A Modified Particle Swarm Optimizer Algorithm

A Modified Particle Swarm Optimizer AlgorithmYang Guangyou(School of Mechanical Engineering, Hubei University of Technology, Wuhan, 430068,China)deals with this issue, yet conceptually it is simple as well as being very easy to implement. In the algorithm, through periodically monitoring aggregation degree of the particle swarm and on the later development of the PSO algorithm, it has been taken strategy of the Gaussian mutation to the best particle’s position, which enhanced the particles’ capacity to jump out of local minima.Abstract: This paper presented a modified particle swarm optimizer algorithm (MPSO). The aggregation degree of the particle swarm was introduced. The particles’ diversity was improved through periodically monitoring aggregation degree of the particle swarm. On the later development of the PSO algorithm, it has been taken strategy of the Gaussian mutation to the best particle’s position, which enhanced the particles’ capacity to jump out of local minima. Several typical benchmark functions with different dimensions have been used for testing. The simulation results show that the proposed method improves the convergence precision and speed of PSO algorithm effectively.2 Standard PSO Algori thmPSO simulates the behaviors of bird flocking and used it to solve the optimization problems. PSO is initialized with a group of random particles (solutions, xi) and then searches for optima by updating generations. In every generation, each particle is updated by following two "best" values. The first one is the best solution (fitness) it has achieved so far. This value is called pbest (p Keywords: Particle Swarm, Aggregation Degree, Mutation, Optimization.1 Introducti onThe particle swarm optimization (PSO) is a new community intelligence optimization method firstly proposed by Kennedy and Eberhart in 1995 [1,2]. To be exactly similar with other global optimization algorithm, the PSO algorithm tends to suffer from premature convergence. In order to overcome the problem of premature convergence, many solutions and improvements have been suggested, they included: changing descend direction of the best particle[3]; renewing state of the whole swarm or the some particles according to the certain standards [4];introducing the concepts about the breeding and subpopulation of GA [5,6,7]; introducing diversity measure to control diversity of the swarm[8]; adopting new position and velocity of particle to renew equation; cooperative particle swarm optimizers [9,10] etc.. In this paper, we presented a new model whichi ). Another "best" value that is tracked by the particle swarm optimizer is the best value, obtained so far by any particle in the population. This best value is a global best and called gbest (p g ).The velocity and positions of each particle is updated according to their best encountered position and the best position encountered by any particle according to the following equation.12**()*()*()*()id id id id gd id v w v c rand p x c rand p x (1)id id id x x v (2)Where v id is the particle velocity in d-dimension, x id is the current particle position (solution) in d-dimension, w is the inertia weight. p and p i g are defined as stated before, rand() is a random function in the range [0,1]. c1 and c2are learning factors, usually c1 = c2 = 2. If the velocity is higher than a certain limit, called Vmax , this limit will be used as the new2-6751-4244-1135-1/07/$25.00 ©2007 IEEE.velocity for this particle in this dimension, thus keeping the particles within the search space._*(1new gbest gbest )KV (4)3 The Modi fi ed PSO Algori thmwhere V is a random number with Gaussian distribution. Thus it could not only produce the lesser disturbance scope with the bigger probability to perform local searching, but also appropriately produce the bigger disturbance scope to jump out of local optima. The initial value of K is set 1.0, K = EK , at f generations interval , E is a random number in the range [0.01, 0.9].Studies indicated that excessively concentrated particle swarm is easy to run into local minima due to loss of diversity of population. If the aggregation degree of particle swarm could be controlled validly, the ability to search the global minimum would be improved.3.1 Aggregation Degree of the Particles Swarm 3.3 MPSO AlgorithmThe aggregation degree of the particle swarm is used to describe the discrete degree of the swarm, namely