Copyright @ 2008 MICParticle swarm optimization (PSO) is an effcient tool for optimization and search problems. However, it is easy to betrapped into local optima due to its in-formation sharing mechanism. Many research works have shown that mutation operators can help PSO prevent prema- ture convergence. In this paper, several mutation operators that are based on the global best particle are investigated and compared for PSO. An adaptive mutation operator is designed. Experimental results show that these mutation operators can greatly enhance the performance of PSO. The adaptive mutation operator shows great advantages over non-adaptive mutation operators on a set of benchmark test problems.This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) of UK under Grant EP/E060722/1

Korejo, I

Li, C

Yang, S

English

Brunel University Research Archive

An Adaptive Mutation Operator for Particle SwarmOptimizationChanghe LiDept of Computer Science,University of LeicesterUniversity RoadLeicester LE1 7RH, UKcl160@mcs.le.ac.ukShengxiang YangDept of Computer Science,University of LeicesterUniversity RoadLeicester LE1 7RH, UKs.yang@mcs.le.ac.ukImtiaz KorejoDept of Computer Science,University of LeicesterUniversity RoadLeicester LE1 7RH, UKiak5@mcs.le.ac.ukAbstractParticle swarm optimization (PSO) is anefficient tool for optimization and searchproblems. However, it is easy to betrapped into local optima due to its in-formation sharing mechanism. Many re-search works have shown that mutationoperators can help PSO prevent prema-ture convergence. In this paper, severalmutation operators that are based on theglobal best particle are investigated andcompared for PSO. An adaptive muta-tion operator is designed. Experimentalresults show that these mutation opera-tors can greatly enhance the performanceof PSO. The adaptive mutation operatorshows great advantages over non-adaptivemutation operators on a set of benchmarktest problems.1 IntroductionParticle swarm optimization (PSO) was firstintroduced by Kennedy and Eberhart in 1995[2, 5]. There are two main models of the PSOalgorithm, called gbest (global best) and lbest(local best), which are different in the way ofdefining particle neighborhood. Kennedy andPoli [6, 10] pointed out that the gbest model hasa fast convergence speed with a higher chancegetting stuck in local optima. On the contrary,the lbest model is less vulnerable to the attrac-tion of local optima but has a slower conver-gence speed than gbest. In this paper, all thePSO algorithms studied are based on the gbestmodel, aiming to reduce the probability of beingtrapped into local optima by introducing muta-tion opertor and keep the advantage of fast con-vergence of the gbest model.There are several major versions of the PSOalgorithms. The following version modified byShi and Eberhart [12] is used in this paper. Eachparticle is represented by a position and a veloc-ity, which are updated as follows:~V ′i = ω~Vi + η1r1(~Pi − ~Xi) + η2r2(~Pg − ~Xi) (1)~X ′i = ~Xi + ~V′i (2)where ~X ′i and ~X represent the current and pre-vious positions of particle i, ~Vi and ~V ′i are theprevious and current velocity of particle i, ~Piand ~Pg are the best-so-far position of particle iand the best position found in the whole swarmso far respectively. ω ∈ (0, 1] is an inertia weightwhich determines how much the previous veloc-ity is preserved, η1 and η2 are acceleration con-stants, and r1 and r2 are random numbers gen-erated from the interval [0.0, 1.0].Ratnaweera et al. [11] stated that the lack ofpopulation diversity in PSO algorithms is un-derstood to be a factor in their convergenceon local optima. Therefore, the addition of amutation operator to PSO should enhance itsglobal search capacity and thus improve its per-formance. There are mainly two type of muta-tion operators: one type is based on particle po-sition [7, 8, 4, 13, 3] and the other type is basedon particle velocity [11, 9, 14, 15]. The formermethod is by far the most common techniquefound in the literature. However, the work of thelatter one is very few. There has been few at-tempt to different mutation operators, and so farno comparison of effectiveness of different muta-tion operators on particle velocity.In this paper, several mutation operatorsthat are based on the global best particle areinvestigated for PSO. An adaptive mutation op-erator is also designed for PSO. An experimentalstudy is carried out to compare the performanceof the investigated mutation operators for PSOon a set of benchmark test problems.2 An Adaptive MutationOperator for PSODifferent mutation operators can be used to helpPSO jump out of local optima. However, a mu-tation operator may be more effective than otherones on a certain type of problems and may beworse on another type of problems. In fact, it isthe same