Using Complex Method Guidance GSO Swarm Algorithm for Solving

Using Complex Method Guidance GSO Swarm Algorithm for Solving High Dimensional Function Optimization Problem Guangwei Zhao, Yongquan Zhou, Yingju Wang

Using Complex Method Guidance GSO Swarm Algorithm for Solving High Dimensional Function Optimization Problem Guangwei Zhao 1 , Yongquan Zhou 1, 2 ,Yingju Wang 1 1 College of Mathematics and Computer Science, Guangxi University for Nationalities Nanning Guangxi, 530006, China 2 Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis, Nanning, Guangxi, 530006, China E-mail:[email protected]

Abstract In order to overcome the basic glowworm swarm optimization (GSO) algorithm in the high dimension space function optimization effect is poor defects. This paper, we introduce the idea of the traditional complex method, with the complex method the worst part of the glowworm guidance for reflection be good glowworm swarm, so as to continuously make the worst glowworm swarm become the better glowworm swarm. Finally, an improved artificial glowworm algorithm solves high dimension space of function optimization effect improvement. And through the and basic glowworm swarm algorithm, artificial fish algorithm (AFSA) and particle swarm optimization (PSO) to eight test function optimization results are compared. The show that the efficiency of the method.

Keywords: Glowworm swarm optimization algorithm; Complex method; reflection; function optimization

1. Introduction Glowworm Swarm Optimization algorithm (GSO) [1]-[4] is a novel swarm optimization algorithm which is proposed by K.N.Krishnanad and D.Ghose. The main idea of this algorithm is derived from natural glowworm’s activities in the night, the glowworm exercise in group in nature, they interaction and interattraction with each other by one’s luciferin. The glowworm emits luciferin more light, it can attract more glowworm move toward it. Through simulate this natural phenomena, combined with the characteristics of natural glowworm populations, each glowworm at the owns field of view in search for the glowworm which release the strongest luciferin, also move to the strongest glowworm .So as to achieve the final optimization results. GSO algorithm has a strong versatility and has the high performance for low-dimensional multimodal functions and has a fast convergence rate. However, there are some drawbacks in the GSO algorithm, for example, premature convergence and search accuracy isn’t enough high, also has the low efficiency in the later iterations. Especially in high-dimensional space the optimization is less effective. In this paper, in allusion to the shortcoming which the glowworm swarm optimization algorithm has poor optimization results in the in high-dimensional space, by introducing reflect thinking of the complex method to guide the glowworm swarm’s search, in order to accelerate the search speed of population, make the worst glowworm which in range of the glowworm’s decision-making domain reflected through the center glowworm, then use the reflection point replace the current worst point. Through constant reflection, ultimately reduce the number of poor glowworms, so that the glowworms gathered to the better glowworm, finally successful search for the optimal solution. Use this idea to improve the basic GSO, make the GSO algorithm’s optimization results in the high-dimensional space obtain greatly improved.

2. Glowworm Swarm Optimization algorithm and complex method 2.1 Basic Glowworm Swarm Optimization (GSO) algorithm

Journal of Convergence Information Technology(JCIT) Volume6, Number11, November 2011 doi:10.4156/jcit.vol6.issue11.40

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Using Complex Method Guidance GSO Swarm Algorithm for Solving High Dimensional Function Optimization Problem Guangwei Zhao, Yongquan Zhou, Yingju Wang

In the GSO algorithm, each glowworm is random distributed in the objective function’s declaration space, these glowworms possess own luciferin, every glowworm also has own vision field, this vision field is called local-decision range. The brightness of each glowworm’s luciferin is corresponding to the fitness of the objective functions. The glowworm with brighter luciferin explains this glowworm’s location corresponding the objection function value is better. The glowworm’s moving pattern is that every glowworm hunting the neighbours in own vision field, from the neighbours to find that glowworm which has the lightest luciferin, and then move toward it. Each time, the traversing direction is changed due to different neighbours. In addition, the local-decision range of the glowworms also are affected by the different glowworm’s number in the neighbours, when the glowworm’s number is too small, the glowworms are able to increase it’s local-decision range in order to find more glowworms, otherwise, the glowworms are able to decrease local-decision range .In the end, most of the glowworms gather on a better position. In a general way, the GSO algorithm includes the initial distribution of glowworms, update luciferin, glowworms’ move, update local-decision range. Generally speaking, the GSO algorithm uses under four formulas to iteration update:

li (t )  (1   )li (t  1)  J ( xi (t ))

pij (t ) 



