Firefly algorithm

The firefly algorithm (FA) was a metaheuristic algorithm, inspired by the flashing behaviour of fireflies. The primary purpose for a firefly's flash is to act as a signal system to attract other fireflies. Yang formulated this firefly algorithm by assuming:

1. All fireflies are unisex, so that one firefly will be attract to all other fireflies;
2. Attractiveness is proportional to their brightness, and for any two fireflies, the less brighter one will attract (and thus move) to the brighter one; however, the brightness can decrease as their 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.

Firefly algorithm is a nature-inspired metaheuristic optimization algorithm.

Algorithm description

The pseudo code can be summarized 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 $\gamma$
While (t<MaxGeneration)
   for i=1:n (all n fireflies);
      for j=1:n (n fireflies)
         if ($I_{j}>I_{i}$), 
          move firefly i towards j;
         end if 
      Vary attractiveness with distance r via $\exp(-\gamma \;r)$;
      Evaluate new solutions and update light intensity;
      end for j
   end for i
   Rank fireflies and find the current best;
end while
Post-processing the results and visualization;

end

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.

A simple demo Matlab code is available matlab code'

Recent studies shows that the firefly algorithm is very efficient, and could outperform other metaheuristic algorithms including particle swarm optimization.