Course Work

Firefly algorithm

Nature inspired algorithms provide interesting ways to optimise complex functions. In one of my courses, Scientific Computing, we were given a function with no analytically known solution and asked to optimise it with the Firefly algorithm.

The Firefly algorithm was proposed by Xin-She Yang in the early 2000’s.

The metaphor

The algorithm is based on the metaphor of a firefly swarm, on the following assumptions:

  1. All flies are unisexual and therefore can be attracted to all other flies.
  2. The attractiveness of a firefly is proportional to its brightness – brightness decreases with distance, so flies are attracted to each other based on an apparent attractiveness.
  3. If a firefly is not attracted to other fireflies during one iteration, it will move randomly.

The brightness of a firefly is associated with the function value.

The pseudocode of the algorithm can thus be described as:

  1. Initiate the objective function f(x)
  2. Generate a population of fireflies
  3. Formulate their intensities, I, associated with the objective function f(x)
  4. Define an absorption coefficient (that decreases the intensity over distance)
  5. while criteria is not met
  6. loop i over number of fireflies
  7. loop j over the rest of the fireflies (from i+1)
  8. calculate two flies’ apparent brightness
  9. compare two flies’ brightness and see which attracts the other
  10. if fly i is brighter than fly j, fly j moves towards i
  11. otherwise fly i will move towards fly j
  12. find the most attractive firefly and repeat until some_criteria is met

Objective functions

These functions were simply provided by one of the lecturers, and I have not been able to find documentation for them. The first, xin_she_yang, has an analytically known solution, while xin_she_yang2 does not. This makes them good for testing – implementing the first function to see if you can find a solution, and then testing your algorithm on the second function.


from numpy import array
def xin_she_yang(x):
# xin_she_young:  The Xin She Young multi-dimensional 
# function, which is notoriously difficult to find the 
# global minimum of.  This (original) version has the 
# analytically know solution (easy to see by inspection)
# x(i)=1 for all i.
    rn=array([0.8444218515250481, 0.7579544029403025, \
     0.420571580830845, 0.25891675029296335, \
     0.5112747213686085, 0.4049341374504143, \
     0.7837985890347726, 0.30331272607892745, \
     0.4765969541523558, 0.5833820394550312, \
     0.9081128851953352, 0.5046868558173903, \
     0.28183784439970383, 0.7558042041572239, \
     0.6183689966753316, 0.25050634136244054, \
     0.9097462559682401, 0.9827854760376531])

    f=0
    for i in range(0,len(x)-1):
        f=f+(1-x[i])**2+100*rn[i]*(x[i+1]-x[i]**2)**2;
    return f


from numpy import array
def xin_she_yang2(x):
# xin_she_young2:  A variant of the Xin She Young multi- 
# dimensional function, which is notoriously difficult to  
# find the global minimum of.  This variant does NOT have 
# and analytically know solution 
    rnd=array([[0.8444218515250481, 0.7579544029403025, \
     0.420571580830845, 0.25891675029296335, \
     0.5112747213686085, 0.4049341374504143, \
     0.7837985890347726, 0.30331272607892745, \
     0.4765969541523558, 0.5833820394550312], \
     [0.9081128851953352, 0.5046868558173903, \
     0.28183784439970383, 0.7558042041572239, \
     0.6183689966753316, 0.25050634136244054, \
     0.9097462559682401, 0.9827854760376531, \
     0.5468815192049839, 0.9575068354342976]])

    f=0
    for i in range(0,len(x)-1):
        f=f+(1-x[i])**2+100*rnd[0,i]*(x[i+1]-rnd[1,i]*x[i]**2)**2;
    return f

Implementation of the algorithm

Start by importing the necessary packages:

 

import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import fmin

exp = np.exp
sqrt = np.sqrt
ones = np.ones

Then, I found it useful to make a script that could simply implement both functions using the value of a single parameter. I use the parameter case to indicate this, and use an if-statement to initialise the object functions.

case=0 #Case=1 works with the original case, other values means we work with the modified case.

