Mishra, SK (2006): Performance of Differential Evolution and Particle Swarm Methods on Some Relatively Harder Multimodal Benchmark Functions.

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Abstract
Our objective in this paper is to compare the performance of the Differential Evolution (DE) and the Repulsive Particle Swarm (RPS) methods of global optimization. To this end, some relatively difficult test functions have been chosen. These functions are: Perm, PowerSum, Bukin, ZeroSum, Hougen, Giunta, DCS, Kowalik, FletcherPowell and some now functions.
Our results show that DE (with the exponential crossover scheme) mostly fails to find the optimum of most of these functions. Of course, it succeeds in case of some functions (perm#2, zerosum) for very small dimension (m), but begins to falter as soon as the dimension is increased. In case of DCS function, it works well up to m (dimension) = 5.
When we use no crossover (only probabilistic replacement) we obtain better results in case of several of the functions under study. Thus, overall, table #2 presents better results than what table #1 does. In case of Perm#1, Perm#2, Zerosum, Kowalik, Hougen and Powersum functions the advantage is clear.
Whether crossover or no crossover, DE falters when the optimand function has some element of randomness. This is indicated by the functions: YaoLiu#7, FletcherPowell, and “New function#2”. DE has no problems in optimizing the “New function#1”. But the “New function #2” proves to be a hard nut. However, RPS performs much better for such stochastic functions. When the FletcherPowell function is optimized with nonstochastic c vector, DE works fine. But as soon as c is stochastic, it becomes unstable. Thus, it may be observed that an introduction of stochasticity into the decision variables (or simply added to the function as in YaoLiu#7) interferes with the fundamentals of DE, which works through attainment of better and better (in the sense of Pareto improvement) population at each successive iteration.
Item Type:  MPRA Paper 

Institution:  NorthEastern Hill University, Shillong (India) 
Original Title:  Performance of Differential Evolution and Particle Swarm Methods on Some Relatively Harder Multimodal Benchmark Functions 
Language:  English 
Keywords:  Repulsive particle swarm; Differential evolution; Global optimization; Stochasticity; random disturbances; Crossover; Perm; zero sum; Kowalik; Hougen; Power sum; DCS; Fletcher Powell; multimodal; benchmark; test functions; Bukin; Giunta 
Subjects:  C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C63  Computational Techniques ; Simulation Modeling C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C61  Optimization Techniques ; Programming Models ; Dynamic Analysis 
Item ID:  1743 
Depositing User:  Sudhanshu Kumar Mishra 
Date Deposited:  11. Feb 2007 
Last Modified:  14. Feb 2013 12:26 
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URI:  https://mpra.ub.unimuenchen.de/id/eprint/1743 