Minimization of
Keane's Bump Function by the Repulsive Particle Swarm and the Differential Evolution Methods
continued.. page 2 of 8
Department of Economics
NEHU, Shillong (India)
II. The Objectives of this Paper: We intend in this paper to optimize Keane's function of different dimensions by the Repulsive Particle Swarm (RPS) and the Differential Evolution (DE) methods of global optimization. Our RPS is endowed with intensive local search ability. Similarly, our DE uses the most recent (available) formulation of crossover scheme suggested by Kenneth Price. We have developed our own computer programs in FORTRAN. Our programs have yielded very good results for quite varied and difficult problems (Mishra, 2006 (a, b & c)). Programs are available on request.
III. Results and Discussion: First we obtain min[f(x)] for the two-dimensional (m=2) Keane's problem. We have DE min[(f)] = -0.364979746; g1(x) = -1.11022302E-016 and g2(x) = -11.9306415 for x1 = 1.60086044 and x2 = 0.468498055. Against this, RPS min[(f)] = -0.364979123; g1(x) = -3.82208509E-007; g2(x) = -11.9310703 for x1 = 1.60025376 and x2 = 0.468675907. These results are comparable with the optimum value obtained by Hacker et al. The DE results are marginally better than the RPS results.
Emmerich (2005,p. 118) obtained about -0.6 as the minimum value of the 10-dimensional Keane problem. Hacker et al. report to have obtained min[f(x)] = -0.6737. We have obtained the DE min[(f)] = -0.747310362; g1(x) = -2.2849167E-011 and g2(x) = -58.5724568 for x given in Table-1(a). Against these we obtained by RPS method min[(f)] = -0.747309014; g1(x) = -4.64514816E-009; and g2(x) = -58.5732418 for x given in Table-1(b). Again, the DE performs better than the RPS.| Table-1(a). Values of Decision Variables of Keane (m=10) obtained by DE | ||||
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3.123889470
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3.069156770
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3.014282390
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2.957588300
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1.466044830
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0.368057943
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0.363481530
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0.359121133
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0.354952823
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0.350967972
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Table-1(b). Values
of Decision Variables of Keane (m=10) obtained by RPS
| ||||
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3.124165150
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3.068864800
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3.015470120
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2.955911380
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1.465639690
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0.367340008
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0.363605407
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0.358758758
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0.354977899
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0.352024977
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For the 10-dimensional problem, evidently, our results are much better than those obtained by Hacker et al. by their hybridized Genetic Algorithm, which, in themselves are better than the (pure) Genetic algorithm results. They have not presented the values taken on by the decision variables (x). However, the values of x in our results indicate, first, that xi > xj ; ∀ j > i . This particular observation is very important, although it is just a conjecture. A sub-optimal solution would not have such a sequence. Secondly, we conjecture that the values form two clusters with almost the same number of members.
For the 15-dimensional Keane's problem we have DE min[(f)] = -0.781647601; g1(x) = -5.18851406E-010; g2(x) = -88.5890866. The decision variables take on the following values given in Table-2(a). Against these values of DE we have RPS min[(f)] = -0.778452035; g1(x) = -1.42940888E-005; g2(x) = -88.6245989 for x given in Table-2(b) below. The RPS results are inferior to the DE results. The noted sequence is not given by the RPS results. The values of x7 and x8 have broken the said sequence.
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Table-2(a). Values
of Decision Variables of Keane (m=15) obtained by DE
| ||||
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3.146293070
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3.106145430
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3.066581520
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3.026848660
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2.986415340
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2.944612650
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1.432725200
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0.414056148
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0.409743392
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0.405635026
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0.401726913
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0.397869284
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0.394212294
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0.390652961
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0.387395467
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Table-2(b). Values
of Decision Variables of Keane (m=15) obtained by RPS
| ||||
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3.147885830
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3.107285280
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3.067863320
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3.027769080
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2.986310220
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2.947132220
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0.418950619
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1.373826510
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0.413106309
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0.406717883
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0.403270323
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0.399151233
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0.394427081
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0.391056701
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0.390648466
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