Minimization of
Keane's Bump Function by the Repulsive Particle Swarm and the Differential Evolution Methods
continued.. page 6 of 8
Department of Economics
NEHU, Shillong (India)
V. Minimization of Keane Function with Odd Number of Variables: Do the properties of Keane problem depend on
whether its dimension is odd or oddly chosen? We have (arbitrarily) chosen some
odd numbers; 17, 23, 27, 35, 43 and 47 to study the behavior of Keane function.
We have ignored small dimensions like 3, 7, etc. For m=5 we have DE
min[(f)=-0.634448687, g1(x) = -4.4408921E-016, g2(x)
=-28.4863914 for x = (3.07581932, 2.99199522, 1.47579373, 0.236691358, 0.233308996). A perusal of the following tables - 10 (a, b) through 15 (a, b) - reveals
that it is not so. Hence, we conclude that the minimum values of the problem
monotonously decline with the increase in the dimension of Keane problem.
|
Table-10(a). Values of Decision Variables of Keane (m=17) obtained by DE Min[f(x)]= -0.79150563; g1(x)= -3.85410481E-010; g2(x)= -100.81833 | ||||
|
3.151991990
|
3.111505760
|
3.071640430
|
3.031343620
|
2.990185690
|
|
2.947234280
|
2.900955860
|
0.475303034
|
0.468048135
|
0.461194008
|
|
0.454953916
|
0.449055912
|
0.443555158
|
0.438405917
|
0.433369424
|
|
0.428717289
|
0.424209287
|
|
|
|
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Table-10(b). Values
of Decision Variables of Keane (m=17) obtained by RPS Min[f(x)]= -0.791503305; g1(x)=
-6.14730171E-006; g2(x)= -100.813254
| ||||
|
3.153072390
|
3.110523170
|
3.074380460
|
3.030577260
|
2.990643470
|
|
2.948230310
|
2.903373010
|
0.475140236
|
0.468015020
|
0.461406996
|
|
0.455065868
|
0.448695305
|
0.443058078
|
0.438121089
|
0.434204415
|
|
0.428833184
|
0.423405586
|
|
|
|
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Table-11(a). Values of Decision Variables of Keane (m=23) obtained by DE Min[f(x)]= -0.810036554; g1(x)=
-4.29345448E-012; g2(x)= -137.329364
| ||||
|
3.169694760
|
3.139982510
|
3.110819230
|
3.081854680
|
3.053000050
|
|
3.023921240
|
2.994482270
|
2.964326540
|
2.933023260
|
0.510288829
|
|
0.504420618
|
0.498649275
|
0.493512848
|
0.488546279
|
0.483523272
|
|
0.479153946
|
0.474755611
|
0.470648484
|
0.466631399
|
0.462846694
|
|
0.459054987
|
0.455553637
|
0.451945688
|
|
|
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Table-11(b). Values of Decision Variables of Keane (m=23) obtained by RPS Min[f(x)]= -0.806443992; g1(x)=
-2.44754988E-005; g2(x)= -135.786281
| ||||
|
3.160189530
|
3.132605360
|
3.101283350
|
3.073486860
|
3.045908980
|
|
3.016721420
|
2.984253490
|
2.957205140
|
2.925348010
|
2.891203450
|
|
0.437316502
|
0.434618030
|
0.429887890
|
0.427719236
|
0.422809225
|
|
0.421139133
|
0.416725599
|
0.413117438
|
0.409371097
|
0.407704275
|
|
0.405107257
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0.401105071
|
0.398892942
|
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Table-12(a). Values of Decision Variables of Keane (m=27) obtained by DE Min[f(x)]= -0.811434228; g1(x)=
