Rev. Téc. Ing. Univ. Zulia. Vol. 39, Nº 7, 289 ‐ 296, 2016
doi:10.21311/001.39.7.36
Autonomous Groups Particle Swarm Optimization Algorithm Based On Exponential Decreasing Inertia Weight Haojun Li* College of Education Science and Technology, Zhejiang University of Technology, Hangzhou 310023, Zhejiang, China *Corresponding author(E-mail:
[email protected]) Zhongfeng Liu College of Education Science and Technology, ZheJiang University of Technology, Hangzhou 310023, Zhejiang, China Wanliang Wang College of Computer Science and Technology,ZheJiang University of Technology ,Hangzhou 310023,Zhejiang,China
Abstract Aiming at the problem that particle swarm optimization algorithm is easy to fall into local optimal solution, this paper proposes autonomous groups particle swarm optimization algorithm based on exponential decreasing inertia weight. The exponential decreasing inertia weight is using an exponential function, to adjust inertia weight decreasingly. The algorithm uses the exponential decreasing inertia weight to replace the linear inertia weight of autonomous groups particle swarm optimization algorithm, to better balance global search and local search of the algorithm for improving the algorithm’s ability of avoiding falling into local optimal solution. Through the simulation experiments on six benchmark functions, the results show that the exponential decreasing inertia weight better enhances autonomous groups particle swarm optimization algorithm performance. The algorithm has higher ability of avoiding falling into local optimal solution, better stability and better performance in solving high dimensional problems. Key words: Particle Swarm Optimization, Autonomous Groups, Exponential Function, Inertia Weight, Global Optimization. 1. INTRODUCTION Particle Swarm Optimization(PSO)(Kennedy and Eberhart, 1995)is proposed by Kennedy and Eberhart inspired by the foraging behavior of birds. Shi et al. proposed the inertia weight selection method which comes into being the current standard PSO(Shi and Eberhart, 1998).Because of few parameters and simple realization, the algorithm is very popular. These advantages make PSO widely used in many fields, such as multi-object optimization(Moon et al., 2014; García et al., 2014; Shokrian and High, 2014). But the application process of PSO is prone to fall into local optimal solution, and the problem with the increase of problem dimensions will be more prominent. There are many studies been committed to solve this problem. Because the parameters of PSO are few and the parameter tuning is simple, using the method of dynamic parameter tuning to improve the quality of algorithm has become a widely used method. Cognitive parameter and social parameter affect particles’ability of learning individual optimal solution and population optimal solution and can balance global search and local search of algorithm. Due to the complexity of optimization problem, the constant or the linear tuning ways for cognitive parameter and social parameter, in many cases, are not good. In order to solve this problem, Ziyu et al. introduced an exponential time-varying function to tune cognitive parameter and social parameter, in order to make the algorithm converge to global optimum(Ziyu and Dingxue, 2009); Bao et al. proposed an asymmetric time-varying acceleration parameter tuning strategy to balance local search and global search(Bao and Mao, 2009);Cui et al. proposed nonlinear time-varying cognitive parameter and time-varying social parameter, and the social parameter is the function of cognitive parameter(Cui et al., 2008).However, these methods used to tune cognitive and social parameter are for every particle. Due of lacking diversity of particles, they cannot be good solutions to the problem of easily converging to local optimal solution. Mirjalili proposed the mathematical model of groups called autonomous groups strategy to form autonomous groups particle swarm optimization(AGPSO);AGPSO uses different slope, breakpoints, and curvature functions to tune cognitive parameter and social parameter, to better balance global 289
Rev. Téc. Ing. Univ. Zulia. Vol. 39, Nº 7, 289 ‐ 296, 2016 search and local search; AGPSO’s linear inertia weight limits the algorithm further improvement of performance(Mirjalili et al., 2014). Because inertia weight is also an important parameter affecting performance of the algorithm, many papers focus on improving PSO based on inertia weight. Larger inertia weight making particles conducive to global exploration and smaller making particles tend to local exploit. The inertia weight tuning strategy generally uses the way of decrease(Chauhan et al., 2013; Nickabadi et al., 2011; Arasomwan and Adewumi, 2013).Shi et al. proposed a strategy reducing inertia weight linearly and the optimization effect of the algorithm is improved, but the linear decreasing strategy cannot reflect the actual optimization process(Shi and Eberhart, 1999);Shi et al. proposed using fuzzy system to dynamically tune inertia weight for improving the average fitness, but more parameters increase the complexity of the algorithm(Shi and Eberhart, 2001);Eberhart et al. proposed a random tuning strategy of inertia weight, which is mainly applied in the problem of optimizing the mutable target(Eberhart and Shi, 2001);Liu et al. proposed improved PSO algorithms, and their global convergence abilities are improved, but convergence speeds are not stable.