Imperialist Competitive Algorithm using

Chaos Theory for Optimization

(CICA)

Helena Bahrami Dept. of Elec., comp. & IT, Qazvin Azad University,

Qazvin,Iran Bahramihelena@yahoo.com

Karim Faez Dept. of EE., Amirkabir University of Tehran,

Tehran, Iran kfaez@aut.ac.ir

Marjan Abdechiri Dept. of Elec., comp. & IT, Qazvin Azad University,

Qazvin,Iran Marjan.Abdechiri@qiau.ac.ir

Abstract—The Imperialist Competitive Algorithm (ICA) that was recently introduced has shown its good performance in optimization problems. This novel optimization algorithm is inspired by socio-political process of imperialistic competition in the real world. In this paper a new Imperialist Competitive Algorithm using chaotic maps (CICA) is proposed. In the proposed algorithm, the chaotic maps are used to adapt the angle of colonies movement towards imperialist’s position to enhance the escaping capability from a local optima trap. The ICA is easily stuck into a local optimum when solving high-dimensional multi-model numerical optimization problems. To overcome this shortcoming, we use four different chaotic map incorporated into ICA to enhance the exploration capability. Some famous unconstraint benchmark functions are used to test the CICA performance. Simulation results show this variant can improve the performance significantly.

Keywords-Imperialist Competitive Algorithm; absorption policy; chaos theory.

I. INTRODUCTION The global optimization problem is applicable in every

field of science, engineering and business. So far, many Evolutionary Algorithms (EA) [1,2], have been proposed for solving the global optimization problem. Inspired by the natural evolution, EA analogizes the evolution process of biological population, which can adapt the changing environments to the finding of the optimum of the optimization problem through evolving a population of candidate solutions. Some of the Evolutionary Algorithms that have been proposed for optimization problem are: the Genetic Algorithm (GA) [3,2,4,5,6,7], at first proposed by Holland, in 1962 [4], Particle Swarm Optimization algorithms (PSO) [8,9] that at first proposed by Kennedy and Eberhart [8], in 1995, Simulated Annealing (SA) [10,11,12,13], Cultural Evolutionary algorithms (CE) [14,15] at first was developed by Reynolds, in the early 1990s [15] and etc. Recently, a new algorithm has been proposed by Atashpaz-Gargari and lucas [16], in 2007 that has inspired from a socio-human phenomenon. In this paper a new method using the chaos theory is proposed to adjust

the angle of colonies movement towards the imperialist’s positions. The proposed method uses some chaotic maps to generate chaotic movement angle. This chaotic movement angle, will act like the mutation operator in genetic algorithm. Using the chaos theory the semi-random variation of movement angle causes the proposed algorithm escape from the local optimums during the search process. We examined the proposed algorithm in several standard benchmark functions that usually tested in Evolutionary Algorithms. The results of applying the proposed algorithm on benchmark functions indicated that the convergence speed and the quality of obtained solution in compare with ICA, PSO using a Sugeno function as inertia weight decline curve[17] and GA algorithm has a good performance.

The simulation result show that using chaotic map instead of uniform distribution in adjusting the angle of colonies movement towards the imperialist’s position may be a possible method to improve the performance of ICA algorithm.

We organized the rest of this paper as follows. Section two, provides an introduction to ICA and Chaos theory algorithms. In section three, Chaotic Imperialistic Competitive Algorithm is proposed. Fourth section is devoted to the empirical results of proposed algorithm implementation and its compression with the results obtained by ICA, PSO and GA algorithms. The last section concludes the paper.

II. INTRODUCTION OF IMPERIALIST COMPETITIVE ALGORITHMS (ICA)

In this section, we introduce ICA algorithm and chaos

theory.

A. Imperialist Competitive Algorithm (ICA) Imperialist Competitive Algorithm (ICA) is a new

evolutionary algorithm in the Evolutionary Computation field based on the human's socio-political evolution. The

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algorithm starts with an initial random population called countries. Some of the best countries in the population selected to be the imperialists and the rest form the colonies of these imperialists. In an N dimensional optimization problem, a country is a 1 array. This array defined as below

, , … , 1

The cost of a country is found by evaluating the cost function f at the variables , , , … , . Then , , … , 2

The algorithm starts with N initial countries and the best of them (countries with minimum cost) chosen as the imperialists. The remaining countries are colonies that each belong to an empire. The initial colonies belong to imperialists in convenience with their powers. To distribute the colonies among imperialists proportionally, the normalized cost of an imperialist is defined as follow

3

Where, is the cost of nth imperialist and is its normalized cost. Each imperialist that has more cost value, will have less normalized cost value. Having the normalized cost, the power of each imperialist is calculated as below and based on that the colonies distributed among the imperialist countries.

∑ 4

On the other hand, the normalized power of an imperialist is assessed by its colonies. Then, the initial number of colonies of an empire will be

. 5

Where, is initial number of colonies of nth empire and is the number of all colonies.