diversity. It is represented as a distance between particles. In this paper, we applied the absolute difference value of each dimensional coordinate to denote the distance, and defined its biggest value as the aggregation degree of the particle swarm. If m is the size of swarm, N is the dimensionality of the problem, 3The modified PSO (MPSO) adds monitoring the aggregation degree of the particle swarm periodically to the standard PSO. Furthermore, the mutation operation for the gbest is performed when it do not reach global optimum point. The processes of implementing MPSO are listed as follows:Step 1. Set current iteration generation Iter=1.Initialize a population including m particles; Set the current position as the pbest position, the gbest is the best particle position of initialization particle swarm.id is the d th value of the i th particle and M id is the d th value of the j th particle. The Aggregation Degree of the swarm is calculated according to the following formula.Step 2. Evaluate the fitness for each particle;Step 3. Compare the evaluated fitness value of each particle with its pbest . If current value is better than pbest , then set the current position as the pbest position. Furthermore, if current value is better than gbest , then reset gbest to the current index in particle array;(){,,=1,2,,;;1,2,}id jd d t m a x x -x i j mi j d N z ""˙ (3)Between distances of particles in particle swarm also are available with Euclidean space mean distance seeing reference [6], but its computation load is relative big.Step 4. Change the velocity and position of the particle according to the equations (1) and (2), respectively;3.2 Strategy of MutationThe mutation operator of the algorithm includes two parts. One is, when monitoring periodically the aggregation degree of particle swarm (for example period=50), if the aggregation degree is less than the given value (d ˄t ˅< e ), then all particles’ position and velocity should be reinitialized, however, the pbest and the gbest would be reserved. The other is that when the PSO algorithm can not searches the global optimal point, the mutation for the gbest would be performed as follow:Step 5. if (Iter%Ie ==0) {Calculate d(t) of aggregation degree according to the equation (3);if d(t) is less then given threshold value e, reinitialize velocities and position of particle; }Step 6. Iter = Iter +1, If a stop criterion is met, end algorithm; else execute mutation operation to the gbest according to equation (4)ˈturn to step 2.2-6764 Results and Discussion4.1. Benchmark FunctionsComparison functions adopted here are benchmark functions used by many researchers. In which x represents a real number type the vector, its dimension is n, but x is ith element.i The function f1 is the generalized Rastrigrin function:211()(10cos(2)10),[5.12,5.12]ni i i i f x x x x S¦The function f 2 is the generalized Griewank function:22111()1,[600,600]4000n n i i i i f x x x¦ <The function f 3 is the Rosenbrock function:2223111()(100()(1)),[30,30]ni i i i f x x x x x¦The function f 4 is the Ackley function:411()20exp exp cos(2)20,n i i f x x n S §§·u ¨¨¸¨©¹©¦[30,30]i x e 4.2 Results and AnalysisFor the purpose of comparison, we had 50 trial runs for every instance. The size of particle swarm is 30; the goal value of all function is 1.0e-10; the maximum number of iteration is 3000. Table 1 lists the Vmax values for all the functions, respectively. Table 2 to 4 lists the results of both standard PSO and modified PSO for the four benchmark functions with 10, 20 and 30 dimensions. Where PSO is the results of standard PSO with a linearly decreasing w which from 0.7 to 0.4 or be fixed as 0.4, respectively. MPSO is the results of the modified PSO, in which inertia weight w is as 0.375 and interval Ie is set as 50. The Avg /Std stands for average value and standard deviation of fitness value of 50 trail runs, the fev als meant the average number of function call, the Ras representedratio of the number of reaching goal to the number of experiment. When the fitness value less than 1.0e-10, it is zero. By comparing the results, it is easy to see that MPSO have better results than PSO for all cases. Fig. 1 to 4 show typical convergence results of PSO and MPSO during 3000 generations for the four benchmark functions with 10 and 30 dimensions, respectively. In each case it is seen that MPSO performs better than PSO.Table 1. The initial range of benchmark function