hybrid artificial glowworm swarm optimization algorithm for - yan yang, yongquan zhou, qiaoqiao gong

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Hybrid Artificial Glowworm SwarmOptimization Algorithm for Solving System of

Nonlinear Equations

ARTICLE · OCTOBER 2010

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3 AUTHORS, INCLUDING:

Yong-Quan Zhou

Guangxi University for Nationalities

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Journal of Computational Information Systems 6:10 (2010) 3431-3438

Available at http://www.Jofcis.com

1553-9105/ Copyright © 2010 Binary Information PressOctober, 2010

Hybrid Artificial Glowworm Swarm Optimization Algorithm for

Solving System of Nonlinear Equati ns

Yan YANG, Yongquan ZHOU†, Qiaoqiao GONG

College of Mathematics and Computer Science Guangxi University for Nationalities， Nanning 530006, China

Abstract

A hybrid artificial glowworm swarm optimization (AGSO) algorithm for solving system of nonlinear equations is proposed,where the Hooke-Jeeves pattern search is a local search operator embedded to AGSO and combined with AGSO to speedup the local search. The problem of solving nonlinear equations is equivalently changed to the problem of functionoptimization, and then a solution is obtained by artificial glowworm swarm optimization algorithm, considering it as theinitial value of Hooke-Jeeves method, a more accurate solution can be obtained. The results show the hybrid optimizationalgorithm has high convergence speed and accuracy for solving nonlinear equations.

Keywords: Artificial Glowworm Swarm Algorithm; Hooke-Jeeves Method; AGSO; System of Nonlinear Equations

1. Introduction

Solving system of nonlinear equations is an important question in scientific calculations and engineeringtechnology field. So far, people have made a lot of researches to solve nonlinear equations using theoretical

and computational methods. But the problem of solving nonlinear equations is still not completely solved

and lacks efficient and reliable algorithms especially for the strong nonlinear engineering calculation

problems. Therefore, studying the efficient algorithms for strong nonlinear equations is a significant work.

At present, some scholars calculate the nonlinear equations with genetic algorithm, evolutionary strategy,

particle swarm algorithm and neural network algorithm [3-10]. However, the initial value selection is

difficult for most traditional algorithms, the solution accuracy need to be improved for intelligent algorithm.

So looking for efficient algorithm has important theoretical significance.

In 2005, Krishnanand and Ghose put forward glowworm swarm optimization (GSO) algorithm [1]. GSO

algorithm is a swarm intelligence bionic algorithm and it has good capacity to search for global extremum

and more extremums of multimodal optimization problems. The GSO algorithm was applied to multimodal

optimization, noise issues, theoretical foundations, signal source locatisation, addressing the problem of

sensing hazards and pursuiting of multiple mobile signal sources problems.

Related work. The artificial GSO does not depend on the initial points and derivative to solve objective

function and the problem of solving system of nonlinear equations can be transformed into a function

† Corresponding author.

Email address: [email protected] (Yongquan ZHOU)

https://www.researchgate.net/publication/265492155_Trust_region_method_for_solving_nonlinear_equations?el=1_x_8&enrichId=rgreq-3c3b4b96-fb54-438a-a260-f4498b9b724a&enrichSource=Y292ZXJQYWdlOzI2NzU0NzU2NTtBUzoyODIyNDQ0ODU3OTU4NDRAMTQ0NDMwMzczNzE4Mg==https://www.researchgate.net/publication/265492155_Trust_region_method_for_solving_nonlinear_equations?el=1_x_8&enrichId=rgreq-3c3b4b96-fb54-438a-a260-f4498b9b724a&enrichSource=Y292ZXJQYWdlOzI2NzU0NzU2NTtBUzoyODIyNDQ0ODU3OTU4NDRAMTQ0NDMwMzczNzE4Mg==https://www.researchgate.net/publication/4037842_Performance_Analysis_of_Partition_Algorithms_for_Parallel_Solution_of_Nonlinear_Systems_of_Equations?el=1_x_8&enrichId=rgreq-3c3b4b96-fb54-438a-a260-f4498b9b724a&enrichSource=Y292ZXJQYWdlOzI2NzU0NzU2NTtBUzoyODIyNDQ0ODU3OTU4NDRAMTQ0NDMwMzczNzE4Mg==


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Y. Yang et al. /Journal of Computational Information Systems 6:10 (2010) 3431-3438 3433

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

−+=+

)()(

)()()()1(

t xt x

t xt xst xt x

i j

i jii

(3)

)(1)1(

t D

r t r

i

sid

∗+=+

β (4)

Where, )}()();()()(:{)( t lt lt r t xt x jt N jiid i ji ε ;

2.(1) (1)

y x= ;

3. 1== jk ;

4. While _ k Max Interations


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3434 Y. Yang et al. /Journal of Computational Information Systems 6:10 (2010) 3431-3438

6. if )()( )()( j j j y f de y f


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make equation 0)( * = X P established, in other words, solve a set of values to make function )( X P

minimum value is 0 .

