Mixed-variable Particle Swarm Optimization Algorithm Based on Competitive Coevolution

doi:10.16182/j.issn1004731x.joss.22-1466

Abstract

Abstract:

For the current algorithm, it is difficult to obtain the available solution due to the irregularity of problem decision space caused by the numerous mixed variable optimization problems during real industrial applications. The coevolution strategy is introduced and a mixed variable particle swarm optimization algorithm(CCPSO) based on competitive coevolution is proposed. The search direction adjustment mechanism based on tolerance is designed to judge the evolution state of particles, adaptively adjust the search direction of particles, avoid falling into local optimum, and balance the convergence and diversity of the population.The learning object generation mechanism is adopted for each particle to generate new learning objects when particle evolution stagnation is detected to promote the evolution of particles and improve the diversity of the population. The prediction strategy based on competitive learning is applied to select the appropriate learning objects for particles, which makes full use of the learning potential of new and old learning objects and ensures the convergence speed of the algorithm. Experimental results show that, CCPSO can obtain the better results compared with the other main mixed variable optimization algorithms.

Key words: mixed variable optimization, coevolution strategy, evolutionary algorithm, particle swarm

CLC Number:

TP273

Zhang Hu, Zhang Heng, Huang Zilu, Wang Zhe, Fu Qingpo, Peng Jin, Wang Feng. Mixed-variable Particle Swarm Optimization Algorithm Based on Competitive Coevolution[J]. Journal of System Simulation, 2024, 36(4): 844-858.

Figures/Tables 12

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Table 1

Algorithm parameter settings

算法名称	算法参数设置
GPSO	$N = 20$ ， $w = 0.4$ ， $c 1 = c 2 = 2$ ， $I m a x$ =10 000
CLPSO^[8]	$N = 20$ ， $w ∈ [0.4,0.9]$ ， $c = 1.494 45$ ， $m = 7$ ， $I m a x$ =10 000
AEDA^[17]	$N = 20$ ， $N P b e s t = 0.45$ ， $g p = 0.5$ ， $I m a x$ =10 000
SLPSO^[10]	$N = 20$ ， $α = 0.5$ ， $β = 0.01$ ， $I m a x$ =10 000
DEmv^[18]	$N = 20$ ， $F = R = 0.2$ ， $I m a x$ =10 000
CCPSO	$N = 20$ ， $w = 0.4$ ， $c = 2$ ， $T = 5$ ， $I m a x$ =10 000

