Incremental Attribute Based Particle Swarm Optimization - IEEE Xplore

2012 8th International Conference on Natural Computation (ICNC 2012)

Incremental Attribute Based Particle Swarm Optimization Wei Bai∗† ,

Shi Cheng‡§ ,

Emmanuel M. Tadjouddine† ,

Sheng-Uei Guan†

∗ Department

of Computer Science, University of Liverpool, Liverpool, UK of Computer Science and Software Engineering, Xi’an Jiaotong-Liverpool University, Suzhou, China [email protected] [email protected] [email protected] ‡ Department of Electrical Engineering and Electronics, University of Liverpool, Liverpool, UK § Department of Electrical & Electronic Engineering, Xi’an Jiaotong-Liverpool University, Suzhou, China [email protected]

† Department

Abstract—An incremental-attribute based particle swarm optimization (IAPSO) which utilizes incremental learning strategy in function optimization is presented in this paper. Traditionally, particle swarm optimization (PSO) searches all the dimensions at the same time. Decomposition strategy is utilized in IAPSO to decompose the whole search space (D-dimension) into D numbers of one-dimensional space. In this approach, incremental learning strategy optimizes the function by searching the Ddimensional space one by one. Experimental results show that IAPSO gets more accurate and stable results than standard PSO in multimodal problems. IAPSO could avoid the “local optima”, i.e., it has better “exploration” ability than standard PSO. Keywords: Particle swarm optimization; Incremental learning; Multimodal function optimization

I. I NTRODUCTION Function optimization is a research area that has been studied for many decades. In mathematics and computer science, optimization means finding maximal or minimal values of some objective function over a given constraint set [11]. Some functions have many local optimal solutions which make it difficult for optimization algorithms to find the global optimum. Particle Swarm Optimization (PSO) [5], [9] algorithm is one of the evolutionary algorithms that obtain solutions to multimodal function optimization tasks. An optimization algorithm’s performance is influenced by its ability of “exploration” and “exploitation” [2]–[4], [12]. Exploration is the act of searching one space for the purpose of finding some information. In PSO, exploration means the ability of this algorithm to explore different areas in the search space to find the optimal value. On the other hand, exploitation means the ability of this algorithm to explore a specific area to find a candidate solution. The standard PSO algorithm can easily get trapped in local optima when solving complex multimodal problems [10]. This phenomenon is called premature convergence. When premature convergence occurs, the exploration ability of PSO is decreased. This means that all the particles in one swarm cannot explore new areas. PSO is also a population based stochastic optimization algorithm, which suffers from the problem of “curse of dimensionality.” It means that, as the dimensionality of the search space increases the performance of the algorithm will deteriorate [13].

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In this paper, the proposed algorithm IAPSO uses Incremental Attribute Learning Strategy [6]–[8] in PSO. For the whole search space (D-dimension), incremental learning strategy optimizes the function by searching the D-Dimensional space one by one. In each dimension, the good value of that dimension will be found and used to update other dimensions’ values. This is a greedy approach to do the optimization. This paper is organized as follows. Section II reviews the canonical PSO algorithm. Section III describes incremental attribute learning strategy and presents incremental attribute based particle swarm optimization algorithm. Experiments are conducted in Section IV, as well as the analysis and discussion on the result. Finally, Section V concludes with some remarks and future research direction. II. PARTICLE SWARM OPTIMIZATION Particle swarm optimization (PSO) is inspired from the observation of social behaviors in various animals groups such as bird flocking and fishing schooling. In these natural swarms, animals seek food in a collaborative way. Each animal gets the information about food from their own and others’ experiences. In the PSO modal, each member of the swarm is treated as a particle. The location of food is regarded as the minimum or maximum value of one function. Each particle has its fitness value and velocity to search the function space to find the optimal value. The canonical equations to update the velocity and position of one particle are presented in the following [5], [9]: vid =wvid + c1 ∗ rand() ∗ (pid − xid ) + c2 ∗ Rand() ∗ (pgd − xid ) xid =xid + vid

(1) (2)

where w is inertia weight, vid represents the velocity of the ith particle, xid represents the position of the ith particle. pid is the personal best position for the ith particle, pgd is the best position for the whole swarm. c1 and c2 are the acceleration constants, rand() and Rand() are two random numbers in the range of [0, 1]. The pseudo code of the canonical PSO is given in Algorithm 1.

