Mathematical Problems in Engineering

Research Article

A Local and Global Search Combined Particle Swarm Optimization Algorithm and Its Convergence Analysis

Table 1

The comparison results of the PSO algorithm and the LGSCPSO algorithm.
 Min./Max./Average ( represents )
 
 LGC0.10.20.30.40.5
Fun 
, 
Dim 
150,
Best 
0;
PS 
150,
EG 
2500		.1
.2
.3
.4
.5
.6
.7
.8 
.9		7.9247/798.1366/119.2735
1.0216/328.5215/36.3621
0.1788/3.6762/0.9580
0.0276/1.0302/0.1732
0.0015/0.2004/0.0345
6.199e  −  4/20.4397/2.1839 
8.7194e  −  4/32.076/3.5118
0.0027/1.0443/0.3794
0.0301/3.1973e  +  5/5.7818e  +  4		4.8684/788.0086/137.7727
0.2372/39.35/5.2925
0.0516/72.579/18.8086
0.0024/16.9442/2.0285
0.0011/6.0067/0.7585 
0.1109/326.9967/38.2332
217.0564/2.5137e  +  3/922.9063
131.8696/1.0093e  +  3/605.7608
3.5023/3.5197e  +  5/7.5222e  +  4		3.4298/466.135/129.4039
0.2165/36.7093/9.1546
0.0446/0.9215/0.3245
0.0081/5.1882/0.6059
0.0021/15.3744/3.103
9.091e  −  4/50.1812/9.9065 
0.1802/416.0314/88.5176
22.395/2.7571e  +  3/760.49
3.3725/3.3151e  +  5/3.3313e  +  4		6.2134/108.3578/34.9678
0.336/297.5513/35.9433
0.0931/2.9157/0.6956
0.0255/12.8562/1.5062
0.0042/1.563/0.276
0.0031/1.3039/0.2172
1.5877e  −  4/2.5115/0.3145
4.9801e  −  5/0.0052/8.8123e  −  4*9.7207e  −  5/390.2679/40.2488		202.8074/3.8168e  +  3/899.1347
25.8671/126.0089/74.3555
2.0381/42.8/14.7393
0.2738/12.3747/2.6693
0.0354/0.9538/0.2124
0.0088/0.3742/0.0659
0.0027/0.3203/0.0554
0.0104/0.1721/0.0678
0.5088/4.2984e  +  5/2.6068e  +  5
 PSO3.5545e  +  5/4.2006e  +  5/3.9996e  +  53.6563e  +  5/4.1904e  +  5/3.9654e  +  53.8295e  +  5/4.1465e  +  5/4.0243e  +  50.5082/4.185e  +  5/3.4273e  +  53.834e  +  5/4.2439e  +  5/4.0552e  +  5
 LGC0.10.20.30.40.5
Fun 
, 
Dim 
30,
Best 
0;
PS 
150,
EG 
2000		.1
.2
.3
.4 
.5
.6
.7
.8
.9		1.5314e  −  5/0.0012/2.3471e  −  4
7.544e  −  6/4.1057e  −  4/1.2064e  −  4
3.3468e  −  6/2.5205e  −  4/8.7442e  −  5
2.2108e  −  5/3.3094e  −  4/9.2148e  −  5
1.4607e  −  5/3.3855e  −  4/9.0081e  −  5
1.4749e  −  4/0.0097/0.0016
2.2105e  −  4/0.0705/0.0232
0.0879/0.9873/0.4732
1.3288/15.724/5.0923		3.4819e  −  8/9.7461e  −  7/2.8081e  −  7
3.6274e  −  9/2.252e  −  7/6.7068e  −  8
1.7849e  −  10/6.8322e  −  8/1.3459e  −  8
9.6468e  −  11/1.3930e  −  7/2.0447e  −  8
