Research Article  Open Access
Hongtao Ye, Wenguang Luo, Zhenqiang Li, "Convergence Analysis of Particle Swarm Optimizer and Its Improved Algorithm Based on Velocity Differential Evolution", Computational Intelligence and Neuroscience, vol. 2013, Article ID 384125, 7 pages, 2013. https://doi.org/10.1155/2013/384125
Convergence Analysis of Particle Swarm Optimizer and Its Improved Algorithm Based on Velocity Differential Evolution
Abstract
This paper presents an analysis of the relationship of particle velocity and convergence of the particle swarm optimization. Its premature convergence is due to the decrease of particle velocity in search space that leads to a total implosion and ultimately fitness stagnation of the swarm. An improved algorithm which introduces a velocity differential evolution (DE) strategy for the hierarchical particle swarm optimization (HPSO) is proposed to improve its performance. The DE is employed to regulate the particle velocity rather than the traditional particle position in case that the optimal result has not improved after several iterations. The benchmark functions will be illustrated to demonstrate the effectiveness of the proposed method.
1. Introduction
Algorithms to tackle optimization problems include not only classical techniques such as dynamic programming, branchandbound, and gradientbased methods, but also more recent techniques such as metaheuristics [1]. Among the existing metaheuristic algorithms, the particle swarm optimization (PSO) algorithm is a populationbased optimization technique developed by Kennedy and Eberhart in 1995 [2]. The PSO has resulted in a large number of variants of the standard PSO. Some variants are designed to deal with specific applications [3–6], and others are generalized for numerical optimization [7–10]. A hierarchical version of PSO (HPSO) has been proposed by Janson and Middendorf [10]. In HPSO, all particles are arranged in a tree that forms the hierarchy. A particle is influenced by its own best position and the best position particle in its neighborhood. It was shown that HPSO performed very well compared to the standard PSO on unimodal and multimodal test functions [10, 11]. HPSO presents the advantage of being conceptually very simple and requiring low computation time. However, the main disadvantage of HPSO is the risk of a premature search convergence, especially in complex multiple peak search problems.
A number of algorithms combined various algorithmic components, often originating from algorithms of other research areas on optimization. These approaches are commonly referred to as hybrid metaheuristics [12]. The surveys on hybrid algorithms that combine the PSO and differential evolution (DE) [13] were presented recently [14, 15]. These PSODE hybrids usually employ DE to adjust the particle position. But the convergence performance is dependent on the particle velocity. Limiting the velocity can help the particle to get out of local optima traps [16, 17]. In this paper, we will combine these two optimization algorithms and propose the novel hybrid algorithm HPSODE. The DE is employed to regulate the particle velocity rather than the traditional particle position in case that the optimal result has not improved after several iterations. The hybrid algorithm aims to aggregate the advantages of both algorithms to efficiently tackle the optimization problem.
The remainder of this paper is organized as follows. Section 2 briefly describes the basic operations of the PSO, HPSO, and DE algorithms. Section 3 presents an analysis of the relationship of particle velocity and convergence. Section 4 provides the hybrid optimization method: HPSODE. Section 5 reveals the simulations and analysis of HPSODE in solving unconstrained optimization problems. Finally, conclusions are given in Section 6.
2. The PSO, HPSO, and DE Algorithms
2.1. The PSO Algorithm
The PSO [18–20] is a stochastic populationbased optimization approach. Each particle is a dimensional vector, and it consists of a position vector , which represents a candidate solution of the optimization problem, a velocity vector , and a memory vector , which is the best candidate solution encountered by the particle. The velocity and position of the particle are updated in every dimension by where is the inertia weight, which determines how much of the previous velocity the particle is preserved. and are positive constants. and are randomly chosen numbers uniformly distributed in the interval . represents the best position achieved by any member of the population.
2.2. The HPSO Algorithm
In HPSO [21], all particles are arranged in a hierarchy. The hierarchy is defined by the height h, the branching degree bd, and the total number of nodes tnn of the corresponding tree.
In HPSO, the iteration starts with the evaluation of the objective function of each particle at its current position. Then, the new velocity vectors and the new positions for the particles are determined. This means that for particle , the value of in (1) equals , with being the particle in the parent node of the node of particle . HPSO uses only when particle is in the root. If the function value of a particle is better than the function value at its personal best position so far, then the new position is stored in . For each particle in a node of the tree, its own best solution is compared to the best solution found by the particles in the child nodes . If the best of these particles is better than particle , then particles and swap their places within the hierarchy.
2.3. The DE Algorithm
The DE [11, 13, 22] is a stochastic parallel direct search method. More specifically, DE’s basic strategy can be summarized as follows.
Initialization. DE begins with a randomly initiated population of dimensional parameter vectors , as a population for each generation . The initial population of the th parameter of the th vector is where and indicate the lower and upper bounds, respectively. is a uniformly distributed random number lying between 0 and 1.
Mutation. DE mutates and recombines the population to produce a population of trial vectors. Specifically, for each individual , a mutant vector is generated according to where , commonly known as scale factor, is a positive real number. Three other random individuals , , and are sampled randomly from the current population such that , and .