even for a specific problem at differentstage of the optimization process. That is, thebest mutation results can not be achieved by asingle mutation operator, instead several muta-tion operators may have to be applied at differ-ent stages for the best performance. This paperdesigns a mutation operator that can adaptivelyselect the most suitable mutation operator fordifferent problems. Before presenting the adap-tive mutation operator, three mutation opera-tors designed for the global best particles aredescribed as follows.2.1 Three Mutation OperatorsA. Cauchy mutation operator~V ′g = ~Vgexp(δ) (3)~X ′g = ~Xg + ~V′gδg (4)where ~Xg and ~Vg represent the position and ve-locity of the global best particle. δ and δg denoteCauchy random numbers with the scale param-eter of 1.B. Gaussian mutation operator~V ′g = ~Vgexp(N) (5)~X ′g = ~Xg + ~V′gNg (6)where ~Xg and ~Vg represent the position and ve-locity of global best particle. N and Ng areGaussian distribution numbers with the mean0 and the variance 1.C. Levy mutation operator~V ′g = ~Vgexp(L(α)) (7)~X ′g = ~Xg + ~V′gLg(α), (8)where L(α) and Lg(α) are random numbers gen-erated from the Levy distribution with a param-eter α. In this paper, α is set to 1.3.2.2 The Adaptive Mutation OperatorThe proposed adaptive mutation operator usesthe three mutation operators described above.All mutation operators have an equal initial se-lection ratio with 1/3. Each mutation operatoris applied according to its selection ratio andits offspring fitness is evaluated. The mutationoperators that result in higher fitness values ofoffspring have their selection ratios increased.The mutation operators that result in lower fit-ness values of offspring have their selection ratiosdecreased. Gradually, the most suitable muta-tion operator will be chosen automatically andcontrol all the mutation behavior in the wholeswarm. Without lose of generality, we discussthe minimization optimization problems in thispaper.First, some definitions are given below: Theprogress value progi(t) of operator i at genera-tion t is defined as follows:progi(t) =Mi∑j=1f(pij(t))−min (f(pij(t)), f(cij(t))),(9)where pij(t) and cij(t) denote a parent and itschild produced by mutation operator i at gener-ation t and Mi is the number of particles thatselect mutation operator i to mutate.The reward value rewardi(t) of operator i atgeneration t is defined as follows:rewardi(t) = exp(progi(t)∑Nj=1progj(t)α+ siMi (1− α))+ cipi(t)− 1(10)where si is the number of particles whose chil-dren have a better fitness than themselves afterbeing mutated by mutation operator i, pi(t) isthe selection ratio of mutation operator i at gen-eration t, α is a random weight between (0, 1),N is the number of mutation operators, and ci isa penalty factor for mutation operator i, whichis defined as follows:ci ={0.9, if si = 0 and pi(t) = maxNj=1 (pj(t))1, otherwise(11)if the previous best operator has no contributionat current generation, then the selection ratio ofthe current best operator will decrease.With the above definitions, the selection ra-tio of mutation operator i is updated accordingto the following equation:pi(t+1) =rewardi(t)∑Nj=1 rewardj(t)(1−N∗γ)+γ, (12)where γ is the minimum selection ratio for eachmutation operator, which is set 0.01 for all theexperiments in this paper. This selection ra-tio update equation considers four factors: theprogress value, the ratio of successful mutations,previous selection ratio, and the minimum selec-tion ratio. Another important parameter for theadaptive mutation operator is the frequency ofupdating the selection ratios of mutation oper-ators. That is, the selection ratio of each mu-tation operator can be updated at a fixed fre-quency, e.g., every Uf generations, instead ofevery generation.The framework of the PSO algorithm withone of the three mutation operators describedabove is given as follows:Step 1: Generate the initial particles by ran-domly generating the position and velocityfor each particle.Step 2: Evaluate the fitness of each particle.Step 3: For each particle i, if its fitness issmaller than the fitness of its previous bestposition (~Pi), update ~Pi.Step 4: For each particle, if its fitness is smallerthan the fitness of the best position (~pg) ofall particles, update ~Pg.Step 5: Update each particle according toEqs. (1) and (2).Step 6: Mutate ~Pg according to one of the threemutation operators for T times (T is thelocal search size for the global best particle).Step 7: Compare the best one ~P ∗g of the mu-tants with ~Pg and select the better one asthe new global best particle.Step 8: Stop if the stop criterion is satisfied;otherwise, go to Step 3.The PSO with the adaptive mutation oper-ator differs from the above PSO algorithms inStep 6. At Step 6, we select one of the threemutation operators according to