(Liu et al., 2007; Tsai et al., 2013; Worasucheep, 2008; Zhou et al., 2010). It is important to balance local search and global search in the whole iterative process for getting global optimal solution(Shi and Eberhart, 2001; Črepinšek et al., 2013).The aforementioned researches focus on inertia weight or cognitive parameter and social parameter, but consider the common effect of them about balancing global search and local search little. In view of this problem, this paper proposed autonomous groups particle swarm optimization algorithm based on exponential decreasing inertia weight (AGWPSO); AGWPSO introduces a new inertia weight tuning method, namely using the exponential function in decreasing way to tune inertia weight, for improving the original linear inertia weight of AGPSO. In this algorithm, the common characteristics of the exponential decreasing inertia weight tuning method and autonomous groups strategy are used to balance global search and local search of algorithm in order to improve the algorithm’s ability of avoiding falling into local optimal solution. 2. STANDARD PSO ALGORITHM PSO is inspired by the foraging behavior of birds in search space and proposed as evolutionary computer technology. Particles in the feasible space fly to find optimal solution. Particles can record the present positions and the historical optimal location and each other shares optimal location information, namely particles are affected by their own optimal solution and population optimal solution. The formula (1) and formula (2) tune speeds and positions of particles in the iterative process. k k Vi,dk+1 = w Vi,dk + c1 rand (pBest i,d - X i,d )+ k c 2 rand (gBest dbest - X i,d )
k+1 k Xi,d = Xi,d + Vi,dk+1
(1) (2)
w controls the stability of PSO which is inertia weight of PSO and is usual among 0.4～0.9; c1 is cognitive parameter which controls the particles’ability of learning individual optimal solution; c 2 is social parameter which controls the particles’ability of learning population optimal solution; c1 and c 2 are both called acceleration k coefficients among (0,2] in general; Vi,dk , Xi,d are the speeds and positions of the d-th dimension of the i-th
particle at the k-th iteration; rand is a random number uniformly from the interval [0,1],which is used to give k is individual optimal position of the d-dimension of the i-th PSO more abilities of random search; pBest i,d particle at k-th iteration; gBest dbest is population best position of the d-th dimension; It can been observed PSO are affected by three parameters: w , c1 and c 2 .Dynamically tuning these parameters gives particles different behavior. 3. AUTONOMOUS GROUPS PARTICLE SWARM OPTIMIZATION ALGORITHM BASED ON EXPONENTIAL DECREASING INERTIA WEIGHT
At present, there are many literatures through improving inertia weight tuning methods to improve particle swarm optimization algorithm(Taherkhani and Safabakhsh, 2016; Chauhan, Deep, and Pant, 2013);This paper introduces the exponential decreasing inertia weight to improve AGPSO by replacing it’s linear inertia weight; Thereby utilize the exponential decreasing inertia weight and autonomous groups strategy to balance local search and global search, in order to make the algorithm have better ability of avoiding falling into local optimal solution. 290
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3.1. Autonomous Groups Particle Swarm Optimization Algorithm The concept of autonomous groups is subject to the inspiration of diversity of animal populations and insect community. In this method, each particle autonomously attempts to search problem space with its own strategy which is based on different tuning methods on c1 and c 2 In AGPSO, firstly, set the number of particles, randomly initialize positions of particles in search space and randomly initialize velocities of particles; Then through fitness function determining fitness value, individual optimal solution and population optimal solution; Particles will be randomly divided into four groups in advance; Particles of different groups update parameters of c1 and c 2 with different methods respectively; Use the formula (3),the formula (4),the formula (5) and the formula (6) to update c1 and c 2 of group 1, group 2, group 3 and group 4;When update their speeds and positions, particles with different updating strategies of c1 and c 2 show different abilities of learning individual optimal and population optimal solutions which will make particles exhibit diversity. Autonomous groups use different strategies to update c1 and c 2 , so the abilities of global exploration and local expolition are different with standard PSO. Dynamic and diverse c1 and c 2 tuning modes give particles different behaviors and abilities of random search ,which increase population diversities, balance local search and global search in the iterative process and improve particles ‘abilities of avoiding falling into local optimal solution(Mirjalili et al., 2014). c1 = 3 - 2 exp[-(4iter/ Max_iter) 2 ] (3) c 2 = 4 - c1 c1 = 3 + 2(
