Study of Parametric Optimization of the Cuckoo Search Algorithm

Arijit Mallick1, Sourya Roy2, Sheli Sinha Chaudhuri3

1,2Department of Electronics and Instrumentation Engineering 3 Department of Electronics and Telecommunication

Engineering Jadavpur University, Kolkata, India

Sangita Roy Department of Electronics and Communication Engineering

Narula Institute of Technology, WBUT Kolkata, India

aribryan@gmail.com souroy099@gmail.com

Abstract—Cuckoo search (CS) is one of the latest and most

efficient optimization techniques developed so far. Several attempts have been made in past in order to improve the efficiency of CS algorithm. In this paper we have tried to exploit several parameters of the CS algorithm in order to increase its efficiency. Cuckoo search is a metaheuristic optimization technique. Its parameters involve the Levy distribution factor beta ( ) and the probability factor (P) with which solutions are replaced with new solutions. Hence for optimum values of the aforesaid parameters, efficiency of CS algorithm can be improved and can be used to solve optimization problems.

Keywords— Cuckoo search; optimization; metaheuristic ; Levy distribution

I. INTRODUCTION The act of obtaining the best result under given

circumstances of a definite problem statement is known as optimization [1]. Classical methods of optimization are generally not used for their impracticality in complicated real life situation. They are generally deterministic in nature. Modern day optimization techniques are a hybrid of deterministic and stochastic algorithm [2]. The goal is to develop more proficient and better optimization techniques that might involve more and more sophistication of algorithm. Nature has remained a great source of inspiration to mankind to develop novel methods of optimization techniques. Bio mimicking of several natural events have given birth to modern day metaheuristic algorithm. The main essence of metaheuristic algorithm is to exploit the method of trial and error. Genetic Algorithm [3] was inspired by the Darwinian theory of selection of the fittest. Ant Colony Optimization [4], Particle Swarm Optimization [5] etc; as the names suggest, are all inspired by natural stochastic phenomenon. The latest of them being Cuckoo Search (CS) Optimization [6]. In this paper we have attempted to find out the special constraints and parametric conditions in which the cuckoo search algorithm works in most efficient manner. Parameters involve the observation of special distribution factor beta ( ) while varying the probability factor. As we will see, for a given probability factor, we will obtain a definite value of for which time of operation will be minimum. Hence point of maximum efficiency will be defined with respect to time constraint.

II. LITERATURE SURVEY CS is comparatively new algorithm. Due to its quick rate of

convergence and fewer parameters to adjust, lot of works is developing recently [21]. Levy stable characteristic function was first observed by Monin 1955 and its distribution and mathematical details are studied [11].CS has been used for solving structural optimization tasks [12]. Springs and welded beam structures are optimally designed using cuckoo search [13].

III. CUCKOO SEARCH VIA LEVY FLIGHT

A. Breeding behaviour CS algorithm is inspired from the breeding behaviour of

cuckoos. Some species of cuckoo lay eggs in a host bird’s nest. In some cases, cuckoos remove the eggs of host bird in order to increase the survival probability of its own eggs. If a host bird somehow detects this anomaly, it either abandons its own nest or throws away the cuckoo’s eggs. Studies have shown that in some cases, cuckoo eggs seems similar to that of the host’s eggs and even some species of cuckoo chicks can mimic the voice of the host bird’s chicks. This increases the probability of survival of cuckoo chicks [7].

B. Levy flight Mathematically speaking, a Levy flight is a special case

Random Walk which has a heavy tailed probability distribution. Studies have shown that some species of insects, even animals such as spider monkeys show movement in Levy Flight in order to find randomly distributed food sources[8][9]. Next step of a foraging animal primarily depends on its present state and the next step length. Hence the Random walk whose step length is drawn from Levy distribution is termed as Levy flight.

C. Mechanism: Each egg in a nest represents a solution while a cuckoo egg

represents a new solution. The target is to replace the host eggs that are replacing old solution with potentially better and new solution. In our algorithm we have used a multiple number of eggs in a nest. Hence for generating potentially better solution,

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we perform a Levy Flight. Let the new solution be xi (t+1) which replaces xi (t) then

(1)

Where ‘ ’ is the step length and is generally drawn from

Levy distribution. The product means entry wise multiplications [2].

IV. PARAMETRIC EVALUATION For optimal performance of the CS algorithm, variation of

with respect to probability factor ‘P’ is observed, after a brief description of the aforesaid parameters.

A. Probability factor (P) For a fixed number of host nests let us assume that a host

bird can discover a cuckoo nest with probability ‘P’. If by any chance the host bird detects the cuckoo egg in its nest, it will either discard the alien egg or will abandon its nest in order to build a new nest. In other words fraction P of the given N nests will be replaced by new nests or random solutions. In case of a maximization task, fitness function might be proportional to the objective function itself [2].

B. Levy distribution factor -Beta ( ) A Levy flight is a random walk in which the step-lengths

have a probability distribution that is heavy-tailed. In general, Levy distribution should be de ned in terms of Fourier transform

Where is a scale parameter.

For a special case, when =1; the above equation represents Cauchy distribution.

