Minimization of
Keane's Bump Function by the Repulsive Particle Swarm and the Differential Evolution Methods
continued.. page 3 of 8
Department of Economics
NEHU, Shillong (India)
For the 20-dimensional Keane's problem we have DE min[(f)] = -0.803619104 ; g1(x) = -4.95659069E-012 ; g2(x) = -119.067416 the decision variables take on the following values given in Table-3(a). We obtain the RPS min[(f)] = -0.785263489; g1(x)= -2.13415866E-005; g2(x)= -117.548076 for the values of x given in Table-3(b).
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Table-3(a). Values of Decision Variables of Keane (m=20) obtained by DE | ||||
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3.162462120
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3.128331570
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3.094791700
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3.061449390
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3.027931110
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2.993829330
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2.958666990
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2.921839070
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0.494829273
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0.488358755
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0.482312620
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0.476648099
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0.471293545
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0.466228022
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0.461417418
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0.456836629
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0.452460752
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0.448267681
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0.444248266
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0.440381976
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Table-3(b). Values of Decision Variables of Keane (m=20) obtained by RPS | ||||
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3.151484020
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3.119328890
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3.086598240
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3.053529180
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3.021145250
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2.985534770
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2.949188660
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2.911211960
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0.418488616
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2.821296710
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0.410584741
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0.405855243
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0.399953254
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0.398307630
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0.394483277
|
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0.391179683
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0.388534529
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0.383874458
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0.382995909
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0.378349313
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We note that the DE results obey the observed rule of sequence while the RPS results, which are sub-optimal, do not obey the said rule. We also note that while Keane (1994) observed that for m=20 the value of min[f(x)] could be about -0.76, we obtain DE min[(f)] = -0.803619104. This result is surely better than the one envisaged by Keane. However, Ong and Keane (2003, p. 12) and Ong et al. (2005 ?) mention that the minimal value obtained by them is approximately - 0.81. If it is so, we have not been able to obtain the minimum value of the function. Keane in his personal letter (email dated 4.5.2007) informed the author that the best value for m=20 known to him till date is -0.803619104.
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Table-4. Values of Decision Variables of Keane (m=30) obtained by DE | ||||
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3.168225530
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3.146211650
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3.124531090
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3.102979710
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3.081823620
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3.060544800
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3.039195590
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3.017679840
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2.995636850
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2.973543750
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2.950666480
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2.927562460
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2.903088710
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0.440434895
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0.437505191
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0.435103340
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0.432460693
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0.429815709
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0.427295405
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0.424698735
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0.422641158
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0.420028735
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0.417678117
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0.415577752
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0.413108742
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0.410869231
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0.408999549
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0.406826514
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0.405042008
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0.402869708
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For the 30-dimensional Keane's problem we have DE min[(f)] = -0.818056222 ; g1(x) = -1.90829275E-009 ; g2(x) = -177.357354. The decision variables take on the values given in Table-4. The RPS results are grossly sub-optimal and hence we do not present them.