iter 2iter )^ 2-2 Max_iter Max_iter
(4)
iter 2iter )^ 2Max_iter Max_iter
(5)
c 2 = 4 - c1
c1 = 2.5 + ( c 2 = 4 - c1
c1 = 2.5 - exp[-(4iter/ Max_iter) 2 ] c 2 = 4 - c1
(6)
3.2. Exponential Decreasing Inertia Weight The formula (7) is a linear inertia weight tuning function in AGPSO; The formula (8) is an exponential decreasing inertia weight tuning function in AGWPSO. w = wMax- iter(wMax- wMin) / Max_iter
(7)
w = wMin(wMax/ wMin)1/ (1+iter/ Max_iter)
(8)
wMax is the maximum inertia weight and set 0.9; wMin is the minimum inertia weight and set 0.4; iter is the current iteration; Max_iter is the maximum iterations. For formula (8), when iter = 0 , w = wMax = 0.9 ; when iter = Max_iter , w = 0.6 ensuring that w is in { wMin , wMax }. 0.9 Formula(7) Formula(8)
Inertia weight
0.8 0.7
0.6 0.5
0.4
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1500
2000
Iteration
Figure 1. Formula (7) and formula (8) curve 291
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From figure 1,it can be observed that formula (8) is different with the uniform changes of formula (7) and it’s changing rate is fast at first and slow later; Using these two functions to tune inertia weight will both give PSO convergence effect; But with formula (7),PSO once falling into local optimal solution is difficult to jump out; With formula (8),at early stage larger changing rate gives PSO rapid global exploration, so as to determine global optimal region and later smaller changing rate gives PSO meticulous local exploit in smaller speed, so as to ensure the convergence of PSO to global optimal solution; Hence using the exponential function to tune inertia weight is more conducive to balance global search and local search. The exponential decreasing inertia weight is using the exponential function as formula (8) shown to tune inertia weight by decrease. This paper proposes autonomous groups particle swarm optimization algorithm based on exponential decreasing inertia weight (AGWPSO), which uses the exponential decreasing inertia weight and autonomous groups strategy to better balance global search and local search, so as to improve the PSO’abilities of avoiding falling into local optimal solution. 3.3. Basic Steps of Autonomous Groups Particle Swarm Optimization Algorithm Based on Exponential Decreasing Inertia Weight Basic steps of AGWPSO: Step 1 set the number of particles, the maximum number of iterations, and the maximum value of inertia weight and the minimum inertia weight. Step 2 random initialization of particles ‘positions in the feasible region and random initialization of particles’speeds. Step 3 obtain fitness values of particles through the fitness function, and record individual optimal solutions and population optimal solutions. Step 4 using formula (8) to tune inertia weight. Step 5 particles are divided into pre assigned group 1,group 2,group 3and group 4 respectively, using formula (3),formula (4),formula (5) and formula (6) to updates cognitive parameter and social parameter of group 1,group 2,group 3 and group 4. Step 6 group 1, group 2, group 3 and group 4, respectively, use formula (1) to update velocities of particles by group. Step 7 use formula (2) to update particles ‘positions. Step 8 determine whether to meet the termination conditions, if satisfied, that is, to achieve the maximum number of iterations, then implement step 9,otherwise return to step 4. Step 9 terminate the operation, output of the particle's global optimal solution and optimal location. 4. EXPERIMENT AND RESULT 4.1. Test Functions Table 1 lists the six benchmark functions, which are used as test functions to assess the performance of AGWPSO and can be divided into three categories: unimodal functions ( F1 and F2 ),multimodal functions ( F3 and F4 ) and fixed dimensional multimodal functions( F5 and F6 );Unimodal and multimodal functions for evaluating the performance of PSO in high dimensional problems and fixed dimensional multimodal functions for a comprehensive study. The parameter Dim points out dimensions of the benchmark functions; The parameter Range gives ranges of search space at the benchmark functions; the parameter fmin is the minimum value of the benchmark functions. Table 1. Benchmark functions and their parameters Function Dim Range fmin F1 (x) = n x 2 i=1 i
300
[-50,50]
0
F2 (x) = n-1[100(x - x 2 )2 + (x -1)2 ] i=1 i+1 i i 1 n 2 F3 (x) = -20 exp(-0.2 x ) n i=1 i 1 - exp( n cos(2 x )) + 20 + e i n i=1 x 1 n 2 n F4 (x) = i=1 xi - i=1 cos( i ) +1 4000 i