7.8029e  −  11/9.93e  −  9/4.5009e  −  9
3.0876e  −  9/4.2278e  −  7/5.6384e  −  8
1.8076e  −  7/1.0146e  −  5/2.1222e  −  6
1.437e  −  6/6.4389e  −  4/2.8104e  −  4
0.0042/0.1077/0.0241		1.6392e  −  9/8.0260e  −  8/2.4483e  −  8
4.3649e  −  11/1.9808e  −  8/3.0528e  −  9
4.3800e  −  11/8.3452e  −  9/2.5903e  −  9
2.9752e  −  12/5.6342e  −  10/1.6065e  −  10* 
7.1437e  −  12/4.1282e  −  9/8.6320e  −  10
2.0796e  −  11/5.0013e  −  9/1.0295e  −  9
1.1123e  −  10/2.6655e  −  8/8.6717e  −  9
8.3474e  −  8/7.0419e  −  6/2.2156e  −  6
7.8655e  −  5/0.0055/0.002		1.9438e  −  8/4.5893e  −  6/1.0199e  −  6
6.9751e  −  9/4.4344e  −  6/5.5416e  −  7
3.0993e  −  10/2.9669e  −  8/6.9531e  −  9
5.1336e  −  11/6.7099e  −  8/9.4611e  −  9
1.2011e  −  10/6.3320e  −  9/1.7701e  −  9
4.0553e  −  10/3.5928e  −  8/5.5263e  −  9
6.2429e  −  9/6.8153e  −  7/1.8691e  −  7
9.0305e  −  7/1.0765e  −  4/1.9495e  −  5
2.6371e  −  4/0.0057/0.0024		0.0016/0.0746/0.0181
4.1506e  −  5/0.002/5.1297e  −  4
1.3450e  −  5/1.3095e  −  4/6.5852e  −  5
1.1121e  −  6/2.0099e  −  5/6.5348e  −  6
1.5491e  −  6/1.6945e  −  5/6.7154e  −  6
3.3047e  −  7/7.1073e  −  5/2.0406e  −  5
2.4160e  −  5/5.8853e  −  4/1.5538e  −  4
6.3392e  −  4/0.0096/0.0042
0.0595/1.9094/0.3682
 PSO364.253/2.4939e  +  4/1.0134e  +  41.3765/4.6947e  +  3/481.84640.0691/2.5876/0.72830.1723/3.9289/1.800517.3828/984.8965/188.1513
 LGC0.10.20.30.40.5
Fun 
,
Dim 
30,
Best 
0;
PS 
150,
EG 
2000		.1
.2
.3
.4 
.5
.6
.7
.8
.9		2.3621e  −  4/0.0013/7.311e  −  4
6.0977e  −  5/4.2674e  −  4/2.1957e  −  4
5.5341e  −  5/7.0551e  −  4/2.1313e  −  4
4.9251e  −  5/3.4147e  −  4/1.4638e  −  4
5.2814e  −  5/0.0019/4.5195e  −  4
4.8841e  −  4/0.0053/0.0029
0.0046/0.0476/0.0191
0.0569/0.5819/0.2266
0.4589/4.2785/1.3995		1.3902e  −  5/2.0080e  −  4/8.8198e  −  5
1.1701e  −  6/2.8489e  −  5/8.0878e  −  6
3.8603e  −  7/8.6707e  −  6/1.7948e  −  6
1.4e  −  7/5.6791e  −  6/1.3055e  −  6
1.6044e  −  7/1.6259e  −  6/5.7405e  −  7
1.7168e  −  7/6.7027e  −  6/2.6332e  −  6
9.6091e  −  6/7.5094e  −  5/2.8845e  −  5
8.2486e  −  5/0.0012/3.9755e  −  4
0.001/0.0436/0.0158		1.6884e  −  5/3.3877e  −  4/1.194e  −  4
7.4212e  −  7/4.1010e  −  6/1.8869e  −  6
5.1647e  −  8/1.5398e  −  6/3.7869e  −  7
1.5353e  −  9/1.8722e  −  7/6.9333e  −  8* 
