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Performance of Differential Evolution and Particle Swarm Methods on Some Relatively Harder Multi-modal 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, Power-Sum, Bukin, Zero-Sum, Hougen, Giunta, DCS, Kowalik, Fletcher-Powell 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, zero-sum) 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, Zero-sum, Kowalik, Hougen and Power-sum 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: Yao-Liu#7, Fletcher-Powell, 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 Fletcher-Powell function is optimized with non-stochastic 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 Yao-Liu#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.

Suggested Citation

  • Mishra, SK, 2006. "Performance of Differential Evolution and Particle Swarm Methods on Some Relatively Harder Multi-modal Benchmark Functions," MPRA Paper 1743, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:1743
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    1. Mishra, SK, 2006. "Some Experiments on Fitting of Gielis Curves by Simulated Annealing and Particle Swarm Methods of Global Optimization," MPRA Paper 465, University Library of Munich, Germany.
    2. Mishra, Sudhanshu, 2006. "Some new test functions for global optimization and performance of repulsive particle swarm method," MPRA Paper 2718, University Library of Munich, Germany.
    3. Mishra, SK, 2006. "Least Squares Fitting of Chacón-Gielis Curves by the Particle Swarm Method of Optimization," MPRA Paper 466, University Library of Munich, Germany.
    4. Mishra, SK, 2006. "Repulsive Particle Swarm Method on Some Difficult Test Problems of Global Optimization," MPRA Paper 1742, University Library of Munich, Germany.
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    Cited by:

    1. Mishra, SK, 2017. "Measuring Degree of Globalization of African Countries on Almost Equimarginal Contribution Principle," MPRA Paper 82614, University Library of Munich, Germany.
    2. Mishra, SK, 2012. "A note on construction of heuristically optimal Pena’s synthetic indicators by the particle swarm method of global optimization," MPRA Paper 37625, University Library of Munich, Germany.
    3. Mishra, SK, 2012. "A maximum entropy perspective of Pena’s synthetic indicators," MPRA Paper 37797, University Library of Munich, Germany.
    4. Sudhanshu K Mishra, 2013. "Global Optimization of Some Difficult Benchmark Functions by Host-Parasite Coevolutionary Algorithm," Economics Bulletin, AccessEcon, vol. 33(1), pages 1-18.
    5. Mishra, SK, 2012. "Global optimization of some difficult benchmark functions by cuckoo-hostco-evolution meta-heuristics," MPRA Paper 40615, University Library of Munich, Germany.

    More about this item

    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;

    JEL classification:

    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis

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