Crossover. DE crosses each vector with a mutant vector: where is called the crossover rate.
Selection. To decide whether or not it should become a member of generation , the trial vector is compared to the target vector using the greedy criterion. The selection operation is described as where is the objective function to be minimized.
3. Relationship of Particle Velocity and Convergence
This section presents an analysis of the relationship of particle velocity and convergence.
Substituting (1) into (2) results in From (2), it is known that Substituting (8) into (7) results in
This recurrence relation can be written as a matrixvector product, so that
The characteristic polynomial of the matrix in (10) is , which has a trivial root of and two other solutions where .
Note that and are both eigenvalues of the matrix in (10). The explicit form of the recurrence relation (9) is then given by where , , and are constants determined by the initial conditions of the system.
Substituting (12) into (8) results in
where , .
Consider
Equation (15) implies that if the PSO algorithm is convergent, the velocity of the particles will decrease to zero or stay unchanged until the end of the iteration.
4. The Proposed HPSODE Algorithm
The main idea of the hybrid HPSODE algorithm is to employ the DE to regulate the particle velocity rather than the traditional particle position in case that the optimal result has not improved after several iterations. If the swarm is going to be in equilibrium, the evolution process will be stagnated as time goes on. To prevent the trend, if the stagnating step of evolution process is larger than threshold value , the particle velocity performs mutation operators. The velocity and position of the particles are updated as follows.
If ( or , ), then where , is a random number in the interval , and and are sampled randomly from .
The procedure for HPSODE algorithm is presented in Algorithm 1.

5. Simulations and Results
In this section, we present a simulation study to validate the proposed HPSODE algorithm. A set of test functions that are commonly used in the field of continuous function optimization is listed in the appendix. They are a set of curvilinear functions for difficult unconstrained minimization problems. For illustration, the landscapes of twodimensional versions of the six functions are depicted in Figure 1. The first two functions (Sphere and Rosenbrock) are unimodal functions, and they have a single local optimum that is also the global optimum. The remaining functions are multimodal, and they have several local optima. Note that the dimensional increase of these scalable functions does not change their basic features.
(a)
(b)
(c)
(d)
(e)
(f)
In our experiments, the HPSO uses the parameter values , and as suggested in [23] for a faster convergence rate. The population size that has been used is . The maximal number of generations uses . The remainder parameters are set as , , , , , and . Thirty independent runs were carried out. The convergence behavior of the HPSO is shown in Figure 2. For comparison purpose, the HPSODE is also given in the same figure. As shown in Figure 2, the convergence performance of the HPSODE is better than the HPSO. HPSODE is compared with HPSO, DE, and PSODE [1] in terms of the selected performance metrics, such as the mean, maximum, and minimum values. In DE, we use DE/rand/1/bin strategy (, ). As shown in Tables 1, 2, and 3, the HPSODE outperforms HPSO, DE, and PSODE. The HPSODE is quite competitive when compared with the other existing methods.



6. Conclusions
In this paper, a new method named HPSODE is proposed to solve optimization problems, which improves the performance of the HPSO by incorporating DE. In HPSODE, when the evolution process is stagnated for several generations, all the particles may lose the ability of finding a better solution. Then, the DE is employed to regulate the particle velocity to avoid wasting too much calculation time for vain search, so the searching efficiency of the HPSODE is improved greatly. The HPSODE is compared on test functions with HPSO, DE, and PSODE. It is shown that HPSODE performs significantly better.
Appendix
Benchmark Functions
Sphere:
Rosenbrock:
Rastrigin:
Griewank:
Ackley:
Schaffer’s F6:
Acknowledgments
This work was supported by the Key Project of Chinese Ministry of Education (no. 212135), the Guangxi Natural Science Foundation (no. 2012GXNSFBA053165), the project of Education Department of Guangxi (no. 201203YB131), and the Doctoral Initiating Project of Guangxi University of Science and Technology (no. 11Z09).
