Firefly algorithm

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 ( x ) , x = ( x 1 , x 2 , . . . , x d ) ; 2) Generate an initial population of fireflies x i ( i = 1 , 2 , … , n ) ;. 3) Formulate light intensity I so that it is associated with f ( x ) (for example, for maximization problems, I ∝ f ( x ) or simply I = f ( 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 ), Vary attractiveness with distance r via exp ⁡ ( − γ 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

The main update formula for any pair of two fireflies x i and x j is

where α t is a parameter controlling the step size, while ϵ t is a vector drawn from a Gaussian or other distribution.

It can be shown that the limiting case γ → 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.