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Algorithm 1 The Pseudo Code of canonical PSO 1: f : denotes the function 2: P.xi : position of ith particle 3: P.bi : personal best position 4: P.ˆ b: previous best position of the whole swarm 5: Initialize a swarm of D-dimensional particles 6: while not find “good enough” solution or not reach the maximum iteration do 7: for each particle i ∈ [1 · · · s]: do 8: if f (P.xi ) < f (P.bi ) then 9: P.bi = P.xi 10: end if 11: if f (P.bi ) < f (P.ˆb) then 12: P.ˆb = P.bi 13: end if 14: Update each dimension using equations (1) and (2) 15: end for 16: end while

III. I NCREMENTAL ATTRIBUTE BASED PARTICLE S WARM O PTIMIZATION (IAPSO) A. Incremental Attribute Learning Strategy A problem faced by PSO is that in a D-dimensional search space, the improvement of fitness in some dimensions could lead to poor solutions in the other dimensions. Because the fitness value is computed only after all the D-dimensional components are updated in canonical PSO. In [13], multiple swarms were used to optimize different components of the D-dimensional search space cooperatively. Guan et al. [6]–[8] applied an incremental attribute learning strategy to improve the performance in supervised or unsupervised learning and function optimization problems. In their incremental learning strategy, only one variable is in focus when the search is started and then other variables are added incrementally. In this work, the incremental attribute learning strategy is used to optimize particle swarm algorithm. Figure 1 is a flowchart of the incremental attribute learning strategy. In this flowchart, one dimension of a function is treated as one attribute. So a D-dimension function has D (D is an integer) attributes. Letter i denotes the ith dimension of one function. Decomposition is a basic strategy in traditional optimization [15]. Our algorithm also utilized decomposition strategy to decrease the hardness of problems. The flowchart starts in the upper left corner and it contains one main loop to optimize each dimension of one function. At the first step, the Ddimensional search space of one function is decomposed to D numbers of 1-dimensional vectors. Then the process starts with the ith dimension, i is initialized to 1. The next step is “Search in dimension i”. In this step, the positions of all the particles in the ith dimension will be updated. When the “Search in dimension i” is satisfied, the next step is “Calculate Fitness (D-dimension)”. All the D dimensions information of one particle is used to calculate each individual’s (particle’s) fitness in this procedure.

Then i, is compared to D. If i is equal to D, then the entire process ends at “Output Global Best Value and Stop.” Otherwise, i, is incremented to i+1. In the new loop, the i+1 dimension will be optimized. Until the last dimension of the function is optimized, the main loop ends. This strategy is similar to the “divide and conquer” strategy. After the D dimensions of one function are divided into D numbers of 1-dimensional vectors, the step “Search In Dimension i” runs, each individual (particle) find its optimal position in that dimension. Then the optimal positions in ith dimension will be used to optimize the position of these individuals in dimension i + 1.This is a “greedy” way to do the optimization. After all the individuals find their optimal positions in each dimension, it could lead to a globally optimal solution. This optimization strategy can solve the problem called “Two Steps Forward, One Step Back” [13] which may occur in PSO. This problem means that when PSO is used to optimize say a 3-dimensional function optimization problem, two dimensions’ improvement may cause the third dimension getting a worse value. By considering dimensions one by one, the bad influence between different dimensions could be reduced, because for every individual, the personal best position in each dimension was recorded. B. Iterative Incremental Learning Strategy The Incremental attribute learning strategy can be used repeatedly during the optimization. Figure 2 is the flowchart of the iterative incremental learning strategy that implements incremental attribute learning strategy repeatedly. The inner loop of this flowchart is identical to that of Figure 1. After all the dimensions of one function have been optimized, the inner loop ends. In the outer loop, j is the current repeat number and it is initialized to 1. The variable G refers to the total number of repeating time. When the outer loop starts, the current repeat number, j, is tested as to whether it equals to G. If it does not, j is incremented and a new loop begins. If j is equals to G, the program outputs the global best value and stops. In this strategy, the total number of repeating time G can be treated as the number of generations in PSO. That is to say, in each generation of these individuals (particles), incremental attribute learning strategy is used once. In jth generation, all of the D dimensions of one function have been updated. Then j + 1 generation starts a new update with jth generation’s result. C. IAPSO algorithm IAPSO introduces Incremental Attribute Learning Strategy to PSO. For the whole search space (D-Dimension), incremental attribute learning strategy is used to optimize the function by searching the D-dimensional space one by one. The flowchart of this algorithm is shown in Figure 3. The first step is to initialize the positions, velocities and personal best value of each particle in the search space and set the global best value in the swarm. Then for each generation of the

