Glowworm swarm optimization algorithm merging simulated annealing strategy*

Xiu-shuang CAO

Department of Information Engineering，Tangshan College，Tangshan 063000，China

Glowworm swarm optimization algorithm merging simulated annealing strategy*

Xiu-shuang CAO†

Department of Information Engineering，Tangshan College，Tangshan 063000，China

Artificial glowworm swarm optimization algorithm is a new research orientation in the field of swarm intelligence recently．The algorithm has achieved success in the complex function optimization，but it is easy to fall into local optimum，and has the low speed of convergence in the later period and so on．Simulated annealing algorithm has excellent global search ability．Combining their advantages，an improved glowworm swarm optimization algorithm was proposed based on simulated annealing strategy．The simulated annealing strategy was integrated into the process of glowworm swarm optimization algorithm．And the temper strategy was integrated into the local search process of hybrid algorithm to improve search precision．Overall performance of the Glowworm swarm optimization was improved．Simulation results show that the hybrid algorithm increases the accuracy of solution and the speed of convergence significantly，and is a feasible and effective method．

Glowworm swarm optimization（GSO），Simulated annealing strategy，Annealing method，Temper strategy，Benchmark

1．Introduction

Glowworm Swarm Optimization（GSO）［1］is proposed by K.N.Krishnanad and D.Ghose in 2005，which is a new bionic swarm intelligence optimization algorithm.The algorithm has strong local search capabilities，but its global search performance is bad.It attracts the attention of scholars increasingly.At present，many scholars have improved traditional GSO in many aspects.A pattern search operator was integrated into GSO in reference［2］，which improves global and local search ability of GSO.But the search result based on pattern search operator depends on the initial point to a large degree；Chaos theory was integrated into GSO in reference［3］.The glowworm’s position of algorithm was initiated based on chaos theory，and mutation was operated by Gaussian disturbances in due time，which improves the capabilities of algorithm in a certain extent；aim at improving optimization structure of GSO by ZHOU Yongquan et al，the algorithms of glowworm swarm optimization based on fluoresce in diffusion and self-adaptive step were proposed，which overcomes the shortcoming of insufficient collaboration between glowworm and slow convergence in the late search disadvantage，respectively；the Swarm intelligence behaviors of bee colony and birds colony were used as move strategy of GSO in reference［4-7］，which improves capability of algorithm in a certain extent.In addition，the standard GSO has also been applied in many areas successfully.The improved algorithm were used to solve function and TSP problems in reference［8-11］respectively，it proves that the algorithm is applicable in the relevant area.

In order to further improve the GSO problems oflocal optimum easily，slow convergence and lower searching precision in optimizing complex functions，simulated annealing strategy is integrated into GSO in this paper.GSO has better local search performance，simulated annealing strategy is used to strengthen the whole search ability of algorithm，and temper strategy is used to promote search accuracy.A glowworm swarm optimization algorithm merging simulated annealing strategy is proposed.

2．GSO principle and simulated annealing search

2．1．Standard GSO

In standard GSO，the‘n’glowworm individual is distributed in whole feasible field random ly，every glowworm will carry a certain amount of fluorescein li．The glowworm releases appropriate fluorescein that will influence on each other around it.Every glowworm individual’s decision-making domains can be set to：．Here，the size of decision making domains is related to the number of neighbors，and the smaller neighbor density，the bigger radius of decision-making domains in order to search more neighbors.Contrary，radius of decision-making domains will decrease.The size of fluoresce in of every glowworm is related to the objective function value of position in search area.The glowworm has the bigger fluoresce in value，and it illustrates that the position in search area is better，so the objective value is better.Every glowworm individual searches the neighbor set Ni()t in decision-making domains，in this set，the neighbor with the bigger fluoresce in value fascinates glowworm individual，which makes glowworm to move in this direction.And the moving direction also will change with the difference of neighbor selected.After moving several times，most glowworms gather in the better solutions position.Initially，the sensing range of every glowworm individual is all rs，the decision-making range is determined by the domain of the object function.After completing the initial settings of the algorithm，the algorithm goes into a loop iteration process，which has 4 processes mainly：fluoresce in update，probability selection，location update and dynamic decision-making domain update.