(1)

l j (t )  li (t )

(2)

l (t )  li (t ) kN ( t ) k i

 x (t )  xi (t ) xi (t  1)  xi (t )  st *  j  x j (t )  xi (t ) 

   

(3)

rdi (t  1)  min{rs , max{ 0, rdi (t )   (nt  Ni (t ) )}}

(4)

 represent decay factor of luciferin,  represent strengthen factor of luciferin, J ( xi (t )) represent in time t at position i correspond the function value; Formula (2) is used to calculate the probability for the glowworm i moved to the glowworm j ; Formula (3) is used to update glowworm’s position, st represent moving step; Formula (4) is to update local-decision Formula (1) is used to update luciferin,

range . The procedure of GSO is summarized as follows: Step1: Initialize the number of glowworms and the dimension of search space, initialization required parameters; Step2: Initialize the glowworm i(i  1 n) position in the search space of objective function; Step3: Apply formula (1) calculates the glowworm

i in the time t corresponding objective function

value, then calculate the luciferin li (t ) ; Step4: Each glowworm within own local-decision rang

rdi (t ) determine a set Ni (t ) , the set include

glowworm whose luciferin is brighter than present glowworm; Step5: Apply formula (2) calculates the probability for the glowworm

i moved to the glowworm j ,

use the roulette wheel selection choose the glowworm j ,and move to it ,use formula (3) to update location; Step6: Apply formula (4) update the glowworm’s local-decision range.

2.2 Complex method Complex method [7] is a traditional algorithm for solving the optimization problem. Assuming the optimization problem as follow:

min f ( x)  f ( x1 , x2 ,

xn )

(5)

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Using Complex Method Guidance GSO Swarm Algorithm for Solving High Dimensional Function Optimization Problem Guangwei Zhao, Yongquan Zhou, Yingju Wang

The main idea of the algorithm is randomly initialized

k (k  n  1) points make up initial

k points value corresponding to the function, to x find the worst point D , that is the largest function value corresponding to point. Then calculate the center of k points, we defined as x . Through the center we make the worst point to reflect get a new point xr , then compare the xr and xD corresponding function value. If f ( xr )  f ( xD ) , replace xD with xr reconstituted complexes, or else abandon the worst point, continuously loop iteration, complexes in the solution space, and then calculates

constantly excluding the worst point and introduce better points, and finally achieve optimal effect. Graphic description of the algorithm shown in Figure 1:

xd

xr

best point

reflect point

x centre

xk

xD worst point Figure 1. Reflection image

The procedure of Complex method is summarized as follows: Step1: Randomly initialized k (k  n  1) points make up initial complexes in the feasible region; Step2: Calculate the objective function value, accordingly determine the worst point xD , and then

k points, defined as x ; Step3: Make the worst point xD reflected along this direction of x , and then obtain a new point xr ; Step4: Compare the xr and xD corresponding function value. If f ( xr )  f ( xD ) replace xD with xr calculate the center of

reconstituted complexes, or else abandon the worst point under certain conditions.

3. The GSO algorithm based on the complex method (CGSO) Based on the characteristics of the Glowworm Swarm Optimization algorithm and Complex method make the idea of the Complex method blend in the Glowworm Swarm Optimization algorithm use the Complex method to guide the glowworm swarm’s search. Through each glowworm in own decision-making domain compare with others glowworms, then find the glowworm that release stronger luciferin and move to it. Our decision is find the best luciferin glowworm and the worst luciferin glowworm in the each glowworm’s decision-making domain, then make the worst glowworm reflect through the center of glowworm, to become better glowworm ,thus affecting the current glowworm or other glowworm’s direction of movement. The glowworm after reflection maybe within the current glowworm’s decision-making domain, then can contribute to the current glowworm carry out the local optimization. May also reflect to the other glowworms’ decision-making domain, thus contributing to the glowworm conduct the global optimization, ultimately achieve optimal results. Use the formulas in the iteration:

x

1 k  xi k i 1

(6)

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Using Complex Method Guidance GSO Swarm Algorithm for Solving High Dimensional Function Optimization Problem Guangwei Zhao, Yongquan Zhou, Yingju Wang

xr  xD   ( x  xD )

(7)

The procedure of CGSO is summarized as follows: Step1: Randomly initialized k (k  2n) glowworms in the feasible region and dimension initialize required parameters; Step2: Initialize the glowworm i(i  1 n) position in the search space of objective function;

k points; i Step4: Each glowworm within own local-decision rang rd (t ) determine a set Ni (t ) , the set include

Step3: Use the formula (6) to find the center point of

glowworms whose luciferin is brighter than present glowworm; Step5: By comparison of luciferin values to find the worst luciferin glowworm, the use the formula (7) for reflection; Step6: Apply formula (2) calculates the probability for the glowworm i moved to the glowworm j , use the roulette wheel selection choose the glowworm j , and move to it, use formula(3)to update location; Step7: Apply formula (4) update the glowworm’s local-decision range.