Table 1

Table 2

Table 3

Experimental results of numerical statistics

函数类型	函数		GPSO	CLPSO	AEDA	SLPSO	DEmv	CCPSO
单峰	f1	误差均值	1.87E+01	2.57E-01	3.89E-02	3.60E+02	4.31E+03	2.36E-07
	f1	误差方差	1.15E+02	2.42E+00	9.83E-01	1.44E+03	3.62E+03	2.16E-06
	f2	误差均值	6.25E+06	5.78E+06	8.60E+07	2.18E+07	1.64E+08	5.38E+06
	f2	误差方差	1.57E+07	1.06E+07	2.99E+08	4.30E+07	1.34E+08	6.61E+06
	f3	误差均值	1.25E+09	1.16E+08	6.33E+08	1.20E+10	7.34E+10	3.26E+08
	f3	误差方差	6.69E+09	6.46E+08	2.36E+09	2.23E+10	7.92E+10	6.04E+08
	f4	误差均值	1.12E+03	6.13E+03	4.80E+04	4.41E+04	9.18E+04	3.86E+03
	f4	误差方差	4.26E+03	7.02E+03	3.08E+04	4.81E+04	4.71E+04	1.37E+03
	f5	误差均值	1.35E+01	1.10E+00	3.69E-01	2.10E+02	7.43E+02	3.68E-06
	f5	误差方差	1.28E+02	7.74E+00	2.33E+00	7.73E+02	4.51E+02	1.98E-05
多峰	f6	误差均值	1.29E+02	1.09E+02	2.10E+02	2.56E+02	6.73E+02	7.31E+01
	f6	误差方差	4.10E+02	1.67E+02	2.12E+02	2.81E+02	4.59E+02	1.07E+02
	f7	误差均值	5.69E+01	2.47E+01	2.55E+01	6.92E+01	1.88E+02	4.32E+01
	f7	误差方差	8.91E+01	4.97E+01	3.21E+01	7.38E+01	1.18E+02	3.08E+01
多峰	f8	误差均值	2.11E+01	2.11E+01	2.13E+01	2.11E+01	2.13E+01	2.11E+01
	f8	误差方差	2.22E-01	2.39E-01	1.71E-01	3.28E-01	2.47E-01	1.62E-01
	f9	误差均值	4.30E+01	2.08E+01	6.40E+01	3.54E+01	6.58E+01	3.27E+01
	f9	误差方差	4.67E+01	2.73E+01	1.11E+02	2.43E+01	9.55E+00	3.11E+01
	f10	误差均值	3.20E+01	4.65E+00	2.77E+01	2.18E+02	1.46E+03	2.75E+00
	f10	误差方差	1.37E+02	1.92E+01	7.31E+01	3.53E+02	8.86E+02	1.24E+00
	f11	误差均值	8.50E+01	4.90E+01	1.57E+02	1.38E+02	3.82E+02	4.40E+01
	f11	误差方差	1.09E+02	1.12E+02	1.11E+02	1.34E+02	1.15E+02	3.64E+01
	f12	误差均值	2.16E+02	2.64E+02	5.70E+02	2.05E+02	6.23E+02	1.66E+02
	f12	误差方差	2.17E+02	2.87E+02	1.01E+02	1.94E+02	1.49E+02	1.60E+02
	f13	误差均值	2.95E+02	3.37E+02	3.71E+02	3.22E+02	6.34E+02	3.12E+02
	f13	误差方差	3.31E+02	2.37E+02	7.72E+01	2.71E+02	1.61E+02	1.04E+02
	f14	误差均值	2.41E+03	4.54E+03	6.47E+03	4.43E+03	6.24E+03	1.27E+03
	f14	误差方差	4.14E+03	1.20E+04	1.68E+03	5.02E+03	1.44E+03	1.28E+03
	f15	误差均值	8.59E+03	1.31E+04	1.36E+04	7.02E+03	1.31E+04	6.97E+03
	f15	误差方差	8.30E+03	3.68E+03	2.04E+03	4.58E+03	1.94E+03	3.55E+03
	f16	误差均值	3.27E+00	3.51E+00	3.73E+00	1.50E+00	3.75E+00	3.15E+00
	f16	误差方差	2.24E+00	1.38E+00	1.27E+00	2.95E+00	2.08E+00	1.03E+00
	f17	误差均值	1.19E+02	9.64E+01	1.69E+02	2.17E+02	6.28E+02	4.72E+01
	f17	误差方差	1.81E+02	2.55E+02	7.76E+01	3.18E+02	2.23E+02	2.70E+01
	f18	误差均值	1.07E+02	4.86E+01	1.76E+02	2.16E+02	6.40E+02	4.65E+01
	f18	误差方差	1.62E+02	1.88E+02	3.62E+02	2.93E+02	1.91E+02	3.65E+01
	f19	误差均值	1.21E+01	2.11E+01	1.62E+01	6.14E+01	3.71E+01	7.42E+00
	f19	误差方差	3.98E+01	1.75E+01	1.15E+01	1.90E+02	6.69E+02	2.93E+00
	f20	误差均值	1.96E+01	2.04E+01	2.16E+01	1.87E+01	2.18E+01	1.85E+01
	f20	误差方差	5.31E+00	2.46E+00	1.32E+00	4.52E+00	1.22E+00	4.54E+00
复合	f21	误差均值	2.01E+02	2.00E+02	2.01E+02	2.37E+02	4.78E+02	2.00E+02
	f21	误差方差	1.11E+01	3.36E-01	1.71E-01	1.02E+02	1.84E+02	6.64E-09
	f22	误差均值	2.56E+03	5.31E+03	6.61E+03	4.72E+03	6.39E+03	1.42E+03
	f22	误差方差	2.95E+03	1.14E+04	1.68E+03	4.39E+03	1.43E+03	1.70E+03
	f23	误差均值	1.29E+04	1.54E+04	1.64E+04	1.21E+04	1.62E+04	1.20E+04
	f23	误差方差	5.56E+03	2.70E+03	9.06E+02	3.43E+03	1.50E+03	4.90E+03
	f24	误差均值	1.05E+03	1.19E+03	1.43E+03	9.93E+02	1.74E+03	9.14E+02
	f24	误差方差	4.93E+02	7.32E+02	2.96E+02	4.93E+02	2.75E+02	3.94E+02
	f25	误差均值	4.72E+02	5.00E+02	5.36E+02	4.65E+02	6.14E+02	4.47E+02
	f25	误差方差	8.24E+01	1.24E+02	6.83E+01	1.15E+02	5.44E+01	7.89E+01
	f26	误差均值	1.20E+03	7.94E+02	7.70E+02	1.65E+03	2.05E+03	1.05E+03
	f26	误差方差	1.15E+03	8.89E+02	8.39E+02	1.60E+03	2.33E+03	5.63E+02
	f27	误差均值	5.35E+03	4.91E+03	5.08E+03	5.49E+03	5.34E+03	5.01E+03
	f27	误差方差	8.66E+02	5.59E+02	5.83E+02	7.93E+02	2.19E+02	3.04E+02
	f28	误差均值	8.56E+03	9.94E+03	1.15E+04	7.68E+03	1.05E+04	7.56E+03
	f28	误差方差	4.27E+03	7.47E+03	1.18E+03	3.56E+03	1.45E+03	2.97E+03

