Abstract
Standard particle swarm optimization 2011 (SPSO2011) is a major improvement of the original particle swarm optimization (PSO) with its adaptive random topology and rotational invariance. Its overall performance has also been improved considerably from the original PSO algorithm, but further improvement is still possible. This study attempts to enhance the exploration ability of SPSO2011 further. The enhancement method conditionally introduces a new genetic mechanism to improve the personal best condition of each particle. This conditional event is called an event-triggered mechanism. Moreover, the new genetic mechanism is utilized to crossover, mutate, and select an improved offspring to improve the condition of the cognitive component and indirectly enhance the exploration ability. The proposed algorithm is called genetic mechanism-enhanced SPSO2011 (SPSO2011_GM). SPSO2011_GM is empirically analyzed with 42 benchmark functions. Results confirm the efficiency of the proposed enhancement method and verify the convergence, exploration, reliability, and scalability of the method.
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Notes
If \({c_1} \ne 0\) and \({c_2} = 0\), then each particle has no external constraints and will explore the search space independently. Therefore, the cognitive component influences the exploration ability. Literally, the exploration ability represents the ability of the swarm to explore the potential optima. The more diversified the swarm is, the stronger the exploration ability is.
If \({c_1} = 0\) and \({c_2} \ne 0\), then each particle has remarkably strong constraints from other particles and will converge directly to the global best position identified so far. Therefore, the social component influences the convergence or exploitation ability. Literally, the convergence ability represents the ability of the swarm to converge to a point. The more gathered the particles are, the stronger the convergence ability is.
Iterative stochastic optimization algorithm is said to converge in probability if \(\forall \varepsilon > 0,\mathop {\lim }\nolimits _{t \rightarrow \infty } P(||\overrightarrow{{x_t}} - \overrightarrow{X} || < \varepsilon ) = 1\) . Where P is the probability measure, \(\overrightarrow{{x_t}} \) is a solution at generation t, \(\overrightarrow{X} \) is a point in the search space, \(\varepsilon \) is a small positive value. If the point \(\overrightarrow{X} \) is a local optimum and the stochastic algorithm guarantees the satisfaction of the previous equation, then the algorithm is said to be locally convergent.
As is shown in Fig. 1, PSO and most of its variants converge fast in the early stages. For the unimodal functions which have only one local (global) optimum and will not get trapped in any other local optimum, the performance of an algorithm is mainly determined by the ability of convergence in the middle and later stages. For the multimodal and composition functions which have many local optima, the performance of an algorithm is mainly determined by the ability of exploration so that it would is more capable of jumping out the local optima. The more complicated the objective function is, the exploration ability matters more.
Therefore, if one algorithm achieves better results than the other with unimodal functions, this algorithm possesses better convergence ability. If one algorithm achieves better results than the other with complicated multimodal or composition functions, this algorithm possesses better exploration ability.
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Acknowledgements
This work was supported by National Natural Science Foundation of China (61590923, 61422303) and “Shu Guang” project supported by Shanghai Municipal Education Commission and Shanghai Education Development Foundation.
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Du, W., Zhang, F. Genetic mechanism-enhanced standard particle swarm optimization 2011. Soft Comput 22, 7207–7225 (2018). https://doi.org/10.1007/s00500-017-2724-2
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DOI: https://doi.org/10.1007/s00500-017-2724-2