4.7963e  −  9/3.1684e  −  7/6.3918e  −  8
2.0730e  −  9/6.1577e  −  7/9.0325e  −  8
3.4465e  −  8/7.9999e  −  6/1.0596e  −  6
1.1441e  −  6/1.3502e  −  5/5.7652e  −  6
2.4963e  −  5/0.0027/6.1308e  −  4		1.5547e  −  5/6.6884e  −  4/1.4984e  −  4
3.5838e  −  7/8.3143e  −  6/4.7977e  −  6
3.9851e  −  8/2.4747e  −  6/7.3211e  −  7
3.4271e  −  8/4.2062e  −  7/1.0447e  −  7
6.6274e  −  9/8.7745e  −  7/1.4709e  −  7
2.0358e  −  9/4.6960e  −  8/1.9028e  −  8
3.9048e  −  8/3.9274e  −  7/1.5350e  −  7
7.5274e  −  7/1.2135e  −  5/3.6172e  −  6
4.4217e  −  5/5.6209e  −  4/1.7621e  −  4		0.001/0.0262/0.0055
1.1206e  −  5/5.1122e  −  4/1.9353e  −  4
3.1555e  −  6/3.1428e  −  5/1.2056e  −  5
1.8118e  −  7/1.3756e  −  5/3.7208e  −  6
2.2951e  −  7/1.1701e  −  6/5.3581e  −  7
8.2804e  −  8/2.1413e  −  5/3.1923e  −  6
1.0194e  −  6/1.0954e  −  5/3.7688e  −  6
9.2958e  −  6/1.813e  −  4/1.0582e  −  4
0.0016/0.0311/0.0073
 PSO0.9442/7.2642/3.71780.1348/0.5193/0.24360.005/0.0675/0.02120.0084/0.043/0.030.2007/0.9216/0.4999
 LGC0.10.20.30.40.5
Fun 
,
Dim 
25,
Best 
0;
PS 
150,
EG 
2500		.1
.2
.3
.4 
.5
.6
.7
.8
.9		1.8571/1.229e  +  3/135.8807
0.1568/2.5441e  +  3/388.2091
0.6066/799.8443/169.6422
0.2485/181.0596/25.4415
0.0237/40.8918/11.1441
0.0253/276.2887/57.0108
0.0916/228.4057/30.7965
0.4745/435.3273/56.9634
4.3957/341.7015/67.9295		6.2873/3.2863e  +  3/709.8222
0.0113/789.7377/115.475
2.3034/796.4533/123.7322
9.6966e  −  5/6.8529/2.0791* 
0.0069/23.873/4.9974
2.547e  −  4/9.0536/4.042
0.0702/14.2596/6.6697
0.0189/16.2452/9.9379
3.2415/838.6487/128.4819		0.2013/285.8471/44.28
4.0693/4.9375e  +  3/562.5381
3.3983e  −  4/302.2135/32.9758
0.2062/179.2151/21.4179
3.5267e  −  4/3.8396/0.8494
3.6341e  −  4/6.6633/3.6567
0.0372/152.4195/22.0972
0.1941/238.9312/38.1225
0.7361/334.4369/49.1472		0.5944/144.3777/23.7188
1.8021/937.5994/153.7682
0.0455/207.5395/40.3132
0.0121/42.4684/5.2882
0.0018/182.335/21.0831
1.4713/10.0475/5.6529
0.1549/12.1124/7.7882
0.5691/1.2973e  +  3/170.297
0.2681/312.4161/40.7414		0.0678/486.002/165.0663
4.3764/7.2066e  +  3/1.0582e  +  3
0.063/197.4164/32.7749
3.4459/962.925/115.1339
0.0125/423.4865/75.3354
0.1299/26.4581/10.2862
0.1671/672.3137/106.7018
0.6944/213.5931/30.5417
0.2055/1.1812e  +  3/161.6274