Table 3

Fig. 7

Table 4

Experimental results of strategy comparison

函数类型	函数		CCPSO-CLP	CCPSO-CCLOG	CCPSO
单峰	f1	误差均值	8.19E+01	1.04E+02	2.36E-07
		误差方差	1.04E+03	1.14E+03	2.16E-06
	f2	误差均值	7.28E+06	1.22E+08	5.38E+06
		误差方差	2.22E+07	3.84E+08	6.61E+06
	f3	误差均值	1.06E+09	2.61E+10	3.26E+08
		误差方差	6.80E+09	4.39E+10	6.04E+08
	f4	误差均值	2.65E+04	3.32E+04	3.86E+03
		误差方差	2.56E+04	1.51E+04	1.37E+03
	f5	误差均值	1.63E+01	1.81E+01	3.68E-06
		误差方差	1.31E+02	1.18E+02	1.98E-05
多峰	f6	误差均值	2.34E+02	3.01E+02	7.31E+01
		误差方差	3.84E+02	5.74E+02	1.07E+02
	f7	误差均值	2.97E+01	1.11E+02	4.32E+01
		误差方差	8.70E+01	6.27E+01	3.08E+01
	f8	误差均值	2.12E+01	2.12E+01	2.11E+01
		误差方差	2.44E-01	1.82E-01	1.62E-01
	f9	误差均值	4.07E+01	5.62E+01	3.27E+01
		误差方差	4.21E+01	2.31E+01	3.11E+01
	f10	误差均值	5.20E+01	2.35E+02	2.75E+00
		误差方差	2.68E+02	1.22E+03	1.24E+00
	f11	误差均值	1.56E+02	2.24E+02	4.40E+01
		误差方差	2.22E+02	4.58E+02	3.64E+01
	f12	误差均值	3.25E+02	5.09E+02	1.66E+02
		误差方差	5.04E+02	2.14E+02	1.60E+02
	f13	误差均值	3.99E+02	5.28E+02	3.12E+02
		误差方差	1.13E+02	9.92E+01	1.04E+02
多峰	f14	误差均值	3.99E+03	9.68E+03	1.27E+03
		误差方差	4.44E+03	1.59E+04	1.28E+03
	f15	误差均值	1.27E+04	1.31E+04	6.97E+03
		误差方差	3.40E+03	2.03E+03	3.55E+03
	f16	误差均值	3.49E+00	3.63E+00	3.15E+00
		误差方差	2.27E+00	2.04E+00	1.03E+00
	f17	误差均值	2.04E+02	3.44E+02	4.72E+01
		误差方差	4.70E+02	7.52E+02	2.70E+01
	f18	误差均值	2.45E+02	3.23E+02	4.65E+01
		误差方差	6.17E+02	8.31E+02	3.65E+01
	f19	误差均值	1.57E+01	7.46E+01	7.42E+00
		误差方差	3.04E+01	6.62E+02	2.93E+00
	f20	误差均值	2.07E+01	2.09E+01	1.85E+01
复合		误差方差	1.79E+00	1.44E+00	4.54E+00
	f21	误差均值	2.06E+02	2.04E+02	2.00E+02
		误差方差	6.15E+01	5.79E+01	6.64E-09
	f22	误差均值	4.07E+03	8.68E+03	1.42E+03
		误差方差	4.51E+03	1.57E+04	1.70E+03
	f23	误差均值	1.57E+04	1.59E+04	1.20E+04
		误差方差	2.40E+03	9.87E+02	4.90E+03
	f24	误差均值	1.45E+03	1.56E+03	9.14E+02
		误差方差	3.55E+02	2.27E+02	3.94E+02
	f25	误差均值	5.33E+02	5.66E+02	4.47E+02
		误差方差	5.40E+01	6.21E+01	7.89E+01
	f26	误差均值	1.57E+03	1.91E+03	1.05E+03
		误差方差	1.99E+03	2.91E+03	5.63E+02
	f27	误差均值	5.31E+03	5.64E+03	5.01E+03
		误差方差	7.39E+02	8.60E+02	3.04E+02
	f28	误差均值	1.08E+04	1.14E+04	7.56E+03
		误差方差	3.86E+03	1.22E+03	2.97E+03