-2.53111421E-010; g2(x)= -159.083939
| ||||
|
3.162640070
|
3.138297520
|
3.114299000
|
3.090456170
|
3.066700220
|
|
3.042930870
|
3.018928310
|
2.994522010
|
2.969607880
|
2.943953290
|
|
2.917228550
|
2.889110810
|
0.422657874
|
0.419707697
|
0.416940414
|
|
0.414145420
|
0.411515639
|
0.408872113
|
0.406368134
|
0.403919424
|
|
0.401489671
|
0.399182553
|
0.396883387
|
0.394646104
|
0.392482705
|
|
0.390330937
|
0.388244176
|
|
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Table-12(b). Values of Decision Variables of Keane (m=27) obtained by RPS Min[f(x)]=
-0.816943304 ; g1(x)=
-1.19269072E-005; g2(x)= -160.653353
| ||||
|
3.173409950
|
3.148240590
|
3.122005090
|
3.099207630
|
3.073407570
|
|
3.050460980
|
3.024861520
|
3.000385800
|
2.975618330
|
2.950741980
|
|
2.920317690
|
0.483482816
|
0.477704153
|
0.475182525
|
0.471337525
|
|
0.467439896
|
0.464773078
|
0.461536009
|
0.459642008
|
0.454704367
|
|
0.450330010
|
0.447936313
|
0.444572224
|
0.440030300
|
0.438913875
|
|
0.436688512
|
0.433716710
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Table-13(a). Values of Decision Variables of Keane (m=35) obtained by DE Min[f(x)]= -0.823249408; g1(x)=
-2.19188578E-009; g2(x)= -207.294331
| ||||
|
3.173022940
|
3.154137450
|
3.135351760
|
3.116948540
|
3.098632110
|
|
3.080316430
|
3.062019850
|
3.043838080
|
3.025530650
|
3.007087470
|
|
2.988219760
|
2.969192990
|
2.949682640
|
2.929865560
|
2.909301360
|
|
0.449087441
|
0.446489526
|
0.443976814
|
0.441598501
|
0.439260019
|
|
0.437147128
|
0.434902785
|
0.432761739
|
0.430545870
|
0.428373333
|
|
0.426403086
|
0.424338982
|
0.422526037
|
0.420436699
|
0.418566482
|
|
0.417004278
|
0.414856581
|
0.413117846
|
0.411460307
|
0.409667854
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Table-13(b). Values of Decision Variables of Keane (m=35) obtained by RPS Min[f(x)]=
-0.825277297 ; g1(x)=
-0.000119139164; g2(x)= -204.505815
| ||||
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6.269726250
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3.154360330
|
3.133103100
|
3.114955030
|
3.098820910
|
|
3.079824170
|
3.060554440
|
3.044914760
|
3.023623240
|
3.005605400
|
|
2.985462770
|
2.965903150
|
2.946515240
|
2.925727920
|
2.905643010
|
|
0.433505755
|
0.431011707
|
0.428275671
|
0.426509492
|
0.424762597
|
|
0.422812865
|
0.420239079
|
0.419822901
|
0.418930795
|
0.413752342
|
|
0.410895308
|
0.410283080
|
0.408714840
|
0.408045066
|
0.405630103
|
|
0.404382321
|
0.402369030
|
0.400914304
|
0.397253255
|
0.391334432
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Table-14(a). Values of Decision Variables of Keane (m=43) obtained by DE Min[f(x)]= -0.83226135; g1(x)=
-8.43495951E-009; g2(x)= -254.377563
| ||||
|
6.279437570
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3.166025720
|
3.149944230
|
3.134193120
|
3.118384230
|
|
3.102904180
|
3.087267970
|
3.071788540
|
3.056379620
|
3.040696730
|
|
3.024949450
|
3.009350790
|
2.993456050
|
2.977385710
|
2.961086450
|
|
2.944514810
|
