The main objective is to solve the well-known problem Rosenbrock function optimization using implementation of Bat Algorithm in Python programming language.

Bat Algorithm is a metaheuristic bat-inspired algorithm introduced by Xin-She Yang in 2010. The algorithm is based on bats skill to locate preys and insects using echolocation. With this sonar, bats can fly dodging all kind of obstacles until they finally reach their prey. Going back to programming world we will define a bat in our program by the following features:

1. Position x_i : Location of the bat among the whole solutions space. This feature is also called location or simply solution.
2. Velocity v_i : Speed at which the bat flies. We will use the velocity to calculate subsequent positions of the bat.
3. Frequency f_i : Value that determines the wavelength of the pulses emitted by the bat. As higher is the frequency, then quickly is the response from where is the prey. Moreover, bats can adjust the frequency of their pulses in a range [f_min,f_max] depending on the proximity of their target.
4. Loudness A_i : Value that reflects bats pulse loudness. Loudness varies from the loudest when a bat is searching for his prey to a quieter base when he is closer to his prey.
5. Pulse rate r_i : Value that refers to the rate that a bat emits his ultrasonic pulse, when bats are flying and searching for a prey, they have lower values pulse rate. When they get closer to his prey, then they need higher values of pulse rate in order to get a great accuracy about the prey position.

Here is the pseudocode of the algorithm, implemented in BatAlgorithm.py with some minor changes whose only aim are the improvement of the algorithm performance

The following images show the equations that Bat Algorithm uses and the pseudocode step in which they are required:

• Adjust frecuency

Where beta is in range [0,1]

• Update velocity

Where x_* is the current global best position

• Update position

• Generate local solution arond the selected best solution

Where epsilon is in range [-1,1] and A^t is the average loudness of all the bats at this time step

• Increase r_i

Where gamma is a constant

• Reduce A_i

Where alpha is a constant

The Rosenbrock function is a typical test for optimize algorithms. It was introduced by Howard H. Rosenbrock in 1960. The global minimum is inside a long, narrow, parabolic shaped flat valley. To find the valley is trivial. To converge to the global minimum, however, is difficult.

The function is defined by

Usually a = 1 and b = 100. This function has a global minimum at (x,y) = (1, 1) --> f(x,y) = (0, 0) which can be generalized for N-dimensions, in this case the minimum for N-dimensions will be (x*) = (1,...,1)

The following example presents a simple use of bat algorithm. function() denotes the objective function that may be changed by the user. Control parameters should be defined within BatAlgorithm() constructor. Order of parameters is as follows: BatAlgorithm(dimensions, populationSize, generations, A, r, fMin, fMax, lower, upper, function) where:

• dimensions denotes dimension of the problem,
• populationSize denotes population size,
• generations denotes number of generations (iterations),
• A parameter denotes loudness,
• r parameter denotes pulse rate,
• fMin parameter denotes frequency minimum,
• fMax parameter denotes frequency maximum,
• lower denotes lower bound,
• upper denotes upper bound and
• function passes objective function.

from BatAlgorithm import *
from time import time

def RosenbrockFunction(dimensions, sol):
    val = 0.0
    for i in range(dimensions-1):
        val += (1 - sol[i]**2)**2 + 100*(sol[i+1]-sol[i]**2)**2
    return val

def main():
    for i in range(10):
        Algorithm = BatAlgorithm (
            dimensions=2, populationSize=40, generations=2000, 
            A=1, r=0, fMin=-1.0, fMax=10.0, lower=-10.0, upper=10.0,
            alpha=0.001, gamma=0.001, function=RosenbrockFunction
        )

        start = time()

        Algorithm.moveBats()

        end = time()

        print("Elapsed time: ", end - start)
        print("===")

if __name__ == "__main__":
    main()

Bugs and extension should be send via Github.

Carlos Freire Caballero

Yang, X.-S. "A new metaheuristic bat-inspired algorithm." Nature inspired cooperative strategies for optimization (NICSO 2010). Springer Berlin Heidelberg, 2010. 65-74.

Fister, I. Jr., Fister, I., Yang, X.-S., Fong, S., Zhuang, Y. "Bat algorithm: Recent advances." IEEE 15th International Symposium on Computational Intelligence and Informatics (CINTI), IEEE, 2014. 163-167.

https://www.researchgate.net/publication/300297266_The_bat_algorithm_and_its_parameters

https://www.researchgate.net/publication/318929033_New_binary_bat_algorithm_for_solving_0-1_knapsack_problem

https://www.researchgate.net/publication/318239029_Study_of_Parameter_Sensitivity_on_Bat_Algorithm