Table 4

Table 5

References 19

1	Sengupta S, Basak S, Peters R A II. Particle Swarm Optimization: A Survey of Historical and Recent Developments with Hybridization Perspectives[J]. Machine Learning and Knowledge Extraction, 2019, 1(1): 157-191.
2	Wang Feng, Zhang Heng, Zhou Aimin. A Particle Swarm Optimization Algorithm for Mixed-variable Optimization Problems[J]. Swarm and Evolutionary Computation, 2021, 60: 100808.
3	Wang Feng, Li Yixuan, Zhou Aimin, et al. An Estimation of Distribution Algorithm for Mixed-variable Newsvendor Problems[J]. IEEE Transactions on Evolutionary Computation, 2020, 24(3): 479-493.
4	Lin Ying, Liu Yu, Chen Weineng, et al. A Hybrid Differential Evolution Algorithm for Mixed-variable Optimization Problems[J]. Information Sciences, 2018, 466: 170-188.
5	Liu Yaxian, Lu Hui, Cheng Shi, et al. An Adaptive Online Parameter Control Algorithm for Particle Swarm Optimization Based on Reinforcement Learning[C]//2019 IEEE Congress on Evolutionary Computation (CEC). Piscataway, NJ, USA: IEEE, 2019: 815-822.
6	Raju Maraboina, Munish Kumar Gupta, Bhanot Neeraj, et al. A Hybrid PSO-BFO Evolutionary Algorithm for Optimization of Fused Deposition Modelling Process Parameters[J]. Journal of Intelligent Manufacturing, 2019, 30(7): 2743-2758.
7	Fatih Ahmet Şenel, Gökçe Fatih, Asım Sinan Yüksel, et al. A Novel Hybrid PSO-GWO Algorithm for Optimization Problems[J]. Engineering with Computers, 2019, 35(4): 1359-1373.
8	Liang J J, Qin A K, Suganthan P N, et al. Comprehensive Learning Particle Swarm Optimizer for Global Optimization of Multimodal Functions[J]. IEEE Transactions on Evolutionary Computation, 2006, 10(3): 281-295.
9	Cao Yulian, Zhang Han, Li Wenfeng, et al. Comprehensive Learning Particle Swarm Optimization Algorithm with Local Search for Multimodal Functions[J]. IEEE Transactions on Evolutionary Computation, 2019, 23(4): 718-731.
10	Cheng Ran, Jin Yaochu. A Social Learning Particle Swarm Optimization Algorithm for Scalable Optimization[J]. Information Sciences, 2015, 291: 43-60.
11	Song An, Chen Weineng, Gu Tianlong, et al. Distributed Virtual Network Embedding System with Historical Archives and Set-based Particle Swarm Optimization[J]. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2021, 51(2): 927-942.
12	Zheng Jie, Wang Ling, Wang Jingjing. A Cooperative Coevolution Algorithm for Multi-objective Fuzzy Distributed Hybrid Flow Shop[J]. Knowledge-Based Systems, 2020, 194: 105536.
13	Stolfi Daniel H, Brust Matthias R, Danoy Grégoire, et al. Competitive Evolution of a UAV Swarm for Improving Intruder Detection Rates[C]//2020 IEEE International Parallel and Distributed Processing Symposium Workshops (IPDPSW). Piscataway, NJ, USA: IEEE, 2020: 528-535.
14	段绍米, 罗会龙, 刘海鹏. 人群搜索和樽海鞘群的混合算法优化PID参数[J]. 系统仿真学报, 2022, 34(6): 1230-1246.
	Duan Shaomi, Luo Huilong, Liu Haipeng. A Hybrid Algorithm Based on Seeker Optimization Algorithm and Salp Swarm Algorithm for PID Parameters Optimization[J]. Journal of System Simulation, 2022, 34(6): 1230-1246.
15	Huang Chen, Zhou Xiangbing, Ran Xiaojuan, et al. Co-evolutionary Competitive Swarm Optimizer with Three-phase for Large-scale Complex Optimization Problem[J]. Information Sciences, 2023, 619: 2-18.
16	Qiu Qicang, Yu Wei, Wang Liping, et al. Preference-inspired Coevolutionary Algorithm Based on Differentiated Resource Allocation Strategy[J]. IEEE Access, 2020, 8: 205798-205813.
17	Shi Wen, Chen Weineng, Lin Ying, et al. An Adaptive Estimation of Distribution Algorithm for Multipolicy Insurance Investment Planning[J]. IEEE Transactions on Evolutionary Computation, 2019, 23(1): 1-14.
18	Liang J J, Qu B Y, Suganthan P N, et al. Problem Definitions and Evaluation Criteria for the CEC 2013 Special Session on Real-parameter Optimization[EB/OL]. [2022-12-03].
	meterOptimization_TechnicalReport.pdf.
19	Sandgren E. Nonlinear Integer and Discrete Programming in Mechanical Design Optimization[J]. Journal of Mechanical Design, 1990, 112(2): 223-229.