4. Simulation experiments 4.1 Experimental design The experimental program testing platform as: Processor: CPU T4400, Frequency: 2.20GHz, Memory: 2.00GB, Operating system: Windows 7, Run software: Matlab7.0. Set the experiment parameter as is shown in table 1. The remaining parameters can take different values depending on the different objective functions. The benchmark of the test functions shows in the Table 2. Specific parameter settings in the experiment as: Population size 100, Number of iterations 200, Dimension of the function set according the nature of the function .Independent operation 20 times to each function.

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Using Complex Method Guidance GSO Swarm Algorithm for Solving High Dimensional Function Optimization Problem Guangwei Zhao, Yongquan Zhou, Yingju Wang

Table 1. Value of each parameter ρ

β

γ

st

nt

l0



0.4

0.08

0.6

0.3

5

5

2

Test function as follows: f1 ( x)  0.5  ((sin( x12  x2 2 )) 2  0.5) / (1  0.001*( x12  x2 2 )) 2

f 2 ( x)  (exp( x1 )  x2 )4  100( x2  x3 )6  (tan( x3  x4 ))4  x18 n

f3 ( x)   xi2 i 1 n

f 4 ( x)   ixi2 i 1 n

f5 ( x)   (0.2 xi2  0.1xi2 sin 2 xi ) i 1

n 1

f 6 ( x)  [100( xi2  xi 1 ) 2  (1  xi ) 2 ] i 1

f 7 ( x)  1 

n n x 1 xi2   cos( i )  4000 i 1 i i 1

n

f8 ( x)   ( xi2  10 cos(2 xi )  10) i 1

Table 2. The benchmark of the test functions Dimension

Search space

Theoretical optimal value

f1

2

xi  [100,100]

0

f2

4

xi [1,1]

0

f3

30

xi [100,100]

0

f4

30

xi [5.12,5.12]

0

f5

50

xi [10,10]

0

f6

50

xi [30,30]

0

f7

50

xi [600,600]

0

f8

50

xi [10,10]

0

Function

4.2 Experimental results and Analysis Table 3 shows the eight functions’ maximum, minimum and average which run independently 20 times by CGSO algorithm, GSO algorithm, AFSA algorithm and PSO algorithm in the literature [9-14].

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Using Complex Method Guidance GSO Swarm Algorithm for Solving High Dimensional Function Optimization Problem Guangwei Zhao, Yongquan Zhou, Yingju Wang

Figure 2-Figure 9 shows the eight functions’ evolution image by CGSO algorithm, GSO algorithm, AFSA algorithm and PSO algorithm. Table 3. CGSO、GSO、AFSA、PSO algorithms comparison of optimization algorithms Function

f1

f2

f3

f4

f5

f6

f7

f8

Algorithm CGSO GSO AFSA PSO CGSO GSO AFSA PSO CGSO GSO AFSA PSO CGSO GSO AFSA PSO CGSO GSO AFSA PSO CGSO GSO AFSA PSO CGSO GSO AFSA PSO CGSO GSO AFSA PSO

Maximum fitness value 1.8628e-005 3.0782e-004 8.4776e-004 0.009734201 8.5414e-014 5.4994e-004 9.3824e-008 0.132260156 5.8398e-016 0.0436215898 8.7596e-013 6.7885e-07 7.3660e-013 0.009776927 8.5621e-011 3.1829e-006 1.4982e-015 1.2872e-006 2.4317e-013 8.5936e-009 1.2239e-005 0.320708927 9.8531e-006 4.1648e+002 2.3426e-014 0.0060338086 8.1704e-012 1.8339e-005 0 0.2252573829 5.1543e-010 8.5566e+01

Figure 2 Evolution curves of function

f1

Minimum fitness value 4.2411e-007 2.6722e-006 9.7157e-007 0.002421574 4.2997e-023 3.4415e-014 9.3176e-013 3.9477e-004 2.0880e-019 2.4729e-004 6.1029e-016 1.4915e-10 1.2881e-016 2.3926e-006 3.62631e-015 5.5349e-015 2.1253e-020 3.3667e-010 9.8891e-015 3.7380e-015 2.1540e-008 3.0266e-006 9.0058e-007 2.6537347 0 2.5464e-006 0 2.7756e-014 0 2.3958e-006 1.7764e-015 2.5868e+01

