In mathematical optimization, the firefly algorithm is a metaheuristic proposed by Xin-She Yang and inspired by the flashing behavior of fireflies.^[1]

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
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.zae

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 metaphors. 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

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