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Fig. 1.

Flowchart of the incremental attribute learning strategy

Fig. 2.

Flowchart of the iterative incremental learning strategy

swarm, in each dimension of the function, update the positions and velocities of these particles based on equation (1) and (2) 4 times. 4 is a heuristic number which can lead to a better optimal result. The fitness of each particle will be calculated after each update. The whole program outputs the global best value and stops when the max generation time is satisfied. IV. E XPERIMENTAL R ESULTS AND D ISCUSSIONS A. Parameter Setting

Fig. 3.

Flowchart of the IAPSO

The parameters used in experiments were set as follows [1]: • Population size: 160. • Inertia weight: w = 0.72984. • Acceleration constants: c1 = c2 = 1.496172. • Function evaluation times: 60000 for D = 50, 120000 for D = 100. • Stop criteria: – Function evaluation times reached. – Minimum value 0 found. Population size is one of the factors that influence the optimal result of IAPSO. On one hand, if the number of particles in one swarm is too small, the “exploration” ability of the swarm will be reduced. On the other hand, if the number is too big, the time complexity of this algorithm will increase. In our experiment, 160 is an heuristic number which can lead to better results while not increase the time complexity too much. For the function evaluation times, in IAPSO, it is equal to G ∗ r ∗ D, where G is the max generation times of the swarm, r is the update time in each dimension of one function

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(r is set at 4 in the experiment) and D is the total number of dimensions in one function. In the experiment, when D = 50, r = 4, function evaluation times is set at 60000. Based on the equation 60000 = G ∗ 4 ∗ 50, the max generation times, G, is 300. The max generation times is also 300 when D = 100. B. Benchmark Function The performance of the proposed algorithm is tested in some multimodal benchmark functions [14]. Because of the limitation of space, four multimodal functions were listed to compare both PSO and IAPSO in this paper. Table I shows the details of these four benchmark functions: C. Experimental Results and Discussions TABLE II N ONCONTINUOUS R ASTRIGIN F UNCTION (D IMENSION = 50, RUN T IME =50)

Result Best Median Worst Mean

PSO 32.0000000000002 120 204 112.640000000002

IAPSO 21.0000003030006 64.0491506226967 134.000020310886 65.4223177420262

Tables II, III, IV, and V show the results of benchmark functions with dimension 50 and Figure 4 display the fitness changing over function evaluations. The middle values of these results were used to draw these figures. Tables VI, VII, VIII, and IX show the results of benchmark functions with dimension 100 and the convergence rate for each algorithm is presented in Figure 5. Each algorithm runs 50 times. From tables II, III, IV, and V, all these functions with D=50, we observe that IAPSO can get best values as PSO does. In Noncontinuous Rastrigin Function, IAPSO gets better results in all aspects. IAPSO gets much better results in Ackley Function than PSO on middle, worst and mean values. IAPSO surpasses PSO on worst and mean results in Griewank Function. In Generalized Penalized Function, IAPSO gets much better results on middle, worst and mean values than PSO. For D = 100, according to tables VI, VII, and VIII, IAPSO gets more accurate results in all aspects except that IAPSO gets the same best value as PSO in Griewank Function. Comparing all these results from tables, IAPSO gets more stable results than PSO when the dimension of these functions has increased. TABLE VI N ONCONTINUOUS R ASTRIGIN F UNCTION (D IMENSION =100, RUN T IME =50)