References
 C. Zhang, J. Ning, S. Lu, D. Ouyang, and T. Ding, “A novel hybrid differential evolution and particle swarm optimization algorithm for unconstrained optimization,” Operations Research Letters, vol. 37, no. 2, pp. 117–122, 2009. View at: Publisher Site  Google Scholar
 J. Kennedy and R. Eberhart, “Particle swarm optimization,” in Proceedings of the 1995 IEEE International Conference on Neural Networks, pp. 1942–1948, Perth, Australia, December 1995. View at: Google Scholar
 J. Zhang, J. Wang, and C. Yue, “Small populationbased particle swarm optimization for shortterm hydrothermal scheduling,” IEEE Transactions on Power Systems, vol. 27, no. 1, pp. 142–152, 2012. View at: Publisher Site  Google Scholar
 R. Bhattacharya, T. K. Bhattacharyya, and R. Garg, “Position Mutated hierarchical particle swarm optimization and its application in synthesis of unequally spaced antenna arrays,” IEEE Transactions on Antennas and Propagation, vol. 60, no. 7, pp. 3174–3181, 2012. View at: Google Scholar
 M. A. Cavuslua, C. Karakuzub, and F. Karakayac, “Neural identification of dynamic systems on FPGA with improved PSO learning,” Applied Soft Computing, vol. 12, no. 9, pp. 2707–2718, 2012. View at: Google Scholar
 M. Han, J. Fan, and J. Wang, “A dynamic feedforward neural network based on gaussian particle swarm optimization and its application for predictive control,” IEEE Transactions on Neural Networks, vol. 22, no. 9, pp. 1457–1468, 2011. View at: Publisher Site  Google Scholar
 F. Peng, K. Tang, G. Chen, and X. Yao, “Populationbased algorithm portfolios for numerical optimization,” IEEE Transactions on Evolutionary Computation, vol. 14, no. 5, pp. 782–800, 2010. View at: Publisher Site  Google Scholar
 X. Li and X. Yao, “Cooperatively coevolving particle swarms for large scale optimization,” IEEE Transactions on Evolutionary Computation, vol. 16, no. 2, pp. 210–224, 2012. View at: Publisher Site  Google Scholar
 M. Li, D. Lin, and J. Kou, “A hybrid niching PSO enhanced with recombinationreplacement crowding strategy for multimodal function optimization,” Applied Soft Computing Journal, vol. 12, no. 3, pp. 975–987, 2012. View at: Publisher Site  Google Scholar
 S. Janson and M. Middendorf, “A hierarchical particle swarm optimizer and its adaptive variant,” IEEE Transactions on Systems, Man, and Cybernetics B, vol. 35, no. 6, pp. 1272–1282, 2005. View at: Publisher Site  Google Scholar
 M. G. Epitropakis, D. K. Tasoulis, N. G. Pavlidis, V. P. Plagianakos, and M. N. Vrahatis, “Enhancing differential evolution utilizing proximitybased mutation operators,” IEEE Transactions on Evolutionary Computation, vol. 15, no. 1, pp. 99–119, 2011. View at: Publisher Site  Google Scholar
 C. Blum, J. Puchinger, G. R. Raidl, and A. Roli, “Hybrid metaheuristics in combinatorial optimization: a survey,” Applied Soft Computing Journal, vol. 11, no. 6, pp. 4135–4151, 2011. View at: Publisher Site  Google Scholar
 K. V. Price, R. Storn, and J. Lampinen, Differential Evolution: A Practical Approach to Global Optimization, Springer, Berin, Germany, 2005.
 M. G. Epitropakis, V. P. Plagianakos, and M. N. Vrahatis, “Evolving cognitive and social experience in particle swarm optimization through differential evolution: a hybrid approach,” Information Sciences, vol. 216, pp. 50–92, 2012. View at: Google Scholar
 B. Xin and J. Chen, “A survey and taxonomy on hybrid algorithms based on particle swarm optimization and differential evolution,” Journal of Systems Science and Mathematical Sciences, vol. 31, no. 9, pp. 1130–1150, 2011. View at: Google Scholar
 H. Liu, X. Wang, and G. Tan, “Convergence analysis of particle swarm optimization and its improved algorithm based on chaos,” Control and Decision, vol. 21, no. 6, pp. 636–645, 2006. View at: Google Scholar
 S. Jiang, Q. Wang, and J. Jiang, “Particle swarm optimization algorithm based on velocity differential evolution,” in Proceedings of the Chinese Control and Decision Conference (CCDC '09), pp. 1860–1865, Guilin, China, June 2009. View at: Publisher Site  Google Scholar
 S. Ghosh, S. Das, D. Kundu, K. Suresh, and A. Abraham, “Interparticle communication and searchdynamics of lbest particle swarm optimizers: an analysis,” Information Sciences, vol. 182, no. 1, pp. 156–168, 2012. View at: Publisher Site  Google Scholar
 X. F. Xie, W. J. Zhang, and Z. L. Yang, “A dissipative particle swarm optimization,” in Proceedings of the IEEE Congress on Evolutionary Computation (CEC '02), pp. 1456–1461, Honolulu, Hawaii, USA, May 2002. View at: Google Scholar
 W. Gao, S. Liu, and L. Huang, “Particle swarm optimization with chaotic oppositionbased population initialization and stochastic search technique,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 11, pp. 4316–4327, 2012. View at: Publisher Site  Google Scholar
 S. Janson and M. Middendorf, “A hierarchical particle swarm optimizer for noisy and dynamic environments,” Genetic Programming and Evolvable Machines, vol. 7, no. 4, pp. 329–354, 2006. View at: Publisher Site  Google Scholar
 F. Neri and V. Tirronen, “Recent advances in differential evolution: a survey and experimental analysis,” Artificial Intelligence Review, vol. 33, no. 12, pp. 61–106, 2010. View at: Publisher Site  Google Scholar
 I. C. Trelea, “The particle swarm optimization algorithm: convergence analysis and parameter selection,” Information Processing Letters, vol. 85, no. 6, pp. 317–325, 2003. View at: Publisher Site  Google Scholar
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Copyright © 2013 Hongtao Ye et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.