函数类型	编号	函数名称	最优解
单峰	f1	Sphere Function	-1 400
	f2	Rotated High Conditioned Elliptic Function	-1 300
	f3	Rotated Bent Cigar Function	-1 200
	f4	Rotated Discus Function	-1 100
	f5	Different Powers Function	-1 000
多峰	f6	Rotated Rosenbrocks Function	-900
	f7	Rotated Schaffers f7 Function	-800
	f8	Rotated Ackleys Function	-700
	f9	Rotated Weierstrass Function	-600
	f10	Rotated Griewanks Function	-500
	f11	Rastrigins Function	-400
	f12	Rotated Rastrigins Function	-300
	f13	Non-Continuous Rotated Rastrigins Function	-200
	f14	Schwefel’s Function	-100
	f15	Rotated Schwefel’s Function	100
	f16	Rotated Katsuura Function	200
	f17	Lunacek-Bi-Rastrigin Function	300
	f18	Rotated Lunacek Bi-Rastrigin Function	400
	f19	Expanded Griewanks plus Rosenbrocks Function	500
	f20	Expanded Scaffers f6 Function	600
复合	f21	Composition Function 1 (n=5, Rotated)	700
	f22	Composition Function 2 (n=3, Unrotated)	800
	f23	Composition Function 3 (n=3, Rotated)	900
	f24	Composition Function 4 (n=3, Rotated)	1 000
	f25	Composition Function 5 (n=3, Rotated)	1 100
	f26	Composition Function 6 (n=5, Rotated)	1 200
	f27	Composition Function 7 (n=5, Rotated)	1 300
	f28	Composition Function 8 (n=5, Rotated)	1 400

