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The velocity updating rule according to an oblique coordinate system with mutation and dynamic scaling for particle swarm optimization

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Abstract

Particle swarm optimization (PSO) has been showing powerful search performance especially in separable and unimodal problems. However, the performance is deteriorated in non-separable problems such as rotated problems. In this study, a new velocity updating rule according to an oblique coordinate system, instead of an orthogonal coordinate system, is proposed to solve non-separable problems. Two mutation operations for the best particle and the worst particle are proposed to improve the diversity of particles and to decrease the degradation of moving speed of particles. In addition, the vectors generated according to the oblique coordinate system are dynamically scaled to improve the robustness and efficiency of the search. The advantage of the proposed method is shown by solving various problems including separable, non-separable, unimodal, and multimodal problems, and their rotated problems and by comparing the results of the proposed method with those of standard PSO.

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Notes

  1. “Separable” means that there are no interactions between decision variables and the optimal solution can be obtained by optimizing the objective function along each dimension separately. A typical separable problem is a sum of functions of one decision variable.

  2. If original solutions are rotated and new solutions generated by an algorithm (or an operation) are also rotated by the same rotation, the algorithm is rotationally invariant.

References

  1. Kennedy J, Eberhart RC (1995) Particle swarm optimization. In: Proceedings of IEEE international conference on neural networks, Perth, pp 1942–1948

  2. Kennedy J, Eberhart RC (2001) Swarm intelligence. Morgan Kaufmann, San Francisco

    Google Scholar 

  3. Sun Y, Halgamuge SK, Kirley M, Munoz MA (2014) On the selection of fitness landscape analysis metrics for continuous optimization problems. In: The 7th international conference on information and automation for sustainability, IEEE, pp 1–6

  4. Spears WM, Green DT, Spears DF (2010) Biases in particle swarm optimization. Int J Swarm Intell Res 1(2):34–57

    Article  Google Scholar 

  5. Hansen N, Ros R, Mauny N, Schoenauer M, Auger A (2011) Impacts of invariance in search: when cma-es and pso face ill-conditioned and non-separable problems. Appl Soft Comput 11(8):5755–5769

    Article  Google Scholar 

  6. Zhang J, Sanderson AC (2009) JADE: adaptive differential evolution with optional external archive. IEEE Trans Evolut Comput 13(5):945–958

    Article  Google Scholar 

  7. Bonyadi M, Li X, Michalewicz Z (2013) A hybrid particle swarm with velocity mutation for constraint optimization problems. In: Proceedings of the 15th annual conference on genetic and evolutionary computation, ACM, New York, pp 1–8. https://doi.org/10.1145/2463372.2463378

  8. Takahama T, Sakai S (2010) Solving nonlinear optimization problems by differential evolution with a rotation-invariant crossover operation using Gram-Schmidt process. In: Proceedings of second world congress on nature and biologically inspired computing (NaBIC2010), pp 533–540

  9. Guo SM, Yang CC (2015) Enhancing differential evolution utilizing eigenvector-based crossover operator. IEEE Trans Evolut Comput 19(1):31–49

    Article  MathSciNet  Google Scholar 

  10. Bonyadi MR, Michalewicz Z (2014) Spso 2011: analysis of stability; local convergence; and rotation sensitivity. In: Proceedings of the 2014 annual conference on genetic and evolutionary computation, ACM, pp 9–16

  11. Eberhart R, Shi Y (2001) Particle swarm optimization: developments, applications and resources. In: Proceedings of the 2001 congress on evolutionary computation, pp 81–86

  12. Engelbrecht A (2013) Particle swarm optimization: global best or local best? In: 2013 BRICS congress on computational intelligence and 11th Brazilian congress on computational intelligence, IEEE, pp 124–135

  13. Shi Y, Eberhart R (1999) Empirical study of particle swarm optimization. In: Proceedings of the 1999 congress on evolutionary computation, pp 1945–1950

  14. Clerc M, Kennedy J (2002) The particle swarm—explosion, stability, and convergence in a multidimensional complex space. IEEE Trans Evolut Comput 6(1):58–73

    Article  Google Scholar 

  15. Eberhart R, Shi Y (2000) Comparing inertia weights and constriction factors in particle swarm optimization. In: Proceedings of the 2000 congress on evolutionary computation, pp 84–88

  16. Sakai S, Takahama T (2018) A study on selecting an oblique coordinate system for rotation-invariant blend crossover in a real-coded genetic algorithm. In: Kadoya A, Teramoto H (eds) Recent studies in economic sciences: information Systems. Kyushu University Press, Project Managements, Economics, OR and Mathematics, pp 65–87

  17. Iorio AW, Li X (2008) Improving the performance and scalability of differential evolution. In: Asia-Pacific conference on simulated evolution and learning, Springer, New York, pp 131–140

    Google Scholar 

  18. Eshelman LJ, Schaffer JD (1993) Real-coded genetic algorithms and interval schemata. In: Whitley LD (ed) Foundations of genetic algorithms 2. Morgan Kaufmann Publishers, San Mateo, pp 187–202

    Google Scholar 

  19. Shang YW, Qiu YH (2006) A note on the extended Rosenbrock function. Evolut Comput 14(1):119–126

    Article  Google Scholar 

  20. Yao X, Liu Y, Lin G (1999) Evolutionary programming made faster. IEEE Trans Evolut Comput 3:82–102

    Article  Google Scholar 

  21. Yao X, Liu Y, Liang KH, Lin G (2003) Fast evolutionary algorithms. In: Ghosh A, Tsutsui S (eds) Advances in evolutionary computing: theory and applications. Springer, New York, pp 45–94

    Chapter  Google Scholar 

  22. Akimoto Y, Nagata Y, Sakuma J, Ono I, Kobayashi S (2009) Proposal and evaluation of adaptive real-coded crossover AREX. Trans Jpn Soc Artif Intell 24(6):446–458 in Japanese

    Article  Google Scholar 

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Authors and Affiliations

  1. Hiroshima City University, Hiroshima, Japan

    Tetsuyuki Takahama

  2. Hiroshima Shudo University, Hiroshima, Japan

    Setsuko Sakai

Authors
  1. Tetsuyuki Takahama
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  2. Setsuko Sakai
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Corresponding author

Correspondence to Tetsuyuki Takahama.

Additional information

This research is supported in part by JSPS KAKENHI Grant Numbers 26350443 and 17K00311.

This work was presented in part at the 2nd International Symposium on Swarm Behavior and Bio-Inspired Robotics, Kyoto, October 29–November 1, 2017.

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Takahama, T., Sakai, S. The velocity updating rule according to an oblique coordinate system with mutation and dynamic scaling for particle swarm optimization. Artif Life Robotics 23, 618–627 (2018). https://doi.org/10.1007/s10015-018-0498-y

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  • DOI: https://doi.org/10.1007/s10015-018-0498-y

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