Function Vmax Function Vmax f110f3100f2600f430Table 2. The results of both PSO and MPSO with 10 dimensionsStdPSO MPSOFun.Avg/Std fevalsRas Avg/Std fevalsRasf1 2.965/1.35690009.001/500/017745.0050/50f20.091/0.03890009.000/500.023/0.083 32814.6044/50f313.375/25.07190030.000/507.102/0.333 90030.000/50f40/066064.8050/500/035025.6050/50Table 3. The results of both PSO and MPSO with 20 dimensionsStdPSO MPSOFun.Avg/Std fevalsRas Avg/Std fevalsRasf115.342/4.973590030.000/500/020259.0050/50f20.029/0.02288887.604/500.029/0.124 24218.4046/50f381.864/179.31388887.604/5017.433/0.235 90030.000/50f40/088155.0048/500/052959.0049/50Table 4. The results of both PSO and MPSO with 30 dimensionsStdPSO MPSOFun.Avg/Std fevals Ras Avg/Std fevals Rasf 40.020/7.86090030.000/500/020380.2050/501f 0.012/0.015 89724.009/500/017225.4050/502f 130.629/280.34990030.000/5027.962/0.464 90030.000/503f 0.019/0.132 90030.000/500/4.80E-10 58921.8046/5042-677101010-510105Rastrigrin function with popu.=30,dim=30generationslo g 10(f i t n e s s )101010-5100105Rastrigrin function with popu.=30,dim=10generationsl o g 10(f i t n e s s )(a) dimension = 10 (b) dimension = 30Fig. 1. Performance comparison on Rastrigin101010-5100105Griewank function with popu.=30,dim=30generationslo g 10(f i t n e s s )101010-810-610-410-2100102Griewank function with popu.=30,dim=10generationslo g 10(f i t n e s s )(a) dimension = 10 (b) dimension = 30Fig. 2. Performance comparison on Griewank1010101010Rosenbrock function with popu.=30,dim=10generationslo g 10(f i t n e s s)101021041061081010Rosenbrock function with popu.=30,dim=30generationslo g 10(f i t n e s s )(a) dimension = 10 (b) dimension = 30Fig. 3. Performance comparison on Rosenbrock2-678101010-810-610-410-2100102Ackley function with popu.=30,dim=10generationslo g 10(f i t n e s s )101010-810-610-410-2100102Ackley function with popu.=30,dim=30generationslo g 10(f i t n e s s )(a) dimension = 10 (b) dimension = 30Fig. 4. Performance comparison on Ackley[4] Xiaofeng Xie, Wenjun Zhang, Zhilian Yang, Dissipative particle swarm optimization[C] In: Proceedings of the 2002 Congress on Evolutionary Computation, CEC’02, vol.2, 2002: l456-l46l5 Conclus i onIn this paper, we presented a modified PSO. Two new features are added to PSO: Two newfeatures are added to PSO: aggregation degree and strategy of the Gaussian mutation. The simulation results show that the proposed method improves the convergence precision and speed of PSO algorithm effectively. The method shows good performance in all test cases than the standard PSO method. The further study on application is the subject of future work.[5] Angeline P J, Evolutionary optimization versus particle swarm optimization: Philosophy and performance differences[C] In: Evolutionary programming VII, 1998: 601-610[6] Lovbjerg M, Rasmussen T K, Krink T, Hybrid particle swarm optimization with breeding and subpopulations[C] In: Proc of the third Genetic and Evolutionary computation conference, San Francisco, USA, 200l [7] Angeline Peter J, Using selection to improve particle swarm optimization[C] In: Proceedings of the IEEE Conference on Evolutionary Computation, ICEC, 1998: 84-89AcknowledgmentsThis work is supported by Hubei Key lab of Manufacturing [8] Riget, J. and Vesterstroem, J. S. A Diversity guided particle swarm optimizer-the ARPSO. [R] Aarhus: University of Aarhus, EVALife, 2002.2.2Quality Engineering(LMQ2005A04).References[9]Yu, H .J.,Zhang, L.P.,and H u,S.X. Adaptive particle swarm optimization algorithm based, [J] Journal of Zhejiang University (Engineering), 2005 Vol.39 No.9 1286-1291[1]J.Kennedy and R.C. Eberhart, Particle Swarm Optimization[C], Proc. on feedback mechanism IEEE Int’l. Conf. on Neural Networks, vol. VI, 1942-1948, IEEE Service Center, 1995[10] Van den Bergh F. Engelbrecht A P, Training product unit networks using cooperative particle swarm optimizers[C] In: Proc of the third Genetic and Evolutionary computation conference, San Francisco, USA, 200l, l26-l3l[2] R. Eberhart, J. Kennedy. A new optimizer using particle swarm theory. Proc. 6th Int. Symposium on Micro Machine and Human Science, 1995: 39-43[3] Thiemo Krink, Jakob S VesterstrOm, Jacques Riget, Particle Swarm Optimization with Spatial Particle Extension[C] In: Proceedings of the 2002 Congress on Evolutionary Computation, vol. 2, 2002: 1474-1479Author BiographyYang Guangyou , Ph.D., Professor. Research interests are inthe area of intelligent computing and control. 2-679。