 PSO4.3221e  +  4/5.1189e  +  4/4.6655e  +  43.9971e  +  4/5.4303e  +  4/4.6593e  +  42.876e  +  4/5.5191e  +  4/4.7462e  +  43.937e  +  4/5.2589e  +  4/4.6619e  +  48.4709/102.7603/23.237
 LGC0.10.20.30.40.5
Fun 
,
Dim 
30
Best 
0;
PS 
100,
EG 
2000		.1
.2
.3 
.4
.5
.6
.7
.8
.9		7.9936e  −  15/4.3521e  −  14/2.256e  −  14
7.9936e  −  15/7.9936e  −  15/7.9936e  −  15
4.4409e  −  15/7.9936e  −  15/7.2831e  −  15
4.4409e  −  15/7.9936e  −  15/5.1514e  −  15
4.4409e  −  15/7.9936e  −  15/5.8620e  −  15
4.4409e  −  15/2.3162/0.2316
4.4409e  −  15/1.1908e  −  9/1.1908e  −  10
6.9209e  −  5/2.2201/0.7328
4.8998e  −  5/1.6462/0.3991		7.9936e  −  15/7.9936e  −  15/7.9936e  −  15
4.4409e  −  15/7.9936e  −  15/5.1514e  −  15
4.4409e  −  15/7.9936e  −  15/5.5067e  −  15* 
4.4409e  −  15/1.6462/0.2801
4.4409e  −  15/2.3162/0.5967
4.4409e  −  15/2.3162/1.0934
7.9936e  −  15/2.887/1.1089
0.0021/3.0268/1.7571
0.0043/2.5791/1.7115		4.4409e  −  15/7.9936e  −  15/7.2831e  −  15
4.4409e  −  15/7.9936e  −  15/5.1514e  −  15
4.4409e  −  15/1.5017/0.1502
4.4409e  −  15/2.0119/0.3167
4.4409e  −  15/1.6462/0.7128
4.4409e  −  15/3.6807/1.3585
4.4409e  −  15/2.4949/1.2564
7.9936e  −  15/2.0133/1.2372
7.9936e  −  15/2.6602/0.8856		7.9936e  −  15/1.5099e  −  14/9.4147e  −  15
4.4409e  −  15/7.9936e  −  15/7.2831e  −  15
4.4409e  −  15/7.9936e  −  15/5.5067e  −  15* 
4.4409e  −  15/2.5795/0.3735
4.4409e  −  15/2.0133/0.5
4.4409e  −  15/1.3404/0.2272
4.4409e  −  15/1.3404/0.134
4.4409e  −  15/2.0133/0.5
4.4409e  −  15/1.1551/0.1155		1.056e  −  11/2.0851e  −  10/7.4168e  −  11
1.8652e  −  14/4.1656e  −  13/9.1838e  −  14
7.9936e  −  15/1.5099e  −  14/8.7041e  −  15
4.4409e  −  15/7.9936e  −  15/5.862e  −  15 
4.4409e  −  15/7.9936e  −  15/5.862e  −  15 
4.4409e  −  15/7.9936e  −  15/6.9278e  −  15
4.4409e  −  15/7.9936e  −  15/7.6383e  −  15
4.4409e  −  15/7.9936e  −  15/7.6383e  −  15
7.9936e  −  15/7.9936e  −  15/7.9936e  −  15
 PSO2.4655e  −  5/2.2315/1.11331.6387e  −  4/2.8859/0.74017.9936e  −  15/2.4959/0.69314.4409e  −  15/7.9936e  −  15/7.2831e  −  157.9936e  −  15/1.5099e  −  14/1.2967e  −  14
 LGC0.10.20.30.40.5
Fun 
,
Dim 
150,
Best 
0;
PS 
150,
EG 
2500		.1
.2
.3
.4
.5
.6
.7
.8 
.9		0.9416/1.4223/1.1635
0.2941/0.9446/0.4842
0.0383/0.2148/0.0994
0.002/0.3372/0.0945
4.6926e  −  4/0.2611/0.0958
