Minimization of
Keane's Bump Function by the Repulsive Particle Swarm and the Differential Evolution Methods
continued.. page 5 of 8
Department of Economics
NEHU, Shillong (India)
For m=60, we obtain DE min[(f)] = -0.835835669;g1(x) = -6.7704109E-010; g2(x) = -352.652189. The decision variables take on the following values given in Table-8(a). In this case, however, the RPS results are better than the DE results. We obtain RPS min[(f)] = -0.837746743. The values of the decision variables are given in Table-8(b).
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Table-8(a). Values of Decision Variables of Keane (m=60) obtained by DE | |||||
|
6.285338790
|
3.167861920
|
3.156781730
|
3.145809150
|
3.134964570
|
3.124223950
|
|
3.113376380
|
3.102748600
|
3.092086930
|
3.081489920
|
3.070871930
|
3.060171490
|
|
3.049418270
|
3.038716460
|
3.028148500
|
3.017316210
|
3.006609820
|
2.995700460
|
|
2.984574080
|
2.973620460
|
2.962402510
|
2.950981780
|
2.939532730
|
2.927839980
|
|
2.915972240
|
2.903798910
|
0.434815891
|
0.433294979
|
0.432069993
|
0.430621744
|
|
0.429421427
|
0.427964211
|
0.426917113
|
0.425579719
|
0.424408800
|
0.423234386
|
|
0.422015631
|
0.420993818
|
0.419629141
|
0.418482726
|
0.417330303
|
0.416204475
|
|
0.415108638
|
0.414248869
|
0.413012358
|
0.411953992
|
0.410849798
|
0.409916159
|
|
0.408821128
|
0.407824210
|
0.406679669
|
0.405756013
|
0.404700677
|
0.403723423
|
|
0.402652061
|
0.401793732
|
0.400705123
|
0.399886726
|
0.398889977
|
0.397946680
|
|
Table-8(b). Values of Decision Variables of Keane (m=60) obtained by RPS | |||||
|
6.286843320
|
3.173492270
|
3.160683060
|
3.149693970
|
3.138712600
|
3.126625110
|
|
3.115790130
|
3.105019480
|
3.093778840
|
3.080946500
|
3.071653830
|
3.061515640
|
|
3.049746000
|
3.036256360
|
3.027641820
|
3.017186050
|
3.006125490
|
2.996929890
|
|
2.982453710
|
2.971695830
|
2.959466880
|
2.951034220
|
2.939888170
|
2.925757420
|
|
2.910715510
|
0.463715298
|
0.462422111
|
0.457899571
|
0.457513834
|
0.456935488
|
|
0.454491776
|
0.452602034
|
0.451457866
|
0.451053248
|
0.447671899
|
0.446884213
|
|
0.445433219
|
0.444898576
|
0.444218376
|
0.441387240
|
0.440728984
|
0.440399367
|
|
0.438915986
|
0.438476776
|
0.436563227
|
0.435000188
|
0.433114197
|
0.432494583
|
|
0.430914958
|
0.429462320
|
0.428974897
|
0.427472923
|
0.426076003
|
0.424951615
|
|
0.423772714
|
0.423230527
|
0.420626442
|
0.419746824
|
0.418076501
|
0.415879864
|
Finally we have run DE to optimize the Keane function for m=100. We would also like to mention that we used 4321 as the initial seed for generating the random numbers; population size = 1000; pcross=0.9; fact=0.05 and 5677 as the second random number seed in the DE subroutine. These inputs/parameters are defined in our computer program. We also used our conjecture to arrange x in a descending order. We obtained DE min[(f)] = -0.844219651;, g1(x) = -2.51972313E-008; g2(x) = -586.659534. Values of x are presented in Table-9(a) below.