To distribute the colonies among imperialist, of the colonies is selected randomly and assigned to their imperialist. The imperialist countries absorb the colonies towards themselves using the absorption policy. The absorption policy shown in Fig.1, makes the main core of this algorithm and causes the countries move towards to their minimum optima. The imperialists absorb these colonies towards themselves with respect to their power that described in (6). The total power of each imperialist is determined by the power of its both parts, the empire power plus percents of its average colonies power.

6

Where is the total cost of the nth empire and is a positive number which is considered to be less than one.

~ 0, 7

In the absorption policy, the colony moves towards the imperialist by x unit. The direction of movement is the vector from colony to imperialist, as shown in Fig.1, in this figure, the distance between the imperialist and colony shown by d and x is a random variable with uniform distribution. Where is greater than 1 and is near to 2. So, a proper choice can be 2. In our implementation is 4 respectively. ~ , (8)

In ICA algorithm, to search different points around the imperialist, a random amount of deviation is added to the direction of colony movement towards the imperialist. In Fig. 1, this deflection angle is shown as , which is chosen randomly and with an uniform distribution. While moving toward the imperialist countries, a colony may reach to a better position, so the colony position changes according to position of the imperialist.

Figure 1. Moving colonies toward their imperialist.

In this algorithm, the imperialistic competition has an important role. During the imperialistic competition, the weak empires will lose their power and their colonies. To model this competition, firstly we calculate the probability of possessing all the colonies by each empire considering the total cost of empire.

9

Where, is the total cost of nth empire and is the normalized total cost of nth empire. Having the normalized total cost, the possession probability of each empire is calculated as below

∑ 10

after a while all the empires except the most powerful one will collapse and all the colonies will be under the control of this unique empire.

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B. Chaos Theory In the chaos theory a behavior between rigid regularity

and randomness based on pure chance is called a chaotic system, or chaos. Chaos appears to be stochastic but it occurs in a deterministic non-linear system under deterministic conditions [18]. Chaotic map is very important for optimization problem. Moreover, it has a very sensitive dependence upon its initial condition and parameter. A chaotic map is a discrete-time dynamical system. Since the ICA algorithm suffers from being trapped at local optima, chaotic local search has been introduced to overcome the local optima and speed up the convergence.

III. CHAOTIC IMPERIALIST COMPETITIVE ALGORITHM (CICA)

In this paper, we have proposed a new Imperialist

Competitive Algorithm using the chaos theory. The primary ICA algorithm uses a local search mechanism as like as many evolutionary algorithms. Therefore, the primary ICA may fall into local minimum trap during the search process and it is possible to get far from the global optimum. To solve this problem we increased the exploration ability of the ICA algorithm, using a chaotic behavior in the colony movement towards the imperialist’s position. So it is intended to improve the global convergence of the ICA and to prevent it to stick on a local solution.

A. Definition of chaotic angle in the movement of colonies towards the imperialist

In this paper, to enhance the global exploration

capability, the chaotic maps are incorporated into ICA to enhance the ability of escaping from a local optimum.

The angle of movement is changed in a chaotic way during the search process. Adding this chaotic behavior in the imperialist algorithm absorption policy we make the conditions proper for the algorithm to escape from local peaks. Chaos variables are usually generated by the some well-known chaotic maps [19,20]. Table 1, shows the mentioned chaotic maps for adjusting parameter (Angle of colonies movement towards the imperialist’s position) in the proposed algorithm.

TABLE I. Chaotic Maps.

Chaotic maps

CM1 1

CM2 sin

CM3 2 sin 2 1

CM4 0 01 1 0,1

Where, is a control parameter. is a chaotic variable in kth iteration which belongs to interval of (0,1). During the search process, no value of is repeated. The CICA algorithm is summarized in Figure2.

Figure 2. The CICA algorithm.

The ICA algorithm performance mainly depends on its parameters, and it often lead to be trapped in local optimum. Larger values of theta θ, lead to a global exploration, while smaller values of the θ, redound fine-tuning of the current search area.

Thus, proper control of θ is very important to find the optimum solution accurately. Therefore we controlled the θ parameter using the chaotic maps.

IV. ANALISIS AND CONSIDERATION OF EMPIRICAL RESULTS

In this paper, Chaotic Imperialist Competitive Algorithm (CICA), was applied to some well-known benchmark functions in order to verify its performance and compared with ICA and PSO using a Sugeno function as inertia weight and GA algorithms. These benchmarks are presented in Table2.

TABLE II. BENCHMARKS FOR SIMULATION.

Range Mathematical representation

(-100,100) 1 ∑ Sphere

(-100,100) 2 ∑ 100 1 Rosenbrock

(-10,10) 3 ∑ 10 cos 2 10 Rastrigin

(-600,600) 4 ∑ ∏ cos √ 1 Griewank

(-32,32) 5(x)=-20exp(-0.2 ∑ -

exp( ∑ cos 2 +20+e Ackley

(0, ) 6 (x)= -∑ sin michalewicz

We made simulations for considering the rate of

convergence and the quality of the proposed algorithm optima solution, in comparison to ICA, PSO using a Sugeno function as inertia weight and GA algorithms that all the benchmarks were tested by 30 dimensions separately. The average of optimum value for 20 trails is obtained. In these

(1) Initialize the empires and their colonies positions randomly. (2) Compute the chaotic θ (colonies movement angle towards the imperialist’s position) using the chaotic maps. (3) Compute the total cost of all empires (Related to the power of both the imperialist and its colonies). (4) Pick the weakest colony (colonies) from the weakest empire and give it (them) to the empire that has the most likelihood to possess it (Imperialistic competition). (5) Eliminate the powerless empires. (6) If there is just one empire then stop else continue. (7) Check the termination conditions.

