithm

If $B$ has center $c$ and radius $r$, then $-B$ has center $-c$ and radius $r$. The norm remains the same.

translate!(B::Ballp, v::AbstractVector)

Translate (i.e., shift) a ball in the p-norm by a given vector, in-place.

Input

• B – ball in the p-norm
• v – translation vector

Output

The ball B translated by v.

Algorithm

We add the vector to the center of the ball.

Notes

See also translate(::Ballp, ::AbstractVector) for the out-of-place version.