their selectionratios to mutate ~Pg for T times and then updatethe selection ratio for each mutation operatoraccording to Eq. (18)at a fixed frequency, e.g.,every Uf generations.3 Experimental StudySeven benchmark functions (f1−f7) are used inthis paper. Functions f1 is unimodal functionswhile f2 − f7 have many local optima. Table1 gives the details of these functions. The pro-posed algorithm are compared with another al-gorithm called FEP[16, 17] on all test problems.Algorithm parameters are set as follows: theacceleration constants η1 = η2 = 1.496180 andthe inertia weight ω = 0.729844 as suggested byden Bergh [1]. In order to have the same num-ber of function evaluations, the population sizeis 50 and T = 10 for PSO for mutation opera-tors. In PSO and FEP, the population size is 60,the tournament size is 6 for selection and initialstandard deviation is 3.0 in FEP algorithm [16].For the adaptive mutation operator, the initialselection ratio is 1/3 and the minimum selec-tion ratio γ is 0.01 for each mutation operator,and the update frequency Uf is set to 5. For allalgorithms, we run 50 times independently tillgeneration 2000 for each test problem.3.1 Performance ComparisonThe average results of six algorithms for thetest problems are summarized in Table 2. Ta-ble 3 also shows the statistical comparison ofthe adaptive mutation operator over other fivealgorithms, using the two-tailed T-test with 98degrees of freedom at a 0.05 level of significance.In Table 3, the performance difference is signifi-cant if the absolute value of the T-test result isgreater than 1.984.From Table 2 and Table 3, it can be seen thatthe algorithms with mutation operator performbetter than PSO for most problems. All thefour algorithms with mutation operators presentmuch better results than PSO on f1, f2, f6 andf7. Especially for multimodal functions, algo-rithms with mutation perform obviously betterthan PSO. FEP performs better than PSO on f6and f7, but worse than PSO on the other prob-lems. The mean values obtained by Levy andAdaptive are much better than FEP on all testproblems.As expected, adaption mutation shows its ad-vantages and presents at least the second bestresult among all the mutation algorithms on alltest problems. To our surprise, the adaptive mu-tation achieves the best results among all thealgorithms on function f2 and f4. In this pointof view, adaptive mutation can probably be re-garded as the one with the average best per-formance for PSO on all test problems. Fig. 1shows the evolutionary process of the averageglobal best particle of PSO, Adaptive, and FEP.Due to page limitation, the results of functionf3, f5 and f6 are not presented. Adaptive mu-tation gives the fastest convergence rate on alltest functions.Fig. 2 presents the results of selection ratio ofCauchy, Gaussian, and Levy at generation 2000for 50 runs in the adaptive mutation.Due to pagelimitation, the results of function f3, f5 and f6are not presented. We can see that the frequen-cies of three mutation operators being selectedare quite different for each test problem. Theselection ratio of Levy for function f1, f2 and f40 500 1000 1500 2000-140-120-100-80-60-40-20020Fitness(ln)Generation(f1) PSO Adapt FEP0 500 1000 1500 2000-7-6-5-4-3-2-1fitness(ln)Generation(f2) PSO Adapt FEP0 500 1000 1500 2000-35-30-25-20-15-10-505fitness(ln)Generation(f4) PSO Adapt FEP0 500 1000 1500 200045678910fitness(ln)Generation(f7) PSO Adapt FEPFigure 1: Evolution process of the average best particle of PSO, Adaptive mutation and FEP.0 500 1000 1500 20000.00.20.40.60.81.0selection ratioGeneration(f1) Cauchy Normal Levy0 500 1000 1500 20000.100.150.200.250.300.350.400.450.500.55selection ratioGeneration(f2) Cauchy Normal Levy0 500 1000 1500 20000.00.20.40.60.81.0selection ratioGeneration(f4) Cauchy Normal Levy0 500 1000 1500 20000.00.20.40.60.81.0selection ratioGeneration(f7) Cauchy Normal LevyFigure 2: The average selection ratio of Cauchy,Gaussian and Levy in the adaptive mutation.Table 1: Details of test functions, where n is the dimension of the function, fminis the minimumvalue of the function, and S ∈ RnTest function n S fminf1(x) =∑ni=1 x2i 30 (−5.12, 5.12) 0f2(x) =(sin2√x21+x22)−0.5(1.0+0.001(x21+x22))2 + 0.5 30 (-100,100) 0f3(x) = 14000∑ni=1(xi − 100)2 −∏ni=1cos(xi−100√i) + 1 30 (-300,300) 0f4(x) = −20 exp(−0.2√1n∑ni=1 x2i )− 30 (-30,30) 0exp( 1n∑ni=1 cos(2pixi)) + 20 + ef5(x) =∑ni=1 (x2i − 10 cos(2pixi) + 10) 30 (-5.12,5.12) 0f6(x) =∑ni=1−xi sin (√|xi|) 30 (-500,500) -12569.5f7(x) = 418.9829 · n+∑ni=1−xi sin (√|xi|) 30 (-500, 500) 0Table 2: Comparison of PSO, Cauchy, Gaussian, Levy, Adaptive and FEP. The results are meanbest function values found at generation 2000 over 50 runsTest function PSO Cauchy Gaussian Levy Adaptive FEPf1 1.24e-40 1.17e-41 2.45e-43 1.40e-53 6.51e-53 1.424f2 0.00252 