Average fitness value 9.2593e-006 1.0352e-005 6.8869e-004 0.008988404 1.7704e-014 8.7403e-005 2. 2393e-008 0.0363207584 1.0067e-016 0.010100340 2.0730e-013 6.6227e-08 1.0523e-013 0.001253592 2.7364e-011 6.7611e-007 1.9725e-016 2.4915e-007 8.0764e-014 1.3785e-009 2.2990e-006 0.044723299 3.4516e-006 8.4122e+001 2.4021e-015 0.002055879 2.8269e-012 12.1290e-006 0 0.032947224 1.5863e-010 4.9888e+01

Figure 3 Evolution curves of function

f2

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Using Complex Method Guidance GSO Swarm Algorithm for Solving High Dimensional Function Optimization Problem Guangwei Zhao, Yongquan Zhou, Yingju Wang

Figure 4 Evolution curves of function

f3

Figure 5 Evolution curves of function

Figure 6 Evolution curves of function

f5

Figure 7 Evolution curves of function

f6

Figure 8 Evolution curves of function

f7

Figure 9 Evolution curves of function

f8

f4

In eight test functions, function 1 and function 2 are low-dimensional multimodal functions. Function 3 and function 4 are high-dimensional unimodal functions, need the algorithm has high optimize accuracy. Function 5-fuction 8 is high-dimensional multimodal functions. Deal with such functions not only require the algorithm has high optimize accuracy, but also the algorithm can avoid falling into local optimum .General approach is easy to fall into local optimum. In the Table 3, compare the optimal results of two low-dimensional and six high-dimensional test functions by CGSO algorithm and basic GSO algorithm, AFSA algorithm, PSO algorithm in the literature [9]. By contrast we can see that GSO algorithm has better performance when optimize lowdimensional multimodal functions, the CGSO algorithm in this paper is slightly better than the GSO algorithm. But with the number of dimensions increases, especially in high-dimensional optimization problems, the GSO algorithm performance is very poor, the effective is far from the PSO algorithm and

358

Using Complex Method Guidance GSO Swarm Algorithm for Solving High Dimensional Function Optimization Problem Guangwei Zhao, Yongquan Zhou, Yingju Wang

AFSA algorithm .However, the CGSO algorithm has overcome this shortcoming, in the convergence accuracy is much better than the PSO algorithm and AFSA algorithm. In particular in the highdimensional multimodal functions such as function 5- function 8, the CGSO algorithm combines the superiority of GSO algorithm has better optimization results in the multimodal functions and the superiority of Complex method’s optimization in the high-dimensional. Making the optimization results is better than other algorithm in the high-dimensional multimodal functions. By the evolution of figure 2-figure 9 we can see that this algorithm in this paper not only has better optimization than the GSO algorithm in the low-dimensional multimodal functions, more importantly, it effectively resolve the shortcoming of GSO algorithm has poor optimization in the high-dimensional space. Make the improved GSO algorithm is better than the PSO algorithm and AFSA algorithm in the overcome fall into local optimum and search accuracy.

5. Conclusion In this paper, based on the GSO algorithm and the Complex method, we present a kind of algorithm which uses the Complex method guidance GSO swarm algorithm (CGSO). This algorithm mainly resolve the problem of the traditional GSO algorithm has poor optimization in the highdimensional space by means of the reflective thinking in the Complex method .Each generation through make the worst glowworm reflect to a good glowworm ,ultimately achieve the purpose of optimizing. Final though test results of eight different types test functions, we can see that the CGSO algorithm not only has better optimization than the GSO algorithm in the low-dimensional multimodal functions ,more importantly, it effectively resolve the shortcoming of GSO algorithm has poor optimization in the high-dimensional space, also effectively avoid falling into local optimum and improve search accuracy.

Acknowledgements The authors are very grateful to the referees for their valuable comments and suggestions. These works are supported by National Science Foundation of China (61165015) and the funded by open research fund program of key lab of intelligent perception and image understanding of ministry of education of China under Grant: IPIU012011001.

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Using Complex Method Guidance GSO Swarm Algorithm for Solving High Dimensional Function Optimization Problem Guangwei Zhao, Yongquan Zhou, Yingju Wang

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