A Velocity-Combined Local Best Particle Swarm Optimization Algorithm for Nonlinear Equations

Table 1

Results of optimizing equ1–5 comparing VCLBPSO, VRCLBPSO, NVRCLBPSO, and EVRCLBPSO with PSO.

equ1, Dim = 2; Ps = 50; Genn = 30
Result	Theory	PSO	VCLBPSO	VRCLBPSO	NVRCLBPSO	EVRCLBPSO
The best	0	2.4942e − 5	3.5688 e − 7	1.9969e − 6	1.1267e − 6	2.2815e − 6
Average	0	0.0074	7.5187e − 4	9.6079e − 4	2.2438 e − 4	0.001
Std	0	0.0406	0.0039	0.005	9.6179 e − 4	0.0055
Solution	(10, 1), (−7.5, 1)	(−7.5, 1)	(−7.5, 1)	(10, 1)	(10, 1)	(10, 1)
equ2, Dim = 6; Ps = 50; Genn = 100
Result	Theory	PSO	VCLBPSO	VRCLBPSO	NVRCLBPSO	EVRCLBPSO
The best	0	0.0183	0.0048	6.31 e − 5	0.0015	0.0035
Average	0	0.3254	0.0337	0.0512	0.0292	0.0618
Std	0	0.2374	0.0265	0.1261	0.0548	0.1325
Solution	(−1, 1−1, 1−1, 1)	(−0.9586, 0.971, −1.0349, 0.9584, −1.0236, 0.9812)	(−0.9901, 0.9918, −1.0077, 0.9902, −1.006, 0.996)	(−1.0001, 1.0001, −0.9998, 1.0003, −0.9999, 1.0001)	(−9975, 0.9976, −1.0014, 0.9979, −1.0016, 0.9992)	(−9933, 0.9948, 1.0086, 0.9896, 1.0041, 0.9972)
equ3, Dim = 3; Ps = 50; Genn = 100
Result	Theory	PSO	VCLBPSO	VRCLBPSO	NVRCLBPSO	EVRCLBPSO
The best	0	6.2134e − 5	3.1343e − 7	2.5109 e − 9	2.9418e − 4	7.5664e − 7
Average	0	2.4947	0.4107	0.6465	1.7852	0.6519
Std	0	1.9988	1.137	1.3551	1.5887	1.2548
Solution	(4, 31)	(4, 3, 1)	(4, 3, 1)	(4, 3, 1)	(4, 3, 1)	(4, 3, 1)
equ4, Dim = 10; Ps = 50; Genn = 100
Result	Theory	PSO	VCLBPSO	VRCLBPSO	NVRCLBPSO	EVRCLBPSO
The best	0	6.0214e − 4	9.7507e − 9	6.4342e − 8	7.398e − 7	2.5128 e − 9
Average	0	168.6747	3.9341e − 4	5.9038e − 4	2.4065 e − 4	0.0012
Std	0	95.7371	0.0018	0.0023	8.4349 e − 4	0.006
equ4, Dim = 100; Ps = 150; Genn = 6000
Result	Theory	PSO	VCLBPSO	VRCLBPSO	NVRCLBPSO	EVRCLBPSO
The best	0	158.0426	28.9963	2.2039	17.3192	0.4519
Average	0	1.8912e + 3	298.0509	609.2726	144.6553	131.3043
Std	0	808.4996	93.358	86.7113	40.3754	15.2786
equ5, Dim = 10; Ps = 50; Genn = 100
Result	Theory	PSO	VCLBPSO	VRCLBPSO	NVRCLBPSO	EVRCLBPSO
The best	0	0.0139	3.8629e − 5	1.1267e − 6	4.3148e − 6	8.342 e − 8
Average	0	1.4829e + 3	0.0079	0.0013	0.0011	0.0057
Std	0	2.793e + 3	0.0277	0.0039	0.004	0.0271
equ5, Dim = 150; Ps = 150; Genn = 6000
Result	Theory	PSO	VCLBPSO	VRCLBPSO	NVRCLBPSO	EVRCLBPSO
The best	0	34.801	1.0661	1.0943	0.8923	0.2137
Average	0	5.3188e + 3	1.8718	1.6356	1.0518	0.7122
Std	0	2.5912e + 4	0.62	0.3316	0.1087	0.5780