3.7473e  −  5/0.1647/0.0541
1.572e  −  4/0.0664/0.0194
1.1609e  −  4/0.2322/0.0541
0.0129/3.4498e  +  3/639.0998		0.5782/1.452/0.9452
0.0804/2.7225/0.5028
0.0036/0.8091/0.1571
6.6171e  −  4/1.0183/0.1775
8.1220e  −  5/2.1079/0.4195
0.2266/6.0141/2.1203
1.6879/24.077/11.2293
1.9272/17.0006/7.1286
0.2582/3.7637e  +  3/668.9854		0.3887/8.8918/1.7237
0.0397/2.1197/0.4602
0.0055/4.4603/0.5501
8.199e  −  4/0.4379/0.122
2.5102e  −  4/0.5375/0.3158
0.0023/3.2548/0.7676
0.2953/4.2413/1.1668
1.3151/17.8887/9.7229
0.0979/12.3986/3.8033		0.9863/3.3689/1.3278
0.3404/0.6563/0.4877
0.0084/0.2726/0.1332
0.0013/0.2891/0.1343
0.0044/0.9635/0.1852
3.0448e  −  4/1.2816/0.2687
0.0074/0.5053/0.1469
1.3352e  −  5/0.061/0.0226
5.0879e  −  6/1.1096/0.1934*		3.0738/12.5482/6.1513
1.1998/2.6794/1.5013
0.545/1.2007/0.8291
0.0217/6.2987/0.8092
0.0061/0.2465/0.0621
0.0015/0.3055/0.0567
2.6726e  −  4/0.1051/0.0241
4.8173e  −  4/0.0609/0.019
0.0283/3.9697e  +  3/2.3201e  +  3
  3.1058e  +  3/3.9496e  +  3/3.6383e  +  32.766e  +  3/3.952e  +  3/3.5857e  +  30.9187/3.9622e  +  3/2.9803e  +  30.1916/3.9592e  +  3/2.8647e  +  33.5377e  +  3/3.9083e  +  3/3.7889e  +  3
 LGC0.10.20.30.40.5
Fun 
,
Dim 
100,
Best 
0;
PS 
150,
EG 
2500		.1
.2
.3
.4
.5
.6
.7 
.8		0.0014/0.1015/0.0268
4.7654e  −  5/0.0311/0.0034
3.3537e  −  5/0.0628/0.0165
5.8595e  −  7/0.0311/0.0036
8.2125e  −  7/0.1685/0.0298
5.8595e  −  7/0.0311/0.0036
0.0022/2.5577/0.8246
30.7659/1.0297e  +  9/3.6691e  +  8		2.1663e  −  5/0.1245/0.0189
6.125e  −  7/0.0622/0.0097
7.1955e  −  9/0.0311/0.0062
4.1191e  −  9/0.1555/0.0249
5.3424e  −  8/0.3743/0.0717
4.1946e  −  8/1.731/0.3794
0.1557/2.8177/0.8995
3.3049e  −  4/2.7669e  +  8/2.7669e  +  7		1.0775e  −  5/0.1866/0.0314
2.6987e  −  7/0.1557/0.0249
9.1232e  −  9/0.1868/0.0374
2.9509e  −  9/0.1868/0.0467
5.2224e  −  7/0.6882/0.2318
5.6598e  −  10/1.3103/0.2872
4.3761e  −  5/1.3392/0.4191
1.3321e  −  5/1.4361/0.5313		3.9207e  −  4/0.3284/0.0888
3.7171e  −  6/0.2498/0.0605
3.4490e  −  7/0.1244/0.0467
3.8808e  −  9/0.499/0.0872
4.4126e  −  9/0.2497/0.0655
3.7644e  −  9/0.2808/0.0779
1.0263e  −  10/0.2490/0.0498* 
8.0965e  −  6/0.2789/0.0448		0.8393/18.623/5.157
0.2215/1.3457/0.613
0.0018/0.5005/0.1601
6.0798e  −  4/0.3451/0.0947
7.0408e  −  6/0.137/0.0287
3.7725e  −  7/0.4987/0.1405