2.927548890
|
0.493388747
|
0.490601423
|
0.487922748
|
|
0.485142273
|
0.482630155
|
0.480188053
|
0.477827961
|
0.475449255
|
|
0.473114249
|
0.470720458
|
0.468456669
|
0.466343132
|
0.464350740
|
|
0.462260514
|
0.460134599
|
0.458287613
|
0.456369695
|
0.454410171
|
|
0.452403380
|
0.450637388
|
0.448870370
|
0.446976988
|
0.445250201
|
|
0.443408596
|
0.441789263
|
0.440188104
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Table-14(b). Values of Decision Variables of Keane (m=43) obtained by RPS Min[f(x)]= -0.832259899; g1(x)=
-6.7151146E-005; g2(x)= -252.749098
| ||||
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6.281229680
|
3.158936610
|
3.148112900
|
3.134537550
|
3.118666760
|
|
3.099927430
|
3.086366350
|
3.070651230
|
3.056637670
|
3.041661580
|
|
3.025227620
|
3.011444720
|
2.994136920
|
2.979632390
|
2.963655390
|
|
2.945970420
|
2.931033600
|
2.915054880
|
0.455455069
|
0.451050569
|
|
0.449413031
|
0.447793497
|
0.444416653
|
0.443580846
|
0.442544677
|
|
0.440354860
|
0.436806643
|
0.436328404
|
0.434712866
|
0.431599120
|
|
0.431060144
|
0.429201676
|
0.428003131
|
0.426853958
|
0.426739990
|
|
0.423883701
|
0.420301219
|
0.418990143
|
0.417206138
|
0.415803124
|
|
0.412705272
|
0.411980526
|
0.411233373
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Table-15(a). Values of Decision Variables of Keane (m=47) obtained by DE Min[f(x)]= -0.833300903; g1(x)=
-5.64001357E-009; g2(x)= -276.077314
| ||||
|
6.279663750
|
3.162847070
|
3.148694640
|
3.134779200
|
3.120871690
|
|
3.107283320
|
3.093539880
|
3.079848380
|
3.066213940
|
3.052490340
|
|
3.038756200
|
3.025031050
|
3.011176820
|
2.997157170
|
2.983039540
|
|
2.968763490
|
2.954266890
|
2.939472010
|
2.924405520
|
2.909122960
|
|
0.443783424
|
0.441847883
|
0.440139288
|
0.438398466
|
0.436734371
|
|
0.435098197
|
0.433369075
|
0.431822240
|
0.430138948
|
0.428540952
|
|
0.426973748
|
0.425434393
|
0.423966985
|
0.422492736
|
0.421050529
|
|
0.419561255
|
0.418206208
|
0.416749228
|
0.415372910
|
0.413967574
|
|
0.412681692
|
0.411387253
|
0.410076023
|
0.408800246
|
0.407530631
|
|
0.406203464
|
0.404934000
|
|
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Table-15(b). Values of
Decision Variables of Keane (m=47) obtained by RPS
Min[f(x)]= -0.82892792; g1(x)=
-5.64914444E-005; g2(x)= -274.399424
| ||||
|
6.280865570
|
3.156188810
|
3.146774090
|
3.131048300
|
3.117603860
|
|
3.105551320
|
3.089657260
|
3.079060790
|
3.064843410
|
3.046563230
|
|
3.040863700
|
3.023017970
|
3.012707150
|
2.997817240
|
2.983747000
|
|
2.970102890
|
2.956072610
|
2.940757960
|
2.926886930
|
2.911972030
|
|
2.899856250
|
0.410299283
|
0.408159859
|
0.407157204
|
0.405598548
|
|
0.403227902
|
0.402294096
|
0.402016870
|
0.400016782
|
0.399578215
|
|
0.398264780
|
0.397064629
|
0.395442428
|
0.393422846
|
0.392503227
|
|
0.392036651
|
0.390980169
|
0.390029983
|
0.388256750
|
0.386166054
|
|
0.384685340
|
0.383143291
|
0.381729759
|
0.380716239
|
0.376113897
|
|
0.375371828
|
0.374340769
|
|
|
|