if case==1:
from xin_she_yang import xin_she_yang #Import test case
f = xin_she_yang #Define function
x0=0.5*ones(10)
xmin = fmin(func=xin_she_yang, x0=x0,xtol=1e-6,maxiter=10000,maxfun=10000,disp=0) #Find firefly in minimum
f_min=f(xmin) #Save minimum for reference
print('Working with the original case') #Print to monitor which case we work with
else:
from xin_she_yang2 import xin_she_yang2 #Import modified case
f=xin_she_yang2 #Define function
x0=0.5*ones(10)
xmin=fmin(func=xin_she_yang2, x0=x0,xtol=1e-6,maxiter=10000,maxfun=10000,disp=0) #Find firefly minimum
f_min=f(xmin) #Save minimum for reference
print('Working with the modified case') #Print to monitor which case we work with

it = 0 #Iteration counter, starts at zero
beta0 = 1 #This value is defined in the assignment text
gamma = 0 #Same with this

alpha=2 #This is Elas opt. alpha value
delta=.9985 #This is Elas opt. value

number_of_fireflies=20

Next, I initialise a generation of 20 fireflies spread out over the function area.

search_area=np.linspace(-alpha,alpha,number_of_fireflies) #Linearly space 20 
X = [search_area[x]*np.ones([10,1]) for x in range(0,number_of_fireflies)] #Define a set of fireflies. These flies are randomly started.

Now, I formulate an intensity for each firefly:

I0=np.zeros([20,1]) #Preallocate space for the intensity at 'its own place'.
for i in range(len(X)): #This function calculates the brightness at 'its own place', I_0.
    x=X[i] #Note: It has proved important to "pick out" an and define i'th vector from the list as x rather than simply call it directly from the list.
    I0[i]=1/(f(x)+1)  #The fly should be "brighter"/more attractive, as the function minimises.

Next, I define a couple of functions to use in my while loop. These functions calculate the distance between fireflies, updates the brightness of a firefly when it has taken a new position, calculates the apparent brightness of a firefly a certain distance away.

def r(i,j): #This function calculates the distance between fireflies.
    xi=X[i]
    xj=X[j]
    r=0
    
    for i in range(len(xi)):
        r+=(xj[i]-xi[i])**2

    r=sqrt(r)
    return r

def I_0(i): #Function to uodate the brightness of a firefly
    x=X[i]
    Iny0=1/(f(x)+1)
    return Iny0

def I(x): #This function calculates the apparent brightness.
    I=x*exp(-gamma*(r(i,j)**2)) #This expression is taken from "Firefly Algorithms for Multimodal Optimization", eq. (3).
    return I

I want my algorithm to minimise the relative error of the function using the fmin function from scipy.optimize to provide an ‘upper limit’ to my problem. Therefore, I initialise a couple of variables to keep track of during my while loop.

best=[]
best_function_value=f(X[np.argmax(I0)])
best.append(best_function_value)
rel_err=1
rel=[]

I then setup a while loop to optimise the function based on the pseudocode I wrote above. I like to keep track of how my algorithm is progressing, so I will print out the current optimal value for each 500th iteration. This way, I will quickly notice, if anything isn’t working.

print('Starting while loop')

while rel_err>10**(-6): #While the relative error is too large
    if it%500 == 0: #Monitor progress
        print(it,'iterations and current best minimum at',best_function_value)
    for i in range(len(X)): #Loop over the number of fireflies.
        for j in range(i+1,len(X)): #Loop over the number of fireflies (but ignore those flies, we've already compared)
            Ii=I(I0[i]) #The apperent intensity for the i'th firefly.
            Ij=I(I0[j]) #The apparent intensity for the j'th firefly.
            
            xi=X[i] #Position of the i'th firefly.
            xj=X[j] #Position of the j'th firefly.
            
            if Ij>Ii: #If the one we compare with is more attractive than the i'th firefly, move the i'th firefly. Here, change the position of the i'th firefly
                p1=(xj-xi)
                p2=alpha*(np.random.rand(10,1)-0.5)
                X[i]=xi+p1-p2 #Update i's position
                I0[i]=I_0(i) #Update i's intensity at 'its own place'

            else:
                p1=(xi-xj)
                p2=alpha*(np.random.rand(10,1)-0.5)
                X[j]=xj+p1-p2 #Update j's position
                I0[j]=I_0(j) #Update j's intensity at 'its own place'
    
    alpha=alpha*delta #Update alpha
    it=it+1 #Update counter
    
    best_function_value=f(X[np.argmax(I0)])
    best.append(best_function_value)
    
    rel_err=(best_function_value - f_min)/f_min
    rel.append(rel_err)

Finally, I make a plot of the progression per iteration to evaluate the efficiency of my implementation. For the modified case, this is how the algorithm progresses:

plt.plot(axe,best)
plt.xlabel('#iterations')
plt.ylabel('Best function value')
plt.title('Firefly optimisation')
plt.tight_layout()

There are minor differences between the original Firefly algorithm and my implementation, but I used the overall analogy of a firefly swarm to actually find good solution for the 

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