300
[-50,50]
0
300
[-32,32]
0
300
[-600,600]
0
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2
[-5,5]
0.398
F6 (x) = [1+ (x + x +1)2 (19 -14 x + 3 x 2 -14 x + 6 x x + 3 x 2 2 1 2 1 1 2 1 2 2 2 2 ×[30 + (2 x - 3 x ) (18 - 32 x +12 x + 48 x - 36 x x + 27 x 1 2 1 1 2 1 2
[-2,2]
3
F5 (x) =（ x 2
5.1 2 5 1 x + x - 6)2 +10(1- ) cosx +10 1 1 1 2 8 4
4.2. Results and Discussion In order to verify the performance of AGWPSO, we use the following algorithms to carry out simulation experiments: (1) Basic particle swarm optimization algorithm, SPSO(Yuhui. Shi and Eberhart 1998); (2) Particle swarm optimization algorithm, TAPSO(Ziyu and Dingxue 2009); (3) Particle swarm optimization algorithm, IPSO(Cui et al. 2008); (4) Autonomous groups particle swarm optimization algorithm, AGPSO(Mirjalili et al. 2014); (5) Autonomous groups particle swarm optimization algorithm based on exponential decreasing inertia weight, AGWPSO; In order to be easy to observe, the maximum number of iterations for benchmark functions F1 , F2 , F3 and F4 are set 2000 and for benchmark functions F5 and F6 the maximum number of iterations are set 50;SPSO,IPSO,TAPSO and AGPSO through formula (7) to tune inertia weight and AGWPSO through the formula (8) to tune inertia weight; wMax, the maxmum value of inertia weight, is 0.9 and wMin, the minimum value, is 0.4;SPSO’cognitive parameter and social parameter are both 2;The remaining algorithms adopt dynamic tuning method to tune c1 and c 2 ,TAPSO by formula (3) updating c1 and c 2 ,IPSO by the formula (4) updating c1 and c 2 ,AGPSO and AGWPSO by formula (3),formula (4),formula (5) and formula (6) updating c1 and c 2 by group. In order to compare the performance of all the algorithms. The data of simulation experiments is collected by 30 independent experiments; After the termination of the iteration, mean and variance values of the optimal solutions are listed in table 2,the optimal results with bold font display. Table 2. Simulation results of algorithm Function SPSO TAPSO IPSO AGPSO AGWPSO Mean 7.054 4E+03 6.278 9E+03 6.7520E+03 4.512 2E+03 3.647 3E+03 F1 Variance 6.561 5E+05 6.574 2E+05 6.4754E+05 2.788 8E+05 1.854 8E+05 Mean 6.845 7E+07 5.375 9E+07 6.7309E+07 3.205 1E+07 2.444 4E+07 F2 Variance 1.086 6E+14 1.851 9E+14 2.5655E+14 7.786 7E+13 2.369 3E+13 Mean F3 1.452 4E+01 1.408 3E+01 1.3826E+01 1.423 4E+01 1.229 5E+01 Variance 1.259 8E+00 1.120 1E+00 1.4022E+00 1.595 3E+00 1.572 0E-01 Mean 1.002 4E+01 1.498 3E+01 5.8296E+00 5.034 8E+00 3.995 5E+00 F4 Variance 1.317 8E+00 2.463 8E+00 3.2234E-01 1.673 3E-01 1.253 3E-01 Mean 3.979 0E-01 3.979 0E-01 3.9790E-01 3.979 0E-01 3.979 0E-01 F5 Variance 0.000 0E+00 0.000 0E+00 0.0000E+00 0.000 0E+00 0.000 0E+00 Mean 3.000 0E+00 3.000 0E+00 3.0000E+00 3.000 0E+00 3.000 0E+00 F6 Variance 0.000 0E+00 0.000 0E+00 0.0000E+00 0.000 0E+00 0.000 0E+00
The unimodal function only has global minima and no local minima, so this kind of function is especially suitable for evaluating convergence ability of algorithms. From the results of the benchmark functions F1 and F2 in table 2 can be seen, the convergence accuracies of AGWPSO are higher than that of AGPSO at the two unimodal functions, and are the best among all algorithms. From the figure 2 and figure 3, it can be observed that AGWPSO has the best convergence ability at the unimodal functions and can keep a fast convergence speed in the later period of iteration. The results of simulation experiments show that AGWPSO can improve the convergence ability of PSO at unimodal functions; The reason is autonomous groups strategy can make particles have diversity which can make particles make full use of the location close to optimal solution and the exponential decreasing inertia weight can make PSO at early stage determine quickly optimal solution region and keep faster convergence speed to global optimum solution at later stage. Contrary to unimodal functions, multimodal functions have several local minima, which increases exponentially with problem dimensions, so it is suitable to test the algorithms ‘ability of avoiding falling into 293
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local optimal solution. The results of simulation experiments of the benchmark functions F3 and F4 in table 2 show that AGWPSO has better convergence precision than AGPSO at the multimodal functions and is the best of all algorithms. From figure 4 and figure 5, it is seen that in the later period of iteration the convergence speeds of AGWPSO are the fastest and AGWPSO is not trapped in local optimal solution. The results of simulation experiments show that AGWPSO improves PSO in avoiding falling into local optimal solution. The reason is that autonomous groups strategy gives PSO more random search and the exponential decreasing inertia weight makes PSO with smaller speed changing rate in later iterations, which gives PSO meticulous local search to avoid falling into local optimal solution.