1）Fluoresce in update

The object function value J xi()( )t of position xi（t）in T iterations about glowworm individual i is converted to correspondent fluorescein li()t：

Where，ρ∈0，()1 is the parameter controlling the size of fluorescein value，andγis the parameter measuring object function value.

2）Probability selection

Within radius rid()t of dynamic decision-making domains，determines the fluoresce in value higher than themselves individuals constitute the individual＇s neighborhood set：

In neighborhood set Ni()t probability of move towards individual j is Pij()t：

3）Location update

where sis move step.

4）Dynamic decision-making domain update

Where，βis constant to control the change range of neighborhood individual，and ntis neighborhood threshold value to control the number of neighbors.

2．2．Introduction of simulated annealing

As early as 1953，Metropolis proposed an algorithm for simulating balance evolution process of solid temperature，which guides the optimization process through Metropolis criterion.When temperature is T，current status i produces corresponding new status. That their energies are Eiand Ejseparately，if Ei≤Ej，the new status j is accepted as current optimum status，otherwise，with a certain probability that is greater than RAND to judge whether or not to accept the new status.The guidelines will be gradually reduced as the temperature，and its probability in accepting inferior solution also reduces.Firefly algorithm is implemented in the process of searching for optimal solutions，when it appears that no more excellent individual is repeatedly found in the corresponding perception range of an individual，it will be decided that this individual fall into local optimum. Or so，this individual’s position can’t move，which reduces search efficiency and leads to fall into local optimum.Now，if the algorithm accepts inferior solu-tions with a certain probability through simulated annealing strategy，it can jump to local optimum to increase convergence speed.

2．3．Annealing method and temper strategy

Annealing method refers to temperature drop speed.In theory，the slower annealing temperature drop，the bigger probability to obtain global optimum.But too slow temperature drop will lead to overlong iteration time of search.The most common annealing method：Tk+1=α*Tk，αinfluences annealing method greatly.This paper adopt following annealing strategy：

Firstly，this annealing strategy is converged by quick speed.When iteration carries out to a certain extent，the temperature will drop under Tmin.Now，raising system temperature to T，and carrying out iteration of algorithm again.Temper strategy range is [Tmin，Tmax]．When the temperature drops under Tminagain，this process is tempering process［12-14］. According to the iteration times，algorithm can be set tempering process time and further improve solution accuracy again.

3．GSO merging into simulated annealing strategy（SA-GSO）

3．1．Basic idea of improved algorithm

On the base of search frame structure about standard GSO，when the algorithm has searched for a certain numbers and the optimum doesn’t change，the algorithm adopts inferior solution for certain probability through simulated annealing，which makes s algorithm to skip out local optimum.So a hybrid algorithm merging into simulated annealing is presented.The idea of hybrid algorithm is as follows：firstly，the glowworm individual location is initialized in the range of feasible region，at the same time，given a certain amount of fluorescein values，which are related to the function value.The sensing range of glowworm individual is determined by the definition domain of function.When the distance between two glowworms is in the range of decision domain，and other glowworm individual has bigger fluorescein value，this glowworm individual chooses excellent individual with some probability，then，moves to better individual according to set step.After a number of iterations，when the neighborhood set of a glowworm individual can’t find more excellent individual，the hybrid algorithm accepts inferior solution with some probability by simulated annealing strategy to skip out local optimum and update glowworm location，and its fluoresce in value is updated according to object function value of new solution.At last，judging the current temperature，if temperature drops under the set value，adopting temper strategy is adopted，and local area in neighborhood radius of optimum is further searched.The global optimum is found through above process repeatly.

3．2．Implementation steps of hybrid algorithm

Step 1 Initializing parameters，initialing temperature T0and temper strategy Tminand Tmaxof simulated annealing，and placing glowworm individual locations randomly.

