Firefly algorithm

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

Metaphor

The primary purpose for a firefly's flash is to act as a signal system to attract other fireflies. Xin-She Yang formulated this firefly algorithm by assuming:

1. All fireflies are unisexual, so that any individual firefly will be attracted to all other fireflies;
2. Attractiveness is proportional to their brightness, and for any two fireflies, the less bright one will be attracted by (and thus move towards) the brighter one; however, the intensity (apparent brightness) decrease as their mutual distance increases;
3. If there are no fireflies brighter than a given firefly, it will move randomly.

The brightness should be associated with the objective function.

Algorithm

In pseudocode the algorithm can be stated as:

Begin
   1) Objective function: $f(\mathbf {x} ),\quad \mathbf {x} =(x_{1},x_{2},...,x_{d})$;
   2) Generate an initial population of fireflies $\mathbf {x} _{i}\quad (i=1,2,\dots ,n)$;.
   3) Formulate light intensity I 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 γ
 
   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

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

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

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

It can be shown that the limiting case $\gamma \rightarrow 0$ 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 an elaborate metaphor. 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).