Best-so-far 4
10
SPSO IPSO TAPSO AGPSO AGWPSO
10
10
Best-so-far
SPSO IPSO TAPSO AGPSO AGWPSO
5
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9
10
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Figure 2. Evolution of fitness at benchmark function F1
500
1000 Iteration
1500
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Figure 3. Evolution of fitness at benchmark function F2
From the convergence curves of unimodal and multimodal functions and the experimental data can be seen, AGWPSO at benchmark functions F1 , F2 , F3 and F4 throughout the iterative process are always in evolution and has the best convergence accuracy. This shows that at these benchmark functions AGWPSO outperforms AGPSO and has obvious advantage compared with other algorithms. The results of simulation experiments at the unimodal and multimodal functions show that AGWPSO can not only improve the convergence speed of PSO, but can also improve the algorithm ‘ability of avoiding falling into local optimal solution. Compared with unimodal functions and multimodal functions, fixed dimensional multimodal functions have some local minima values. The results of simulation experiments at the benchmark functions F5 and F6 in table 2 show that all the algorithms are equivalent at the convergence of the two fixed dimensional multimodal functions, that is, all the algorithms converge to global optimal solution. Figure 6 and Figure 7 show that the convergence curves of all algorithms are similar and the AGWPSO convergence speed is fast. From experimental data and convergence figures can be seen, because of the low dimensional character of these benchmark functions, differences of experimental results are not great, illustrating the performances of AGWPSO in high dimensional problems are better and more easily observed. Data in table 2 show that the AGWPSO’s mean values of function evolutions at six benchmark functions are the best illustrating the convergence precision of AGWPSO performs best among all the algorithms, and the variances of PSO are minimum illustrating it’s stability at these algorithms is best. These show that the combination of the exponential decreasing inertia weight and autonomous groups strategy can better improve PSO algorithm in convergence accuracy and stability of performance. From the evolution figures of the six benchmark functions can be seen, AGWPSO in the whole iterative process maintains faster convergence speed and AGWPSO evolutionary speed is higher than AGPSO. Except for low dimensional functions F5 and F6 at which all algorithms converge to the global optimum, AGWPSO and AGPSO at the whole iterative process of surplus functions always keep evolution, but the other algorithms quickly fall into local optimal optimization. This shows that at all the algorithms, AGWPSO is best in term of avoiding falling into local optimal solution, namely AGWPSO better balances the global search and local search and improves PSO’ability of avoiding falling into local optimal solution.
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1.3
SPSO IPSO TAPSO AGPSO AGWPSO
SPSO IPSO TAPSO AGPSO AGWPSO
3
10 Best-so-far
Best-so-far
10
1.2
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1000 Iteration
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Figure 5. Evolution of fitness at benchmark function F4
SPSO IPSO TAPSO AGPSO AGWPSO
SPSO IPSO TAPSO AGPSO AGWPSO
2
10 Best-so-far
Best-so-far
Figure 4. Evolution of fitness at benchmark function F3
0
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20 30 Iteration
40
Figure 6. Evolution of fitness at benchmark function F5
50
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20 30 Iteration
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Figure 7. Evolution of fitness at benchmark function F6
5. CONCLUSIONS
According to the problem that PSO is easy to fall into local optimal solution, this paper has proposed the AGWPSO. In order to test whether the proposed algorithm is effective, six benchmark functions have been simulated and the results compared with other algorithms shows AGWPSO is better in avoiding falling into local optimal solution, especially in high dimensional problems. AGWPSO allowing particles to have different individual and social behaviors which makes the algorithm have more random search, coordinates cognitive coefficient, social coefficient and inertia weight to improve the algorithm ‘ability of avoiding trapping into local optimal solution. In the future research, problems, such as tuning the number of groups and applying this algorithm to solve practical problems, have yet to be explored. Acknowledgements
This work was supported by the National-sponsored Social Sciences Funding Program in China under Grant 16BTQ084, the humanities and social science project of the Ministry of Education in China under Grant 15YJCZH023.
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