算法	最优	最差	平均	方差
GPSO	2.658 55	3.487 27	3.128 16	2.2492E+00
CLPSO	2.628 52	2.628 56	2.628 54	4.8647E-15
AEDA	2.628 52	11.357 50	4.686 63	1.0079E+01
DEmv	2.658 55	2.800 16	2.673 12	2.3224E-01
SLPSO	2.658 55	3.487 27	3.321 53	1.8156E+01
CCPSO	2.628 52	2.628 52	2.628 52	0

[1]	Cheng Jie, Zheng Yuan, Li Chenglong, Jiang Bo. Multi-UAV Collaborative Trajectory Planning Algorithm for Urban Ultra-low-altitude Air Transportation Scenario [J]. Journal of System Simulation, 2024, 36(1): 50-66.
[2]	Lizhen Du, Tao Ye, Yuhao Wang, Yajun Zhang, Zifeng Xuan. Improved Particle Swarm Algorithm of Unrelated Parallel Batch Scheduling Optimization [J]. Journal of System Simulation, 2023, 35(7): 1549-1561.
[3]	Chuanchao Zhao, Rui Zheng, Li Gong, Xiaolu Ma. Particle Swarm Optimization for New Energy Truck Scheduling in Network Environment [J]. Journal of System Simulation, 2023, 35(6): 1337-1350.
[4]	Hongcai Ouyang, Dinghui Wu, Junyan Fan, Jing Wang. Multi-media Energy Planning Optimization of Steel Based on Improved MOEA/D [J]. Journal of System Simulation, 2023, 35(3): 568-578.
[5]	Jiawei Zhou, Xin Du, Youcong Ni, Hu Zhang, Hao Zhang, Haoran Ni, Feng Wang. Uniform Experimental Design with Constrained Region Based on Fruit Fly Algorithm [J]. Journal of System Simulation, 2023, 35(1): 110-122.
[6]	Wanjie Hu, Jianjun Dong, Rui Ren, Zhilong Chen. Layout Planning of Metro-based Underground Logistics System Network Considering Fuzzy Uncertainties [J]. Journal of System Simulation, 2022, 34(8): 1725-1740.
[7]	Feibo Feng, Xingde Yan, Baoqiang Zheng, Xiaofeng Yin, Mengzhen Zhou, Xin Jiang. Bi-Level Optimization of Distribution Network for Hybrid Energy Storage System of Storage Battery and Hydrogen Storage [J]. Journal of System Simulation, 2022, 34(7): 1405-1416.
[8]	Lixin Ma, Ying Cheng. Optimal Operation for Park Integrated Energy System Considering Interruptible Loads [J]. Journal of System Simulation, 2022, 34(4): 817-825.
[9]	Kaiqing Zhang, Qichun Ji. Research on Multi-depot Half-open Vehicle Routing Problem with Time-varying Speed [J]. Journal of System Simulation, 2022, 34(4): 836-846.
[10]	Kui Chen, Li Bi, Wenya Wang. Research on Integrated Scheduling of AGV and Machine in Flexible Job Shop [J]. Journal of System Simulation, 2022, 34(3): 461-469.
[11]	Xiaoan Sun, Xiaoli Luan, Fei Liu. Electronic Solid Waste Prediction Based on Intelligent Optimization Grey Model [J]. Journal of System Simulation, 2022, 34(3): 536-542.
[12]	Fuzhen Zhang, Yaoqin Zhu. Task Allocation Method for Multi-UAV Cooperative Reconnaissance in Complex Environment [J]. Journal of System Simulation, 2022, 34(10): 2293-2302.
[13]	Li Zhang, Huizhen Zhang, Dong Liu, Yuxin Lu. Particle Swarm Algorithm for Solving Emergency Material Dispatch Considering Urgency [J]. Journal of System Simulation, 2022, 34(09): 1988-1998.
[14]	Shao Liangshan, Wang Zhen, Li Changming. Optimization Algorithm of Mine Ventilation Based on SA-IPSO [J]. Journal of System Simulation, 2021, 33(9): 2085-2094.
[15]	Wei Tengfei, Pan Tinglong. Short-term Power Load Forecasting Based on LSTM Neural Network Optimized by Improved PSO [J]. Journal of System Simulation, 2021, 33(8): 1866-1874.