Particle Swarm Optimization (PSO) is a recently developed optimization method, which has attracted interest of researchers in various areas due to its simplicity and effectiveness, and many variants have been proposed. In this paper, a novel Particle Swarm Optimization algorithm is presented, in which the information of the best neighbor of each particle and the best particle of the entire population in the current iteration is considered. Meanwhile, to avoid premature, an abandoned mechanism is used. Furthermore, for improving the global convergence speed of our algorithm, a chaotic search is adopted in the best solution of the current iteration. To verify the performance of our algorithm, standard test functions have been employed. The experimental results show that the algorithm is much more robust and efficient than some existing Particle Swarm Optimization algorithms.
This paper considers the following global optimization problem:
Many real-world problems, such as engineering and related areas, can be reduced to formulation (
Among these stochastic algorithms, PSO is a population-based and intelligent method, which is inspired by the emergent motion of a flock of birds searching for food [
Although PSO algorithm has been applied successfully in solving many difficult optimization problems, it also has difficulties in keeping balance between exploration and exploitation when solving complex multimodal problems. In order to get a better performance for PSO algorithm, many variants of PSO have been developed. For example, by using random value of inertia weight, Eberhart and Shi proposed a modified PSO, which can track the optima in a dynamic environment [
In this paper, by utilizing the information of the the best neighbor of each particle and the best particle of the entire population in the current iteration, a new Particle Swarm Optimization algorithm is proposed, which is named NPSO. To avoid premature, an abandoned mechanism is presented in our algorithm. Furthermore, for improving the global convergence speed, a chaotic search is implemented in the best solution of each iteration.
The remainder of this paper is organized as follows. Section
Assume that the search space is
In the original PSO, since each particle moves in the search space guided only by its historical best solution
From (
Firstly, we explain how to define the neighbors and to determine the best neighbor
In addition, we can also use a more general and flexible definition to determine a neighbor of
After determining the the best neighbor
In our algorithm, it is obvious that before each particle moves, it first watches the region which is centered by itself, selects the best neighbour, and then uses (
To avoid premature, in our algorithm, an abandoned mechanism is proposed.
Assume that “
From (
To improve the global convergence of NPSO, a chaotic search operator is adopted. Next, we give the details.
Let
By (
Based on the abovementioned explanation, the pseudocode of the NPSO algorithm is given in Algorithm
(1) Initialize a population of velocities (2) Set (3) (4) (5) for (6) for (7) By ( (8) By ( (9) end for (10) if (11) (12) else (13) set (14) end if (15) if (16) set (17) end if (18) end for (19) for (20) if (21) By ( (22) end if (23) end for (24) By ( (25) (26) end while
In this subsection, the performance of NPSO algorithm is compared to PSO algorithm by evaluating convergence and best solution found for 14 benchmark functions, where
Benchmark functions used in experiments.
Functions | Dimension | C | Range | Optimal value |
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30 | US |
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0 |
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30 | US |
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0 |
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30 | UN |
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0 |
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30 | MS |
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0 |
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30 | MS |
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0 |
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30 | UN |
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0 |
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30 | MN |
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0 |
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30 | MS |
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0 |
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30 | US |
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0 |
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30 | UN |
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0 |
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30 | MS |
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0 |
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30 | UN |
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−450 |
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30 | MS |
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−330 |
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30 | US |
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90 |
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C: characteristic, U: unimodal, M: multimodal, N: nonseparable, and S: separable.
The proposed algorithm NPSO and PSO are coded in Matlab 7.0, and the experiments’ platform is a personal computer with Pentium 4, 3.06 GHz CPU, 512 M memory, and Windows XP.
The parameters of algorithms are given as follows. The common parameters are the dimension
NPSO performance comparison with PSO.
Function | Max iteration | Algorithm | Mean | SD | Min |
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1000 | PSO |
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NPSO |
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1000 | PSO |
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NPSO |
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1000 | PSO |
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NPSO |
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1000 | PSO |
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NPSO | 0 | 0 | 0 | ||
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1000 | PSO |
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NPSO | 0 | 0 | 0 | ||
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1000 | PSO |
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NPSO |
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1000 | PSO |
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NPSO |
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1000 | PSO |
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NPSO |
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1000 | PSO |
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NPSO |
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1000 | PSO |
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NPSO |
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1000 | PSO |
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NPSO |
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1000 | PSO |
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NPSO |
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1000 | PSO |
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NPSO |
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1000 | PSO |
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NPSO |
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Convergence rates on test functions.
From Table
In this subsection, to further test the efficiency of NPSO, it is compared with other five algorithms, that is, CPSO [
Twelve benchmark functions are used for the comparison. The characteristics, dimensions, initial range, and formulations of these functions are listed in Table
Benchmark functions used in experiments.
Functions | Dimension ( |
C | Range | Optimal value |
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30 | US |
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0 |
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30 | UN |
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0 |
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30 | UN |
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0 |
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30 | MS |
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0 |
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30 | UN |
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0 |
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30 | US |
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0 |
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30 | MS |
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0 |
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30 | US |
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0 |
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30 | MN |
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0 |
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30 | MS |
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0 |
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30 | MS |
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30 | US |
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0 |
C: characteristic, U: unimodal, M: multimodal, N: nonseparable, and S: separable.
In order to make a fair comparison, the maximum number of function evaluations (maxFEs) is set to 2
The mean and standard deviation of the best solutions of six PSO variants on 12 test problems in 200,000 function evaluations.
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CPSO |
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CLPSO |
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FIPS |
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Frankenstein |
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AIWPSO |
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NPSO |
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CPSO |
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CLPSO |
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FIPS |
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Frankenstein |
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AIWPSO |
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NPSO |
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CPSO |
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CLPSO |
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FIPS |
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Frankenstein |
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AIWPSO |
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NPSO |
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CPSO |
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CLPSO |
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FIPS |
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Frankenstein |
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AIWPSO |
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NPSO |
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From Table
In this paper, by utilizing the information of the best neighbor of each particle and the best solution in the current iteration, we presented a new move equation. After that, based on the other two improvement strategies, a novel Particle Swarm Optimization algorithm NPSO was proposed. The performance of NPSO was compared with the standard PSO and other five variants of PSO. The results showed that NPSO presents promising results for considered problems.
In the future, the adaption of the parameters in NPSO can be studied to improve its performance.
The authors declare that there is no conflict of interest regarding the publication of this paper.
The research was supported by NSFC (U1404105, 11171094); the Key Scientific and Technological Project of Henan Province (142102210058); the Doctoral Scientific Research Foundation of Henan Normal University (qd12103); the Youth Science Foundation of Henan Normal University (2013qk02); Henan Normal University National Research Project to Cultivate the Funded Projects (01016400105); the Henan Normal University Youth Backbone Teacher Training.