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experiments, all the simulations were done during 1000 generations for Sphere and Rosenbrock uni-modal functions and Rastrigin, Griwank, Ackley and michalewicz multi-modal functions. In these simulations, we set the parameters 2, a 0.5 and b = 0.2. The number of imperialists and the colonies are set respectively to 8 and 80. In PSO algorithm the parameters and are fix to 1.5 and the number of the particle is 80. Determining this amount for c1 and c2 we have given equal chance to social and cognition components take part in search process. In GA the population size is 80, the mutation and crossover rate are respectively set to 0.01 and 0.5. The results of these experiments are presented in Table 3 and 4. As we can see in the below charts, the proposed algorithm has better results than the ICA , PSO using a Sugeno function as inertia weight and GA algorithms.

In the Fig.3 which belongs to Sphere it is observed that the quality of global optima solution and the convergence velocity towards the optima point has improved in compare with the other three algorithms.

Figure 3. The cost of Sphere function with 30 dimension in 1000 generation.

In Rosenbrock uni-modal function the performance of

PSO and GA is better than ICA algorithm but the CICA proposed algorithm indicates better performance both in convergence speed and in obtained optima solution quality in compare with the other three algorithms.

Figure 4. The cost of Rosenbrock function with 30 dimension in 1000

generation.

As we can see in Fig.5, for Rasrigin multi-modal function the ICA algorithm has better performance rather than the PSO algorithm and the proposed algorithm has shown a good performance in this function and has been able to escape from the local peaks and reach to global optima.

Figure 5. The cost of Rastrigin function with 30 dimension in 1000

generation.

In Fig.6, Griewank multi-modal function at the first 20 iterations, PSO algorithm has better convergence speed than the ICA and CICA algorithms, but from the 20th iteration to the end the proposed algorithm has had remarkable improved in this function both in optima solution quality and in convergence speed rather than the ICA, PSO and GA algorithms.

Figure 6. The cost of Griewank function with 30 dimension in 1000 generation.

In Fig.7, Ackley multi-modal function the proposed algorithm has better performance in this function both in optima solution quality and in convergence speed rather than the ICA, PSO and GA algorithms.

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Figure 7. The cost of Ackley function with 30 dimension in 1000

generation.

In Fig.8, Michalewicz multi-modal function the porposed algorithm has shown good performance.

Figure 8. The cost of Michalewicz function with 30 dimension in 1000 generation.

In Table 3, the average of optimum value for 20 trails which is obtained from proposed algorithm, ICA, PSO and GA are shown. The benchmarks were tested by 30 dimensions and the stop condition was 1000 generations. The numerical results show that the proposed algorithm has recovered the global optima solution remarkably.

TABLE III. AVERAGE OPTIMUM VALUE FOR 20 TRAILS FOR BENCHMARKS.

CICA ICA GA PSO F1 2.5027 10 28.7678 18.8398 9.0371

F2 20.7391 3.6425 10 1.711210 32.6318

F3 6.8273e 10 0.0994 19.2644 4.6395

F4 -2.3712 -0.8319 -0.3702 -0.3687

F5 6.7944 10 1.9738 2.6671 0.1395

F6 -24.0333 -18.4130 -18.7074 -20.1516

In Table 4, the comparative result of using different chaotic maps is shown.

TABLE IV. COMPARATIVE RESULT OF USING DIFFERENT CHAOTIC MAPS.

CM1 CM2 CM3 CM4

F1 2.5420 10 2.5420 10 2.6700 10 2.5027 10

F2 29.4260 27.8044 27.8742 20.7391

F3 6.8273 10 0.0390 0.0018 0.0083

F4 -2.3712 -2.3712 -2.3712 -2.3711

F5 0.9313 0.0128 0.0128 6.7944 10

F6 -21.1684 -20.7368 -24.0333 -19.9290

In Table 4, figures in bold represent the best chaotic map

that used for the benchmarks.

V. CONCLUSION In this paper an improved imperialist algorithm is

introduced called the chaotic imperialist competitive algorithm (CICA). We have used chaos theory in determining parameter for increasing the global exploration ability of primary ICA algorithm. The proposed method uses some chaotic maps to generate chaotic movement angle. This chaotic movement angle, will act like the mutation operator in genetic algorithm. The empirical results found by applying proposed algorithm to some famous benchmarks indicated that the quality of global optimum solution and the convergence speed towards the optimum point has remarkably increased in the proposed algorithm in comparison to the primary ICA, PSO and GA algorithms.

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