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Table-9(a). Values of Decision Variables of Keane (m=100) obtained by DE | ||||
|
9.421209850
|
6.281568910
|
3.172221960
|
3.165330690
|
3.158407070
|
|
3.151790900
|
3.144986000
|
3.138420350
|
3.131239320
|
3.124581260
|
|
3.118033120
|
3.111275770
|
3.104712750
|
3.098148350
|
3.091554370
|
|
3.084750280
|
3.078285450
|
3.071863590
|
3.065027900
|
3.058509040
|
|
3.051894330
|
3.045239320
|
3.038673780
|
3.031942040
|
3.025294780
|
|
3.018717050
|
3.011990360
|
3.005180740
|
2.998547060
|
2.991581570
|
|
2.984878120
|
2.978015990
|
2.971031220
|
2.964043880
|
2.956997030
|
|
2.950130590
|
2.942822710
|
2.935704570
|
2.928359500
|
2.920978890
|
|
2.913649490
|
2.906208780
|
0.453854241
|
0.452817124
|
0.452100592
|
|
0.451067421
|
0.450422860
|
0.449256343
|
0.448279024
|
0.447387629
|
|
0.446720503
|
0.445666150
|
0.444731075
|
0.444080540
|
0.443086457
|
|
0.442474034
|
0.441448740
|
0.440789329
|
0.439877311
|
0.439219450
|
|
0.438206790
|
0.437293937
|
0.436692347
|
0.435877182
|
0.435154888
|
|
0.434306103
|
0.433404136
|
0.432766335
|
0.431938810
|
0.431135854
|
|
0.430542180
|
0.429982791
|
0.428823204
|
0.428462627
|
0.427640123
|
|
0.427102876
|
0.426255391
|
0.425483535
|
0.424786864
|
0.424002049
|
|
0.423462434
|
0.422801656
|
0.422098679
|
0.421377911
|
0.420720915
|
|
0.420010667
|
0.419362836
|
0.418770729
|
0.418116471
|
0.417478448
|
|
0.416845509
|
0.416097790
|
0.415462624
|
0.414916930
|
0.414249639
|
|
0.413555041
|
0.412891312
|
0.412286270
|
0.411823787
|
0.411201187
|
We have run the RPS with random seed=7337. The RPS min[(f)] = -0.84246233; g1(x) = -0.000523752645; g2(x) = -587.939518, which is inferior to the DE min[(f)]. The values of x are presented in Table-9(b). It appears that arranging the values of decision variables in descending order affects the performance of the DE as well as the RPS unpredictably, although it is difficult to generalize such an observation. Further, we do not claim that for m=100 our DE results are the best, although one may note that Liu and Lewis (2002) obtained - 0.84421 and -0.8448539 for m=200 and m=1000 respectively. Our results for m=100 (min(f) = -0.844219651) is smaller than Liu-Lewis' -0.84421 for m=200, which is anomalous. Thus, Liu-Lewis' min(f) for m=200 is grossly sub-optimal.
|
Table-9(b). Values of Decision Variables of Keane (m=100) obtained by RPS | ||||
|
6.295905660
|
6.283013840
|
3.170384370
|
3.165686900
|
3.157958600
|
|
3.151463240
|
3.142404470
|
3.139385180
|
3.125719650
|
3.121996200
|
|
3.114943680
|
3.111719990
|
3.102821160
|
3.098702340
|
3.091716510
|
|
3.088209640
|
3.074270960
|
3.071114580
|
3.064350170
|
3.054508540
|
|
3.054326000
|
3.049031790
|
3.039193270
|
3.030651010
|
3.027172050
|
|
3.021449410
|
3.010442630
|
3.010060930
|
2.996908920
|
2.994202160
|
|
2.985068020
|
2.977596560
|
2.971414430
|
2.968615030
|
2.960406750
|
|
2.954659140
|
2.943754670
|
2.936470690
|
2.928697320
|
2.928315880
|
|
2.915114150
|
2.903913760
|
2.903039740
|
0.440651023
|
0.439306900
|
|
0.439129498
|
0.438640185
|
0.437377084
|
0.436875468
|
0.436572276
|
|
0.435786420
|
0.435199692
|
0.433733227
|
0.432524090
|
0.431707741
|
|
0.430846454
|
0.430573021
|
0.430461588
|
0.430113172
|
0.428261855
|
|
0.428255325
|
0.426487033
|
0.424873853
|
0.424032799
|
0.423253553
|
|
0.423011175
|
0.423006205
|
0.421760827
|
0.421633427
|
0.420970330
|
|
0.419523842
|
0.418669684
|
0.418180749
|
0.417961379
|
0.417258479
|
|
0.417211605
|
0.416822491
|
0.415255666
|
0.414534559
|
0.414523647
|
|
0.414407966
|
0.414029761
|
0.412823195
|
0.411574032
|
0.410124136
|
|
0.409683366
|
0.409510891
|
0.408945793
|
0.407934372
|
0.406994044
|
|
0.406725228
|
0.405931032
|
0.405341784
|
0.404075017
|
0.403745876
|
|
0.403488194
|
0.400118948
|
0.399434791
|
0.397844945
|
0.395982412
|