ExtremalOptimization

for functions of continuous variables.

This package implements the basic mechanism of Extremal Optimization (τ-EO) as described in Boettcher, Stefan; Percus, Allon G. (2001-06-04). "Optimization with Extremal Dynamics". Physical Review Letters.

The only twist w.r.t. classical EO is an affine invariant update equation for the worst performing solutions,

where X₁, X₂, X₃ are chosen random inside the pool of candidate solutions, this update mechanism allows EO to work on continuous spaces, and be invariant w.r.t. affine transformations of X and monotonous tranformations of the cost function.

API:

function optimize(
    f,
    s,
    N;
    reps_per_particle = 100,
    β = 1.5,
    A = 1.0,
    atol = 0.0,
    rtol = sqrt(eps(1.0)),
    f_atol = 0.0,
    f_rtol = sqrt(eps(1.0)),
    verbose = false,
    rng = Random.GLOBAL_RNG,
    callback = state -> nothing,
)

• f : cost function to minimize, whose argument is either a scalar or a vector, must returns a scalar value.
• s : function whose input is the particle number and output is a random initial point to be ranked by f.
• N : number of particles to use, choose a number greater than d+4 where d is the number of dimensions.
• reps_per_particle : maximum number of iterations per particle.

Usage example:

using ExtremalOptimization
rosenbrock2d(x) = (x[1] - 1)^2 + 100*(x[2] - x[1]^2)^2
initpoint(i) = randn(2)
optimize(rosenbrock2d, initpoint, 20)

output

(x = [1.0000008, 1.0000016], fx = 6.50e-13, f_nevals = 1109)

as expected the algorithm has found the optimum at (1, 1), up to the specified tolerance.