Minimization of
Keane's Bump Function by the Repulsive Particle Swarm and the Differential Evolution Methods
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Department of Economics
NEHU, Shillong (India)
VI. Conclusion: A not-so-exhaustive survey of literature on optimization of Keane's function suggests us that many researchers avoid mentioning the values of objective function, constraints and decision variables that they obtained in their works.
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They mention that the program, method or algorithm used by them was repeated so-and-so many times. However, they have hesitated to mention the range - the upper and the lower limits - within which they had obtained their results. Measures like mean, median or standard deviation only blur the findings and perhaps conceal much more than they reveal. If Emmerich (2005) is right in stating that the true minima of Keane's function for different dimensions are unknown, it is required that the research efforts of each of us are recorded clearly, accurately and with necessary details so that the next research worker knows what his (or her) efforts are yielding. |
In this paper we have provided the details to set the records right. A summary of our results is presented in Table-16.
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We also conjectured that values of the decision variables diminish with the increasing index, that is, xi > xj ; ∀ j > i and they form two distinct clusters with almost equal number of members. These regularities indicate whether the function could attain a minimum or (at least) has reached close to the minimum. They may also be exploited to seek the minimum values of the function. The Differential Evolution (DE) program(developed by us) has gone a long way to obtain the optimum (if so!) results. | ||||||||||||||||||||||||||||||||||||||||||||||||
Application of the Particle Swarm optimization program (developed by us) has clearly failed to minimize Keane's function of any considerable size, without arranging the variable values in a descending order. On ordering the variable values, the RPS results are comparable with those of the DE. The DE also needed ordering of variable values to perform when the size of the problem was larger. Our two findings are notable: (i) Keane's envisaged min(f) = -0.835 for 50-dimensional problem is realizable by the RPS as well as the DE; (ii) Liu-Lewis' min(f) = -0.84421 for 200-dimensional problem is grossly sub-optimal.
Finally, although Keane disfavors ordering of variables to obtain a minimum (since such 'tricks' lead to over-estimation of the efficiency of an optimizing program), such a trick, nonetheless, gives us new lower bounds of the function that might stimulate further improvements in different optimization procedures. We hold that our investigation should have two objectives: (a) obtaining the most efficient optimizers; (b) obtaining new lower bounds of the function for different dimensions.