Firefly algorithm

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 firefly algorithm (FA) somehow looks like the PSO, but it has significant differences. First, the PSO updating equations are linear, while FA has a nonlinear equation. Second, as a result of nonlinearity, FA can have rich characteristics than those arisen from a linear system in PSO. Third, due to the fact that short-distance attraction is stronger than long-distance attraction in the FA, the population in FA can automatically subdivide into subgroups or subswarms, which give rise to a multi-swarm system.

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 : n (n fireflies)
            if ($I_{j}>I_{i}$),
               move firefly i towards j;
               Vary attractiveness with distance r via $\exp(-\gamma \;r)$;
               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

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.

See also

• Swarm intelligence