Minimization of
Keane's Bump Function by the Repulsive Particle Swarm and the Differential Evolution Methods
continued.. page 4 of 8
Department of Economics
NEHU, Shillong (India)
For the 40-dimensional Keane's problem we have DE min[(f)] = -0.826624404 ; g1(x) = -4.67459549E-009 ; g2(x) = -237.241084. The decision variables take on the following values given in Table-5. The RPS results are grossly sub-optimal and hence we do not present them.
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Table-5. Values of Decision Variables of Keane (m=40) obtained by DE | ||||
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3.176402220
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3.159781680
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3.143267960
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3.126970650
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3.110958340
|
|
3.094833420
|
3.078715320
|
3.062493030
|
3.046925820
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3.030744620
|
|
3.014489370
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2.998191690
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2.981676650
|
2.964750240
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2.947671240
|
|
2.930427940
|
2.912463350
|
0.455727042
|
0.453400834
|
0.451294809
|
|
0.448984473
|
0.446911098
|
0.444660768
|
0.442769918
|
0.440588651
|
|
0.438735980
|
0.436742512
|
0.435131699
|
0.433072143
|
0.431025920
|
|
0.429591143
|
0.427762652
|
0.425920239
|
0.424344581
|
0.422578191
|
|
0.420945959
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0.419335333
|
0.417769877
|
0.416132722
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0.414726294
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Optimization of the 50-dimensional Keane's problem was problematic. We had to do some fine-tuning of the DE parameters and some trial and error too. Finally, [for RX1=0.5, RX2=0.7 and F =0.05: these are detailed out in the DE program written by us] we have DE min[(f)] = -0.83078783;g1(x) = -2.55134022E-008; g2(x) = -297.149824. The decision variables take on the following values given in Table-6. The RPS results are grossly sub-optimal and hence we do not present them
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Table-6. Values of Decision Variables of Keane (m=50) obtained by DE | ||||
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2.984331850
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2.970850220
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2.957100110
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2.943168300
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2.929067060
|
|
2.914580720
|
0.465266845
|
0.463396171
|
0.461448453
|
0.459610933
|
|
0.457759997
|
0.456007425
|
0.454318803
|
0.452430869
|
0.450861344
|
|
0.449163331
|
0.447302715
|
0.445867968
|
0.444297375
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0.442654703
|
|
0.440993800
|
0.439496353
|
0.438148497
|
0.436730303
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0.435083505
|
|
0.433749816
|
0.432327620
|
0.430812420
|
0.429525278
|
0.428145716
|
|
0.426877925
|
0.425525920
|
0.424203890
|
0.422892481
|
0.421473178
|
|
2.984331850
|
2.970850220
|
2.957100110
|
2.943168300
|
2.929067060
|
|
2.914580720
|
0.465266845
|
0.463396171
|
0.461448453
|
0.459610933
|
|
0.457759997
|
0.456007425
|
0.454318803
|
0.452430869
|
0.450861344
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Keane (1994) mentioned that for m=50 min[(f)] could be around - 0.835. The number obtained by us is -0.83078783 (and the decision variables satisfy the sequence noted earlier). We cannot claim that the sequence conjectured by us provides the necessary and sufficient condition for optimality of values taken on by the decision variables, although the condition appears to be necessary. Keane in his personal letter (email dated 4.5.2007) informed the author that the best value known to him for m=50 till date is: -0.835262348.
IV. A Useful Application of Our Conjecture: Now we make an attempt to (possibly) take advantage of our conjecture on the pattern exhibited by the values of decision variable so far. That is: xi > xj ; ∀ j > i. Before every function evaluation we arrange the values of the decision variables in a descending order and then evaluate the Keane's function (and the constraints). We find, to our pleasant surprise, that it works well and gives a new minimum (for m=50), which is very close to the value (≈ - 0.835) envisaged by Keane. Now, after incorporating our conjecture in the computation, we obtain: DE min[(f)] = -0.834985126 and g1(x) = -9.86513138E-009; g2(x) = -294.382754. The decision variables take on the values given in Table-7. The RPS results are close to the DE results, but sub-optimal. The RPS min[(f)]= -0.83181972; g1(x)= -3.97261395E-05; g2(x) = -292.704914.
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Table-7. Values of Decision Variables of Keane (m=50) obtained by DE | ||||
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6.281881470
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3.165864960
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3.152419640
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3.139117420
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3.125910180
|
|
3.112937380
|
3.099820120
|
3.086810400
|
3.073914590
|
3.060912370
|
|
3.047946650
|
3.034998900
|
3.021821370
|
3.008627600
|
2.995345870
|
|
2.982032700
|
2.968429500
|
2.954671080
|
2.940647550
|
2.926380720
|
|
2.911877500
|
0.454055821
|
0.452282938
|
0.450327194
|
0.448694221
|
|
0.446908360
|
0.445253680
|
0.443532095
|
0.441920750
|
0.440236963
|
|
0.438736885
|
0.437136832
|
0.435642454
|
0.434122273
|
0.432614935
|
|
0.431118544
|
0.429681262
|
0.428261041
|
0.426880934
|
0.425427247
|
|
0.424099015
|
0.422711491
|
0.421441660
|
0.420066322
|
0.418684131
|
|
0.417438227
|
0.416297458
|
0.415067783
|
0.413740624
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0.412497031
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It has been mentioned earlier that Ong and Keane (2003, p. 12) and Ong et al. (2005 ?) report that the minimal value obtained by them is approximately - 0.81 (for m=20). With the success experience due to incorporating our conjecture (in case of m=50) we are tempted to re-compute the min[(f)] for m=20. However, we have not been able to obtain any better results (than -0.803619104, reported earlier).