TABLE III ACKLEY F UNCTION (D IMENSION =50, RUN T IME =50)


PSO 1.4654943925E-14 1.15514850270983 14.9902685922491 2.57774800078812


IAPSO 1.46549439250E-14 2.17603712826E-14 4.30766533554E-14 2.38209452163E-14

PSO 140.000000000005 288.000000000097 603.000000001311 298.620000011377

IAPSO 133.004257747039 249.006493314756 455.000264519153 262.54740515019

TABLE VII ACKLEY F UNCTION (D IMENSION =100, RUN T IME =50) TABLE IV G RIEWANK F UNCTION (D IMENSION =50, RUN T IME =50)


PSO 0 0.00739604033411 90.7470094566445 3.62979602487837


IAPSO 0 0.0073960403341150 0.0660715062296813 0.0089994622849508

PSO 9.4226462131E-33 0.06220141215029 1.75006860865319 0.21195792378163


IAPSO 9.42264621310E-33 3.88522167545E-32 0.062201412150297 0.001244028243005

This section presents the experimental results of both traditional PSO and IAPSO. The best, median, worst and mean values of 50 run times are calculated with the same function evaluation time in each algorithm. The middle values of these results are selected to draw the convergence figures.

IAPSO 5.101385980E-11 4.304863132E-10 1.4235364879729 0.0778188378343

TABLE VIII G RIEWANK F UNCTION (D IMENSION =100, RUN T IME =50)

TABLE V G ENERALIZED P ENALIZED F UNCTION (D IMENSION = 50, RUN T IME =50).


PSO 1.72661744742322 12.9718350132823 18.1503080702469 11.2111404237847

PSO 0 90.9803161636051 360.979343546945 126.470042547612

IAPSO 0 0 0.0705993830033305 0.0090633125100222

By observing the convergence figures of IAPSO and PSO, we find that PSO gets trapped in local optima very quickly while IAPSO searches the search space dimensions one by one and could jump out of local optima. These figures also show that IAPSO also improves the swarm’s “exploration” ability.

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(a)

(b)

(c)

(d)

Fig. 4. Convergence of IAPSO and PSO on benchmark functions (D=50): (a) Noncontinuous Rastrigin, (b) Ackley, (c) Griewank, and (d) Generalized Penalized.

(a)

(b)

(c)

(d)

Fig. 5. Convergence of IAPSO and PSO on benchmark functions (D=100): (a) Noncontinuous Rastrigin, (b) Ackley, (c) Griewank, and (d) Generalized Penalized.

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TABLE I T HE BENCHMARK FUNCTIONS USED IN OUR EXPERIMENTAL STUDY, WHERE n IS THE DIMENSION OF EACH PROBLEM , fmin IS THE MINIMUM VALUE OF THE FUNCTION , AND SEARCH SPACE S ⊆ Rn .

Function Noncontinuous Rastrigin Ackley Griewank Generalized Penalized

Test Function n P [yi2 − 10 cos(2πyi ) + 10] f1 (x) = n i=1 50/100 xi |xi | < 12 yi = round(2xi ) 1 |xiq |≥ 2 2 P 2 f2 (x) = −20 exp −0.2 n1 n x i 50/100 i=1 P − exp n1 n i ) + 20 + e i=1 cos(2πx P Q n n x 2 1 √i f3 (x) = 4000 50/100 i=1 xi − i=1 cos( i ) + 1 P n−1 2 π f4 (x) = n {10 sin (πy1 ) + i=1 (yi − 1)2 50/100 ×[1 10 sin2 (πyi+1 )] + (yn − 1)2 } P+ n + i=1 u(xi , 10, 100, 4) yi = 1 + 14 (xi +  1)  k(xi − a)m xi > a, 0 −a < xi < a u(xi , a, k, m) =  k(−x − a)m xi < −a i

TABLE IX G ENERALIZED P ENALIZED F UNCTION (D IMENSION =100, RUN T IME =50).


