Firefly algorithm

In pseudocode the algorithm can be stated as:

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

    While (t < MaxGeneration)
        for i = 1 : n (all n fireflies)
            for j = 1 : i (n fireflies)
                if (${\displaystyle I_{j}>I_{i}}$),
                    Vary attractiveness with distance r via ${\displaystyle \exp(-\gamma \;r)}$;
                    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

    Post-processing the results and visualization;

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 ${\displaystyle \mathbf {x} _{i}}$ and ${\displaystyle \mathbf {x} _{j}}$ is

${\displaystyle \mathbf {x} _{i}^{t+1}=\mathbf {x} _{i}^{t}+\beta \exp[-\gamma r_{ij}^{2}](\mathbf {x} _{j}^{t}-\mathbf {x} _{i}^{t})+\alpha _{t}{\boldsymbol {\epsilon }}_{t}}$

where ${\displaystyle \alpha _{t}}$ is a parameter controlling the step size, while ${\displaystyle {\boldsymbol {\epsilon }}_{t}}$ is a vector drawn from a Gaussian or other distribution.

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

Nature-inspired metaheuristics in general have attracted criticism in the research community for hiding their lack of novelty behind an elaborate metaphor. The firefly algorithm has been criticized as differing from the well-established particle swarm optimization only in a negligible way.[2][3][4]

• Swarm intelligence

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