0.001748 0.00136 0.00116 0.00097 0.00968f3 0.01865 0.01062 0.01877 0.01454 0.01298 1.2003f4 1.28691 1.586e-14 1.19315 1.486e-14 1.45e-014 6.84345f5 42.0469 33.1918 46.564 27.1624 28.0578 69.283f6 -7804.39 -12523.7 -8249.72 -12569.5 -12533.9 -10522.4f7 4700.71 101.543 4239.15 2.36915 65.9316 2099.74Table 3: The T-test results between adaptive mutation and the other five algorithmsTest function PSO Cauchy Gaussian Levy FEPf1 -1.75169 -2.73526 -1.84775 2.33678 -5.44373f2 -2.10759 -1.14885 -0.61043 -0.31655 -8.68015f3 -1.68669 0.862596 -1.65334 -0.526 -8.06837f4 -10.4887 -1.28531 -10.6313 -0.29368 -16.0647f5 -7.65198 -2.62769 -8.36267 0.49368 -14.1893f6 -74.0754 -0.70909 -55.7929 4.20043 -28.7369f7 -62.7898 -1.42823 -52.0554 4.09011 -28.9074keeps at high level before population converges,it dominates the mutation behaviour during thisperiod. However, the selection ratio of Gaussiansurpasses Levy after 500 generation for functionf7. The selection ratio of Cauchy almost keepsat a very low level during the optimization runfor all test problems.The experimental results validate our expec-tation that one mutation operator may performbetter than other ones on a certain type of prob-lems but worse on another type of problems.Even for a particular problem, different muta-tion operators are needed at different evolvingstage to achieve a good result.4 ConclusionsThis paper investigates three mutation opera-tors that are based on the particle velocity forPSO. A technique of how to design an adaptivemutation operator is presented in this paper andan adaptive mutation operator is proposed forPSO. By introducing mutation, PSO greatly im-proves its global search capability without losingits fast convergence property. Though differentmutation operators give a different performanceon different test problems, the adaptive muta-tion operator shows a balanced performance onall test problems and it gives the best perfor-mance on some problems. That is, the adaptivemutation is more robust than any other muta-tion operators investigated in this paper. Hence,the integration of adaptive mutation operators isa promising work for improving the performanceof PSO.AcknowledgementThis work was supported by the Engineering andPhysical Sciences Research Council (EPSRC) ofUK under Grant EP/E060722/1.References[1] F. van den Bergh. An Analysis of Parti-cle Swarm Optimizers. PhD thesis, Depart-ment of Computer Science, University ofPretoria, South Africa, 2002.[2] R. C. Eberhart and J. Kennedy. A new opti-mizer using particle swarm theory. Proc. ofthe 6th Int. Symp. on Micro Machine andHuman Science, pp. 39-43, 1995.[3] S. C. Esquivel and C. A. Coello Coello. Onthe use of particle swarm optimization withmultimodal functions, Proc. of the 2003IEEE Congr. on Evol. Comput., pp. 1130-1136, 2003.[4] N. Higashi and H. Iba. Particle swarm op-timization with Gaussian mutation, Proc.of the 2003 IEEE Swarm Intelligence Sym-phosium, pp. 72-79, 2003.[5] J. Kennedy and R. C. Eberhart. Par-ticle Swarm Optimization. Proc. of the1995 IEEE Int. Conf. on Neural Networks.pp. 1942-1948, 1995.[6] J. Kennedy and R. C. Eberhart. SwarmIntelligence. Morgan Kaufmann Publishers,2001.[7] R. A. Krohling. Gaussian particle swarmwith jumps. Proc. of the 2005 IEEECongr. on Evol. Comput., pp. 1226-1231,2005.[8] R. A. Krohling and L. dos Santos Coelho.PSO-E: Particle swarm with exponen-tial distribution. Proc. of the 2006 IEEECongr. on Evol. Comput., pp. 1428-1433,2006.[9] C. Li, Y. Liu, L. Kang, and A. Zhou. AFast Particle Swarm Optimization Algo-rithm with Cauchy Mutation and NaturalSelection Strategy. ISICA2007, LNCS4683,pp. 334-343, 2007.[10] R. Poli, J. Kennedy, and T. Blackwell. Par-ticle Swarm Optimization: An overview.Swarm Intelligence , 1(1): 33-58, 2007.[11] A. Ratnaweera, S. K. Halgamuge, andH. C. Watson. Self-organizing hierarchicalparticle swarm optimizer with time-varyingacceleration coefficients. IEEE Trans. onEvol. Comput., 8(3): 240-255, 2004.[12] Y. Shi and R. C. Eberhart. A Modified Par-ticle Swarm Optimizer. Proc. of the IEEEInt. Conf. on Evol. Comput., pp. 69-73,1998.[13] A. Stacey, M. Jancic, and I. Grundy.Particle swarm optimization with muta-tion, Proc. of the 2003 IEEE Congr. onEvol. Comput., pp. 1425-1430,2003.[14] H. Wang, Y. Liu, C. Li, and S. Zeng,A Hybrid Particle Swarm Algorithm withCauchy Mutation, Proc.of the 2007 IEEESwarm Intelligence Symposium, 2007.[15] H. Wang, Y. Liu, S. Zeng, and C. Li.Opposition-based Particle Swarm Algo-rithm with Cauchy Mutation. Proc. of the2007 IEEE Congr. on Evol. Comput., 2007.[16] X. Yao and Y. Liu. Fast evolutionaryprogramming, Proc. of the Fifth AnnualConference on Evolutionary Programming(EP’96), pp. 451-460, 1996.[17] X. Yao, Y. Liu, and G. Lin. Evolutionaryprogramming made faster. IEEE Trans. onEvol. Comput., 3(1): 82-102, 1999.