PSO 4.71132310655E-33 0.031100706075148 0.788995697611049 0.127699223048925

IAPSO 4.889018957578E-22 1.316591049873E-19 0.0932891936157191 0.0105739815733561

V. C ONCLUSIONS In this paper, an improved version of conventional PSO named IAPSO was proposed for multimodal function optimization. PSO are easily to have premature convergence when solving multimodal functions. By using incremental attribute learning strategy, IAPSO makes use of the information in each dimension that helps a swarm jump out of local optima areas. In every dimension of an objective function, all particles update their positions and velocities several times to reach a better value in that dimension. In this way, IAPSO gets good value from each dimension and this “greedy” approach leads to a better optimal result than PSO that use all dimensions’ information to update function’s value at one time. This strategy also increases the exploration ability of PSO that makes this algorithm search more new space. From the experiments, we observe that when the dimension of one function became larger, from 50 to 100 in this paper, IAPSO can get more stable results than canonical PSO in multimodal function optimization. In the strategy of incremental attribute learning, the D dimensions of one function are divided into D numbers of 1-dimension vectors, the different update order of these vectors will be considered in our future work. ACKNOWLEDGMENT The authors’ work is partially supported by National Natural Science Foundation of China under grant No.61070085, No.60975080.

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R EFERENCES [1] D. Bratton and J. Kennedy. Defining a standard for particle swarm optimization. In Proceedings of the 2007 IEEE Swarm Intelligence Symposium (SIS 2007), pages 120–127, 2007. [2] S. Cheng and Y. Shi. Diversity control in particle swarm optimization. In Proceedings of 2011 IEEE Symposium on Swarm Intelligence (SIS 2011), pages 110–118, Paris, France, April 2011. [3] S. Cheng and Y. Shi. Normalized population diversity in particle swarm optimization. In Advances in Swarm Intelligence, volume 6728 of LNCS, pages 38–45. Springer Berlin/Heidelberg, 2011. [4] S. Cheng, Y. Shi, and Q. Qin. Experimental Study on Boundary Constraints Handling in Particle Swarm Optimization: From Population Diversity Perspective. International Journal of Swarm Intelligence Research (IJSIR), 2(3):43–69, July-September 2011. [5] R. Eberhart and J. Kennedy. A new optimizer using particle swarm theory. In Proceedings of the Sixth International Symposium on Micro Machine and Human Science, pages 39–43, 1995. [6] S.-U. Guan and S. Li. Incremental learning with respect to new incoming input attributes. Neural Processing Letters, 14(3):241–260, December 2001. [7] S.-U. Guan and W. Mo. Incremental evolution strategy for function optimization. International Journal of Hybrid Intelligence Systems, 3:187–203, 2006. [8] S.-U. Guan and F. Zhu. An incremental approach to genetic algorithms based classification. IEEE Transactions on Systems, Man and Cybernetics Part B, 35(2):227–239, April 2005. [9] J. Kennedy and R. Eberhart. Particle swarm optimization. In Proceedings of IEEE International Conference on Neural Networks (ICNN), pages 1942–1948, 1995. [10] J. J. Liang, A. K. Qin, P. N. Suganthan, and S. Baskar. Comprehensive learning particle swarm optimizer for global optimization of multimodal functions. IEEE Transactions on Evolutionary Computation, 10(3):281– 295, June 2006. [11] R. K. Sundaram. A First Course in Optimization Theory. Cambridge University Press, 1996. [12] F. van den Bergh and A. P. Engelbrecht. Effect of swarm size on cooperative particle swarm optimizers. In Proceedings of the Genetic Evolutionary Computation Conference, pages 892–899, 2001. [13] F. van den Bergh and A. P. Engelbrecht. A cooperative approach to particle swarm optimization. IEEE Transactions on Evolutionary Computation, 8(3):225–239, June 2004. [14] X. Yao, Y. Liu, and G. Lin. Evolutionary programming made faster. IEEE Transactions on Evolutionary Computation, 3(2):82–102, July 1999. [15] Q. Zhang and H. Li. MOEA/D: A multiobjective evolutionary algorithm based on decomposition. IEEE Transactions on Evolutionary Computation, 11(6):712–731, 2007.

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