V. SIMULATIONS AND RESULTS

A. Procedure In order to evaluate the efficiency of the CS algorithm, we

took 19 distinct and equally spaced values of P (from .05 to .95).For each value of P, we varied beta (within 0 to 2) and simulated the CS, to measure the time requirement at each beta. Three common stochastic test functions [4.2] were used for our simulation purpose. Because CS is a random search algorithm, to get information closer to the accurate values, for same value of P, we ran the search algorithm 10 times and computed the average value of the time required for optimization. For a

particular stochastic function, a fixed search terminating condition or tolerance value is used for each simulation so that the quality of solutions remain same in each run.

B. Stochastic test functions There are several stochastic test function which are used as

benchmark for testing the efficiency of any optimization algorithm [10]. In our simulations, we have used the following test functions:

Rastrigin’s test function

2

1( ) 10 [x 10cos(2 )]

d

i ii

f x d xπ=

= + − (3)

, has a global minimum at, = (0, 0… 0) and its global minimum value is zero, where i =1, 2… d.

Fig. 1. 3-D plot of Rastrigin’s function

Stochastic sphere function

2

1( ) [ 1] , [ 5.12,5.12]

d

i ii

f x x x=

= − ∈ − (4)

, whose global minimum value is zero and occurs at = (1, 1… 1) and its global minimum value is zero, where i =1, 2,…,d.

2014 International Conference on Control, Instrumentation, Energy & Communication(CIEC) 768

Fig. 2. 3-D plot of Sphere function

Griewangk’s test function

2

1 1

1( ) cos( ) 14000

ddi

ii i

xf x x

i= =

= − +∏ (5)

, has global minimum of zero at = (0, 0… 0) for all .

Fig. 3. 3-D plot of Griewangk’s function

C. Results Results from each function optimization case have been

shown (Fig. 4 to Fig. 15) separately below. Though we have simulated CS for 19 distinct values of P, we have presented here results for only 4 values of P (mentioned below) due to space insufficiency. Results for other values of P are similar in nature.

1) Results from optimization of Rastrigin’s function (d=10)

Fig. 4. Time vs. Beta plot at P=0.15

Fig. 5. Time vs. Beta plot at P=0.4

Fig. 6. Time vs. Beta plot at P=0.65

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Fig. 7. Time vs. Beta plot at P=0.90

2) Results from optimization of Sphere function (d=15)

Fig. 8. Time vs. Beta plot at P=0.15

Fig. 9. Time vs Beta plot at P=0.4

Fig. 10. Time vs. Beta plot at P=0.65

Fig. 11. . Time vs Beta plot at P=0.9

3) Results from optimization of Griewangk’s function (d=10)

Fig. 12. Time vs. Beta plot at P=0.15

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Fig. 13. Time vs. Beta plot at P=0.4

Fig. 14. Time vs. Beta plot at P=.65

Fig. 15. Time vs. Beta plot at P=0.9

VI. DISCUSSIONS A brief comparison of Cuckoo Search algorithm has been

made with respect to other popular optimization algorithms. Convergence has been denoted by the variable (fmin) in Table 1. It can be easily concluded that though time of execution is

faster is Genetic Algorithm (GA), convergence efficiency takes its toll. Whereas Particle Swarm Optimization (PSO) has least time efficiency (T). In other words, overall performance of tuned CSO algorithm is more than impressive. Figures of fitness vs. iterations for each function optimization are shown in Fig.16-18 as example of saturation.

TABLE I. COMPARISON OF DIFFERENT ALGORITHMS

Functions Rastrigin’s function

Griewangk’s function

Sphere function Algorithm

CSO(tuned)

T=1.42secs fmin= 7.6779e-006

T=.5 sec, fmin=8.8415e-006

T=.35sec fmin=9.2921e-006

PSO

T=2.5 sec, fmin=2.89

T=4.75 sec, fmin= 6.69e-006

T=2.15 sec, fmin=2.16e-6

GA

T=.13 sec Fmin=3.16

T=.12 sec ,Fmin=0.0056

T=0.13 sec fmin=.052

0 2000 4000 6000 8000 10000 120000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

No. of Iterations

Fitn

ess

Val

ue

Fig. 16. Fitness Value vs. no. of Iterations for Rastringin’s Function

0 2000 4000 6000 8000 10000 120000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

No. of Iterations

Fitn

ess

Val

ue

Fig. 17. Fitness Value vs. no. of Iterations for Griewangk’s Function

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0 2000 4000 6000 8000 10000 12000 14000 16000 180000

10

20

30

40

50

60

70

80

90

Fitn

ess

Val

ue

No. of Iterations

Fig. 18. Fitness Value vs. no. of Iterations for Sphere’s Function

VII. CONCLUSIONS In this paper our objective was to find a smaller range of

beta, for a given value of P, for which the time constraint is minimum. Results show that, for a specific P, there is range of Beta, where optimization process is faster. In case of Rastrigin’s function optimization, we have obtained a very sharp peak point of beta where optimization process is fastest. Since it is a stochastic function, we should take a range of beta values instead of a single value. This range of beta values (0.4 to 1; for all function) is significantly smaller than initial parameters (0 to 2). Hence according to aforesaid simulation results the performance of CS algorithm has been significantly improved. Using these optimum parametric values we can solve optimization problems more efficiently with the help CS algorithm.

ACKNOWLEDGEMENT Matlab program by Xin-she Yang

(http://www.mathworks.in/matlabcentral/fileexchange/29809-cuckoo-search-cs-algorithm) for Cuckoo search algorithm has been used, and edited in some cases. Genetic Algorithm toolbox of MATLAB has been utilized in order to make a comparison study.

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