4.2. Specific Implementation Steps of the Algorithm

Set objective function )( X P as fitness function in the hybrid optimization algorithm. The algorithm

terminates when the value of the best point is less than a given error in the moving process of glowworm,

or the algorithm achieves the prescriptive maximum iterations. The processes of hybrid AGSO algorithm

are as follows:

Step1. Initialize a population and the population has N glowworms. Set the initial luciferin 0l , initial

decision domains range 0 Rd , circular sensor range s R , neighbour number nei . Randomly initialize the

location of glowworms. Set step as moving step of glowworm and s Interation Max _ as the

maximum iterations of Hooke-Jeeves method.

Step2. (Implement AGSO algorithm) According to the formulas (1) and (3), calculate fitness value and

current location of each glowworm, gain a corresponding luciferin value and obtain the glowworm’s

current optimal location U by comparing the values of luciferins.

Step3. (Implement Hooke-Jeeves method) Set the glowworm’s current optimal location U and

corresponding fitness value as the initial value of Hooke-Jeeves method. When Hooke-Jeeves method

achieves the maximum iterations, record the local optimal value T searching by Hooke-Jeeves method

and exit the Hooke-Jeeves operation.

Step4. (Judge termination conditions) Judge T whether it satisfies termination conditions. If T meets

the termination conditions, the algorithm ends and outputs optimal solutions. Otherwise, turns to Step2.

5. Numerical Simulation Experiments

In order to verify the feasibility and validity of the algorithm, code the algorithms in Matlab7.0 and test the

equations for 30 times. The computer configuration is Celeron(R) Dual-Core, 1.8 GHz and 1GB memory.

Main parameters of hybrid algorithm are setted as follows:

The swarm size N is fixed to 50 . Set the initial luciferin 50 =l , moving step-length 01.0=step ,

initial decision domains range 30 = R and neighbour number 5=nei , )7.0,2.0(∈ ρ ,

)09.0,05.0(∈ β and maximum iterations 1000max _ =iter . The error precision is 1010 −= eh .

Example 1[3]

⎩⎨⎧

=−=

=+−=

0)5.0cos()(

01)(

212

2211

x x x f

x x x f

π


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Where, ]2,2[−∈ x , the exact values are T T x x )1,0(,)5.1,2/1( =−= ∗∗

Example 2 [4]

⎪⎩

⎪⎨⎧

=−−+−=

=−−=

01)5.0()2()(

01)(

22

212

2211

x x x f

x x x f

where, ]2,0[∈ x , the exact values are T x )391174.1,546342.1(=∗ and

T x )139460.0,067412.1(=∗ .

Example 3 [5]

⎪⎩

⎪⎨⎧

=−++=

=+++=

022)(2

013)(

2

21

23111

xe x x x f

x x x x f

The approximate roots are 445178.0,451123.0 21 =−= x x

Example 4 [6]

⎪⎪

⎩

⎪⎪

⎨

⎧

≤≤≤≤≤≤=

=−−+

=−−−

=−−+

}25.0,42,53|),,{(

02

060

0855

321321

231

3231

32121

13

23

12

x x x x x x D

x x x

x x x

x x x x x

x x

x x

x x

The exact values are

T

x )1,3,4(=∗

.

The results of equations are in Table 1.

Table 1 Calculation Results of Examples 1 to 4

Exam-

ple

Equations

roots

Exact values Average approximate roots of hybrid AGSO Literature results

1 x 2/1− 0 -0.730839333990662 1.429129869894717e-006 -0.707724 0.000114

1 2 x 1.5 1 1.542360512171837 1.000113980653129 1.500668 0.999817

1 x 1.546342 1.067412 1.546443398564858 1.066851999307000 1.546314 1.067307

2 2 x 1.391174 0.139460 1.390631993359393 0.139425973209928 1.391152 0.139012

1 x A/N -0.631192941509398 -0.451123

3 2 x A/N 0.494703417321859 0.445178

1 x 4 4.001406721890630 3.994 0

2 x 3 3.011350221428261 3.0079

4 3 x 1 0.999234835117343 1.007 9

https://www.researchgate.net/publication/265492155_Trust_region_method_for_solving_nonlinear_equations?el=1_x_8&enrichId=rgreq-3c3b4b96-fb54-438a-a260-f4498b9b724a&enrichSource=Y292ZXJQYWdlOzI2NzU0NzU2NTtBUzoyODIyNDQ0ODU3OTU4NDRAMTQ0NDMwMzczNzE4Mg==https://www.researchgate.net/publication/265492155_Trust_region_method_for_solving_nonlinear_equations?el=1_x_8&enrichId=rgreq-3c3b4b96-fb54-438a-a260-f4498b9b724a&enrichSource=Y292ZXJQYWdlOzI2NzU0NzU2NTtBUzoyODIyNDQ0ODU3OTU4NDRAMTQ0NDMwMzczNzE4Mg==


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From Table 1, it is observed that hybrid AGSO algorithm can obtain the approximate roots of equations,

and the calculation accuracy achieves 1510 −e . Compared with the results of references, the hybrid

algorithm can gain the approximate roots of example 1, 2 and 4, and have small errors. But for the example

3, the hybrid algorithm gets final solution T x )218594947034173.0,093986311929415.0(−=∗ , which

has greater errors compared with the results in references.

