Consensus.jl
Consensus.jl is a lightweight, gradientfree, stochastic optimisation package for Julia. It uses ConsensusBased Optimisation (CBO), a flavour of Particle Swarm Optimisation (PSO) first introduced by R. Pinnau, C. Totzeck, O. Tse, and S. Martin (2017). This is a method of global optimisation particularly suited for rough functions, where gradient descent would fail. It is also useful for optimisation in higher dimensions.
This package was created and is developed by Dr Rafael Bailo.
Usage
The basic command of the library is minimise(f, x0)
, where f
is the function you want to minimise, and x0
is an initial guess. It returns an approximation of the point x
that minimises f
.
You have two options to define the objective function:
x
is of typeReal
, andf
is defined asf(x::Real) = ...
.x
is of typeAbstractVector{<:Real}
, andf
is defined asf(x::AbstractVector{<:Real}) = ...
.
A trivial example
We can demonstrate the functionality of the library by minimising the function
using Consensus
f(x) = x^2;
x0 = 1;
x = minimise(f, x0)
to obtain
julia> x
1element Vector{Float64}:
0.08057420724239409
Your x
may vary, since the method is stochastic. The answer should be close, but not exactly equal, to zero.
Behind the scenes, Consensus.jl is running the algorithm using N = 50
particles per realisation. It runs the M = 100
realisations, and returns the averaged result. If you want to parallelise these runs, simply start julia with multiple threads, e.g.:
$ julia threads 4
Consensus.jl will then automatically parallelise the optimisation. This is thanks to the functionality of StochasticDiffEq.jl, which is used under the hood to implement the algorithm.
Advanced options
There are several parameters that can be customised. The most important are:

N
: the number of particles per realisation. 
M
: the number of realisations, whose results are averaged in the end. 
T
: the run time of each realisation. The longer this is, the better the results, but the longer you have to wait for them. 
Δt
: the discretisation step of the realisations. Smaller is more accurate, but slower. If the optimisation fails (returnsInf
orNaN
), making this smaller is likely to help. 
R
: the radius of the initial sampling area, which is centred around your intiial guessx0
. 
α
: the exponential weight. The higher this is, the better the results, but you might need to decreaseΔt
ifα
is too large.
We can run the previous example with custom parameters by calling
julia> x2 = minimise(f, x0, N = 30, M = 100, T = 10, Δt = 0.5, R = 2, α = 500)
1element Vector{Float64}:
0.0017988128895332278
For the other parameters, please refer to the paper of R. Pinnau, C. Totzeck, O. Tse, and S. Martin (2017). You can see the default values of the parameters by evaluating Consensus.DEFAULT_OPTIONS
.
Nontrivial examples
Since CBO is not a gradient method, it will perform well on rough functions. Consensus.jl implements two wellknown test cases in any number of dimensions:
 The Ackley function.
 The Rastrigin function.
We can minimise the Ackley function in two dimensions, starting near the point
julia> x3 = minimise(AckleyFunction, [1, 1])
2element Vector{Float64}:
0.0024744433653736513
0.030533227060295706
We can also minimise the Rastrigin function in five dimensions, starting at a random point, with more realisations, and with a larger radius, by running
julia> x4 = minimise(RastriginFunction, rand(5), M = 200, R = 5)
5element Vector{Float64}:
0.11973689657393186
0.07882427348951951
0.18515501300052115
0.06532360247574359
0.13132340855939928
Auxiliary commands
There is a maximise(f, x0)
method, which simply minimises the function g(x) = f(x)
. Also, if you're that way inclined, you can call minimize(f, x0)
and maximize(f, x0)
, in the American spelling.