Step 2 Updating individual fluorescein.

Step 3 Computing neighborhood set and corresponding move probability，and determining moving direction.

Step 4 Mobile phase，according to probability formula（2）in neighborhood set to choose next excellent individual to move.

Step 5 Determining the fluoresce in value of better individual whether don’t change in specified iteration or not.If yes，turn to Step 6；If no，turn to Step 7.

Step 6 The simulated annealing strategy：producing new solution in neighborhood of current excellent individual，and accepting inferior solution through Metropolis criterion.

Step 7 Under current optimum，determine whether the temperature drops to that below tempering temperature.If it is smaller than Tmin，adopt temper strategy to search；otherwise，turn to Step 8.

Step 8 Updating the location of glowworm individual and corresponding neighborhood radius，if the finished condition is fulfilled，exit a loop，if not，turn to Step 2.

4．Test function simulation and analysis

4．1．Testing platform

This algorithm is realized through programming on Matlab2009a.Tests depending on the platform arethe dual-core 3G and memory 2G based on XP system PC.

4．2．Test function

In order to verify efficiency of improved algorithm，the typical Benchmark functions（as formula（7）～（12））are used to test improved algorithm in this paper：

1）Rosenbrock function

n=30，-30≤xi≤30，global minimum is 0，optimum point（1，1，1…1），typical non-convex，pathological unimodal function.

2）Shaffer’s f7function

-100＜xi＜100，global minimum is 0，optimum point（0，0），multi modal function，it includes a number of local optimum which consists of concentric circles，so it is easy to fall into local optimum.

3）Schaffer’s f6 function

-100＜xi＜100，global minimum is 0.There are some infinite local minimums that these points are located in the range of about 3.14 from global optimum，these point shock strongly，and the requirement is very high for algorithm performance to search global optimum.

4）Freudenstein-Roth

-10＜xi＜10，global minimum is 0，optimum point（5，4）.

5）Sphere function

global minimum is 0，optimum point（0，0，0…0）. 6）Griewan function

n=30，-100＜xi＜100，global minimum is 0.

4．3．Parameters setting

In GSO and SA-GSO，glowworm individual is 50，iteration times N=500，initial fluoresce in value of glowworm individual are also set l（0）=5，moving step sizes=2，affecting individual firefly selected neighbor number neighborhood threshold nt=5，parametersβ=0.6 that controls neighboring change ranges，initial value of decision domain range is 10，ρ=0．3，γ=0.6，initial temperature of simulated annealing T0=2 000，and temper strategy range is [500，1 000] according to trial and error.

4．4．Experimental results and analysis

Improved GSO（SA-GSO）is tested through above test functions.In order to overcome accidental error because of placing glowworm individual locations randomly，each function is optimized for 20 times.Compared with basic GSO，the best，worst，average and average iterations times are listed as shown in Table 1.

From Figure 1 to Figure 6，these figures are compared after and before improved algorithm between optimum and iterations times.

As can be seen from Table 1，compared with standard GSO，after merging into simulated annealing strategy，the search capability of hybrid GSO is further strong，the optimum is also closer to standard value.The search capability of improved GSO in High-dimensional space is stronger than before.For high-dimensional function f1，the solution accuracy through SA-GSO is superior to standard GSO greatly，its average number of iterations also reduce dramatically.For functions f2that are hard to find optimum，the solution of SA-GSO is nearer to optimum in theory，the number of iterations also reduces in a certain degree，which shows that SA-GSO is rapider to converge than before.For function f3which has unlimited local extremum，SA-GSO finds the optimum that is closer to the optimum in theory in the similar iterations，which reflects that the convergence speed of SA-GSO is quick.For a two-dimensional function f4，all of the optimization results of algorithm are better，however，the optimum accuracy through SA-GSO is improved substantially，its optimum is 9.552 5e-06，optimum point（5.986 4，3.961 4）.And the optimum of GSO is 0.003 1，optimum point（5.593 0，3.815 6）.For high-dimension function f5，the accuracy of solution also increases within a factor of ten.The accuracy of function f6is increased lightly，however，the number of iteration reduces，and it equals to indicate that Glowworm swarm optimization algorithm merging simulated annealing strategy is effective.