Example 5 [7]

⎪⎪

⎩

⎪⎪

⎨

⎧

−∈

=−++

=−++++

=−++

]732.1,732.1[,,

03

05

03

22

222

z y x

z y x

y x xy y x

z y x

The theoretical values are 1=== z y x .

Example 6 [7]

⎪⎪

⎩

⎪⎪

⎨

⎧

−∈

=−+

=−

=−+

]2,2[,,

02

01

02

22

z y x

y x

xy

z y x

The theoretical values are 1=== z y x .

The results of example 5 and 6 are in Table 2.

Table 2 Calculation results of Examples 5 and 6

From Table 2, it is observed that hybrid AGSO algorithm can get the approximate roots of system of

equations and have smaller errors compared with the results in references. The experimental results show

that the hybrid algorithm is effective.

6. Conclusions and Future Work

In this paper, we put forward a hybrid artificial glowworm swarm optimization algorithm for solving

system of nonlinear equations. The hybrid algorithm does not need to restrict the forms of equations; also

Exam-

ple

Roots of

equations

Exact

values

Average approximate

roots of hybrid AGSO

Average errors of hybrid

AGSO

Average

errors of

PSO-LM

Average

errors of

basic PSO

x 1 0.982508798371925 0.017491201628075 0.0773 0.0877 y 1 1.014819818424846 0.014819818424846 0.0773 0.08775

z 1 0.995253075035268 0.004746924964732 0.0773 0.0877

x 1 1.014531658579457 0.014531658579457 0.0492 0.0493

y 1 0.978259911528236 0.021740088471764 0.0492 0.0493 6 z 1 0.998709998385293 0.001290001614707 0.0492 0.0493


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equations do not need to be continuous and differentiable. The hybrid algorithm is able to solve the

problem that it’s difficult for traditional algorithms to select initial values. The results of equations are

accurate and this paper provides an effective algorithm for solving system of nonlinear equations. In the

future work, we will study the more effective AGSO method to solving non-smooth function equations.

Acknowledgement

This work is supported by the Grants 08GX01 from Science Research Foundation of State Ethnic Affairs

Commission of China, the project Supported by Grants 0832082; 0991086 from Guangxi Science

Foundation, and this work is supported by the Innovation Project of Guangxi Graduate Education

(gxun-chx2009091).

References

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functions[J]. Int. J. Computational Intelligence Studies, 2009,1(1):93-119

[2] Garald Recktenwald. Numerical methods and realization and application using MATLAB [M]. Mechanical

Industry Press. 2007

[3] Jianke Zhang, Xiaozhi Wang, Sanyang Liu. Particle swarm optimization for solving nonlinear equation and

system [J]. Computer Engineering and Applications. 2006,42(7): 56-58

[4] Xifeng Xue, Zhidong, Hongyun Meng. Trust region method for solving nonlinear equations [J](in Chinese).

Journal of Northwest University, 2001,31(4): 289-291

[5] Huamin Zhao, Kaizhou Chen. Neural network for solving systems of nonlinear equations [J] (in Chinese). ACTA

Electronic Sinica,2002,30(4):601-614

[6] Yi Zeng. The Application of improved genetic algorithms to solving nonlinear equation Group [J] (in Chinese).

Journal of East China Jiao tong University, 2004,21(4):132-134

[7] Wan-an Yang, An-piing Zeng. New method of solving complicated nonlinear equation group. Computer

Engineering and Applications [J] (in Chinese).Computer Engineering and Application. 200945(28): 41-42

[8] Juan Du, Zhigang Liu. A new hybrid quantum genetic algorithm for solving nonlinear equations [J] (in Chinese).

Microcomputer Applications, 2008,29(7): 1-5

[9] Zhen He, Yi Wang, Changhao Zhou. Nonlinear problems of the Ball-and-Beamsystem [J] (in Chinese). ACTA

Automatica Sinica. 2007,33(5):550-553

[10] Geng Yang. Analysis of Parallel Algorithms for solving nonlinear systems of equations [J]. Chinese Journal of

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[11] Huanwen Tang. Practical optimization methods [M]. Dalian Polytechnic University Press.2003.

hybrid artificial glowworm swarm optimization algorithm for - yan yang, yongquan zhou, qiaoqiao gong

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