6.0553e  −  7/0.5295/0.0861
0.4766/3.2293e  +  3/326.8969
  2.2171e  +  9/2.6638e  +  9/2.3786e  +  92.05e  +  9/2.6165e  +  9/2.2883e  +  91.9676e  +  9/2.8563e  +  9/2.3233e  +  91.3125e  +  9/2.6573e  +  9/2.1146e  +  91.8631e  +  9/2.8019e  +  9/2.2317e  +  9
 LGC0.10.20.30.40.5
Fun 
,
Dim 
100,
Best 
0;
PS 
150,
EG 
2500		.1
.2
.3
.4
.5
.6 
.7
.8		0.0014/0.1008/0.0187
2.0521e  −  4/0.0132/0.0039
9.8083e  −  5/0.9506/0.1062
3.0775e  −  7/0.0037/5.5843e  −  4
3.3459e  −  6/4.3813e  −  4/1.47e  −  4
2.891e  −  6/2.5064/0.497
7.7979e  −  4/9.103/2.623
2.0949e  +  9/2.7924e  +  9/2.4913e  +  9		6.9482e  −  5/0.1026/0.0114
8.9263e  −  7/8.6653e  −  5/2.0482e  −  5
2.1133e  −  7/4.1364e  −  5/9.8935e  −  6
1.8204e  −  8/5.2730e  −  5/1.0100e  −  5
4.9008e  −  8/6.3249e  −  4/8.0700e  −  5
1.1649e  −  7/1.8877/0.5009
2.1485e  −  4/2.7725/0.8736
2.1328e  +  /2.8009e  +  9/2.5344e  +  9		1.9512e  −  5/0.0023/7.8123e  −  4
1.3466e  −  6/0.2978/0.0307
2.7567e  −  8/1.1156e  −  4/2.2848e  −  5
8.9806e  −  10/1.4026e  −  4/2.3733e  −  5
1.4968e  −  9/0.5878/0.0688
4.0322e  −  7/3.0954/0.8618
2.6362e  −  6/6.9698/2.0345
1.816e  +  9/2.8674e  +  9/2.4126e  +  9		2.037e  −  4/0.1195/0.0183
6.6231e  −  5/0.0019/5.873e  −  4
1.1378e  −  6/0.0015/2.6495e  −  4
1.8884e  −  7/0.0905/0.0091
5.5391e  −  10/8.5755e  −  5/9.8527e  −  6
4.9993e  −  10/7.1912e  −  5/8.2796e  −  6* 
3.5393e  −  9/0.0011/1.6126e  −  4
1.6523e  −  7/2.1877/0.3866		5.8424/86.3769/26.7152
0.4244/9.4505/2.6056
0.0034/3.0968/0.581
7.1911e  −  5/1.4867/0.1498
7.1442e  −  7/0.0052/0.0017
1.0077e  −  6/4.2974e  −  4/7.1607e  −  5
1.0331e  −  5/0.2203/0.0486
1.2518/2.391e  +  9/2.391e  +  8
 PSO1.8688e  +  9/2.8661e  +  9/2.3999e  +  92.05e  +  9/2.6165e  +  9/2.2883e  +  91.5174e  +  9/2.7626e  +  9/2.2677e  +  91.3125e  +  9/2.6573e  +  9/2.1146e  +  91.8631e  +  9/2.8019e  +  9/2.2317e  +  9
Remark: In Table 1, Fun and Dim denote function and its dimension, respectively; the Best is this function’s optimum value; PS and EG indicate the population size and algorithm terminate generation. PSO and LGC denote the original PSO and LGSCPSO algorithm. The better solutions and the parameters found in LGSCPSO are illustrated with bold letters, and the best solutions are shown in data with asterisk.