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: [math]\displaystyle{ f(\mathbf{x}), \quad \mathbf{x}=(x_1,x_2,...,x_d)  }[/math];
    2) Generate an initial population of fireflies [math]\displaystyle{  \mathbf{x}_i \quad (i=1,2,\dots,n) }[/math];.
    3) Formulate light intensity I so that it is associated with [math]\displaystyle{ f(\mathbf{x}) }[/math]
       (for example, for maximization problems, [math]\displaystyle{ I \propto f(\mathbf{x}) }[/math] or simply [math]\displaystyle{ I=f(\mathbf{x}) }[/math];)
    4) Define absorption coefficient γ

    while (t < MaxGeneration)
        for i = 1 : n (all n fireflies)
            for j = 1 : i (n fireflies)
                if ([math]\displaystyle{ I_j\gt I_i  }[/math]),
                    Vary attractiveness with distance r via [math]\displaystyle{  \exp(-\gamma \; r)  }[/math];
                    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 [math]\displaystyle{ \mathbf{x}_i }[/math] and [math]\displaystyle{ \mathbf{x}_j }[/math] is

[math]\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 }[/math]

where [math]\displaystyle{ \alpha_t }[/math] is a parameter controlling the step size, while [math]\displaystyle{ \boldsymbol{\epsilon}_t }[/math] is a vector drawn from a Gaussian or other distribution.zae

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