Firefly algorithm

In mathematical optimization, the firefly algorithm is a metaheuristic proposed by Xin-She Yang and inspired by the flashing behavior of fireflies.

Algorithm

In pseudocode the algorithm can be stated as:

Begin
    1) Objective function: $f(\mathbf{𝐱}),\mathbf{𝐱}=(x_1,x_2,...,x_d)$;
    2) Generate an initial population of fireflies ${\mathbf{𝐱}}_i(i=1,2,\dots ,n)$;.
    3) Formulate light intensity Template:Mvar so that it is associated with $f(\mathbf{𝐱})$
       (for example, for maximization problems, $I\propto f(\mathbf{𝐱})$ or simply $I=f(\mathbf{𝐱})$;)
    4) Define absorption coefficient Template:Mvar

    while (t < MaxGeneration)
        for i = 1 : n (all n fireflies)
            for j = 1 : i (n fireflies)
                if ($I_j>I_i$),
                    Vary attractiveness with distance r via $\exp (-\gamma r)$;
                    move firefly i towards j;                
                    Evaluate new solutions and update light intensity;
                end if 
            end for j
        end for i
        Rank fireflies and find the current best;
    end while
end

Note that the number of objective function evaluations per loop is one evaluation per firefly, even though the above pseudocode suggests it is n×n. (Based on Yang's MATLAB code.) Thus the total number of objective function evaluations is (number of generations) × (number of fireflies).

The main update formula for any pair of two fireflies ${\mathbf{𝐱}}_i$ and ${\mathbf{𝐱}}_j$ is $${\mathbf{𝐱}}_i^{t+1}={\mathbf{𝐱}}_i^t+\beta \exp [-\gamma r_{ij}^2]({\mathbf{𝐱}}_j^t-{\mathbf{𝐱}}_i^t)+{\alpha }_t{\mathit{ϵ}}_t$$ where ${\alpha }_t$ is a parameter controlling the step size, while ${\mathit{ϵ}}_t$ is a vector drawn from a Gaussian or other distribution.

It can be shown that the limiting case $\gamma \to 0$ corresponds to the standard particle swarm optimization (PSO). In fact, if the inner loop (for j) is removed and the brightness $I_j$ is replaced by the current global best $g^{*}$, then FA essentially becomes the standard PSO.

Criticism

Nature-inspired metaheuristics in general have attracted criticism in the research community for hiding their lack of novelty behind metaphors. The firefly algorithm has been criticized as differing from the well-established particle swarm optimization only in a negligible way.

See also

• Swarm intelligence

External links

• [1] Files of the Matlab programs included in the book: Xin-She Yang, Nature-Inspired Metaheuristic Algorithms, Second Edition, Luniver Press, (2010).