As can be seen from Figure 1，SA-GSO can converge to the optimum fastly in iteration of 100 or so.But the optimum of GSO after250 iterations is inferior to SA-GSO still.In addition，it falls into the local optimum for a long time in iteration of 100 to 250.As can be seen from Figure 2 to Figure 6 too，the convergence speed of SA-GSO is superior to GSO.When GSO falls into local optimum in a certain iteration times，it is hard to jump local optimum.But because SA-GSO is merged into simulated annealing，it can jump to local optimum and converge quickly. Besides，because of the temperature strategy，the solution accuracy is improved substantially.As can be seen from Figure 6，SA-GSO obtains the optimum 0.047 71 after the iteration of129，but GSO only obtains the optimum 0.056 62 after the iteration of239. It indicates that convergence speed and solution accuracy are improved greatly.

Table 1．The com parison of experimental results

Figure 1.The iterative curve of Rosenbrock

Figure 2.The iterative curve of Shaffer’s f7

Figure 3.The iterative curve of Shaffer’s f6

Figure 4.The iterative curve of Freudenstein

Figure 5.The iterative curve of Sphere

Figure 6.The iterative curve of Griewank

5．Conclusion

Aiming at falling into local optimum easily and slow convergence later stage in optimizing complex function by GSO，a glowworm swarm optimization algorithm merging simulated annealing strategy is proposed，which improves global search capability of algorithm.In order to further improve search accuracy，temper strategy is adopted，which searches local area further at certain temperatures.Simulation test show that the algorithm is improved in search speed and search accuracy.So the improvement of algorithm is effective and feasible.But，because the parameters in algorithm are set through many experiments，the optimization of algorithm parameters and other application need to study further，which is also the next job.

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摘要：人造草坪机的连杆是使机构中的簇针、成圈钩、割刀完成各自往复运动的重要连接部件。利用了ANSYS Workbench有限元分析软件对人造草坪机连杆建立了有限元模型。根据其在实际中的受约束以及载荷情况，进行静力分析。考虑到机床运转时振动对连杆的影响，对其进行模态分析。根据静力学分析的结果对连杆进行拓扑优化设计，为连杆以及人造草坪机的其他部件的进一步优化提供了理论依据及实现方法。

关键词：ANSYS Workbench；连杆；静力分析；模态分析；优化设计

中图分类号：TP114

融合模拟退火策略的萤火虫优化算法*

曹秀爽†

唐山学院信息工程系，河北唐山 063000

萤火虫算法是群智能领域近年出现的一个新的研究方向，该算法虽已在复杂函数优化方面取得了成功，但也存在着易于陷入局部最优且进化后期收敛速度慢等问题，而模拟退火机制具有很强的全局搜索能力，结合两者的优缺点，提出一种融合模拟退火策略的萤火虫优化算法。改进后的算法在萤火虫算法全局搜索过程中融入模拟退火搜索机制，在局部搜索过程中采用了回火策略，改善寻优精度，改进了萤火虫算法的全局搜索性能和局部搜索性能。仿真实验结果表明：改进后的算法在收敛速度和解的精度方面有了显著地提高，证明了算法改进的可行性和有效性。

萤火虫算法；模拟退火策略；退火方式；回火策略；Benchmark

TP312

基于ANSYS Workbench的人造草坪织机连杆的有限元分析及优化设计

吴士昊†，戴惠良，宋佳玲

东华大学机械工程学院，上海 201620

10.3969／j.issn.1001-3881.2014.18.020

2014-05-02

*Project supported Education Department of Hebei Province（No：QN20132019，Science and Technology Planning Project of Tangshan city（No：131302118a）

†Xiu-shuang CAO，E-mail：379511725＠qq.com