A Fast Particle Swarm Optimization Algorithm

A Hybrid Particle Swarm Algorithm

A Modi¯ed Particle Swarm Optimizer.

A new optimizer using particle swarm theory.

An Analysis of Particle Swarm Optimizers.

Evolutionary programming made faster.

Fast evolutionary programming,

Gaussian particle swarm with jumps.

Krohling and L. dos Santos Coelho. PSO-E: Particle swarm with exponential distribution.

On the use of particle swarm optimization with multimodal functions,

Opposition-based Particle Swarm Algorithm with Cauchy Mutation.

Particle swarm optimization with Gaussian mutation,

Particle swarm optimization with mutation,

Particle Swarm Optimization: An overview.

Particle Swarm Optimization.

Self-organizing hierarchical particle swarm optimizer with time-varying acceleration coe±cients.

Swarm Intelligence.

An adaptive mutation operator for particle swarm optimization

Particle swarm optimization (PSO) is an e cient tool for optimization and search problems. However, it is easy to be trapped into local optima due to its information sharing mechanism. Many research works have shown that mutation operators can help PSO prevent premature convergence. In this paper, several mutation operators that are based on the global best particle are investigated and compared for PSO. An adaptive mutation operator is designed. Experimental results show that these mutation operators can greatly enhance the performanceof PSO. The adaptive mutation operator shows great advantages over non-adaptive mutation operators on a set of benchmark test problems

An adaptive mutation operator for particle swarm optimization

Abstract

Similar works

Full text

Available Versions

Brunel University Research Archive

De Montfort University Open Research Archive