Minimization of Keane’s Bump Function by the Repulsive Particle Swarm and the Differential Evolution Methods

Mishra, SK (2007): Minimization of Keane’s Bump Function by the Repulsive Particle Swarm and the Differential Evolution Methods.

Abstract

Keane’s bump function is considered as a standard benchmark for nonlinear constrained optimization. It is highly multi-modal and its optimum is located at the non-linear constrained boundary. The true minimum of this function is, perhaps, unknown. We intend in this paper to optimize Keane’s function of different dimensions (2 to 100) by the Repulsive Particle Swarm and Differential Evolution methods. The DE optimization program has gone a long way to obtain the optimum results. However, the Repulsive Particle Swarm optimization has faltered. We have also conjectured that the values of the decision variables diminish with the increasing index values 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. We have used this conjecture to incorporate ordering of variable values before evalution of the function and its optimization at every trial. As a result, the performance of DE as well as the RPS has improved significantly. Our results are comparable with the best results available in the literature on optimization of Keane function. Our two findings are notable: (i) Keane’s envisaged min(f) = -0.835 for 50-dimensional problem is realizable; (ii) Liu-Lewis’ min(f) = -0.84421 for 200-dimensional problem is grossly sub-optimal.Computer programs (written by us in Fortran) are available on request.

• Item Type: MPRA Paper
• Institution: North-Eastern Hill University, Shillong (India)
• Language: English
• Keywords: Nonlinear; constrained; global optimization; repulsive particle swarm; differential evolution; Fortran; computer program; Hybrid; Genetic algorithms
• Subjects: C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C61 - Optimization Techniques ; Programming Models ; Dynamic Analysis; C - Mathematical and Quantitative Methods > C8 - Data Collection and Data Estimation Methodology ; Computer Programs > C88 - Other Computer Software

References

· Emmerich, MTM: Single- and Multi-objective Evolutionary Design Optimization Assisted by Gaussian Random Field Metamodels, Dissertation for Doctoral Degree in Natural Sciences, University of Dortmund, Dortmund. 2005. · Hacker, KA, Eddy, J and Lewis, KE: “Efficient Global Optimization using Hybrid Genetic Algorithms,” in 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Atlanta, Georgia, 4-6 September 2002. · Keane, AJ: “Bump: A Hard(?) Problem” http://www.soton.ac.uk/~ajk/bump.html, 1994. · Keane, AJ: "Experiences with optimizers in structural design," in Parmee, IC (ed) Proceedings of the 1st Conf. on Adaptive Computing in Engineering Design and Control, University of Plymouth, UK, pp. 14-27, 1994. · Liu, P and Lewis, MJ: “Communication Aspects of an Asynchronous Parallel Evolutionary Algorithm”, Proceedings of the Third International Conference on Communications in Computing (CIC 2002), pp. 190-195, Las Vegas, NV, June 24-27, 2002. http://grid.cs.binghamton.edu/papers/LiuAPEAComm_CIC.pdf · Mishra, SK (a): "Global Optimization by Particle Swarm Method: A Fortran Program" (August 1, 2006). Available at SSRN: http://ssrn.com/abstract=921504 · Mishra, SK (b): "Global Optimization by Differential Evolution and Particle Swarm Methods: Evaluation on Some Benchmark Functions" (September 30, 2006). Available at SSRN: http://ssrn.com/abstract=933827 · Mishra, SK (c): "Performance of Differential Evolution and Particle Swarm Methods on Some Relatively Harder Multi-Modal Benchmark Functions" (October 13, 2006). Available at SSRN: http://ssrn.com/abstract=937147 · Ong, YS and Keane, AJ: “Meta-Lamarckian Learning in Memetic Algorithms” http://ntu-cg.ntu.edu.sg/ysong/journal/IEEE_EC_Ysong2003.pdf, 2003. Ong, YS, Lim, MH, Zhu, N and Wong, KW: “Classification of Adaptive Memetic Algorithms: A Comparative Study”, Un-dated (possibly written in 2005) working paper, http://ntu-cg.ntu.edu.sg/ysong/journal/AdaptiveMA.pdf