Soft Computing

, Volume 22, Issue 6, pp 1993–2014 | Cite as

An evolutionary algorithm using spherical inversions

  • Juan Pablo Serrano-Rubio
  • Arturo Hernández-Aguirre
  • Rafael Herrera-Guzmán
Methodologies and Application


This paper introduces an evolutionary algorithm which uses reflections and spherical inversions for global continuous optimization. Two new geometric search operators are included in the design of the algorithm: the inversion search operator and the reflection search operator. The inversion search operator computes inverse points with respect to hyperspheres, and the reflection search operator redistributes the individuals on the search space of the fitness function. The nonlinear geometric nature of the inversion search operator furnishes more “aggressive” search and exploitation capabilities for the algorithm. The performance of the algorithm is analyzed through a benchmark of 28 functions. Statistical tests show the competitive performance of the algorithm in comparison with current leading (geometric) algorithms such as particle swarm optimization and four differential evolution strategies.


Geometric search operators Evolutionary algorithm Continuous optimization 



This research is partially supported by a Grant of National Council of Science and Technology of Mexico CONACYT (256126). The first author would like to thank the University of Exeter for its hospitality. The third author would like to thank the International Centre for Theoretical Physics (ICTP) and the Institut Des Hautes Etudes Scientifiques (IHES) for their hospitality and support.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


  1. Ardia D, Mullen K, Peterson B, Ulrich J, Boudt K (2016) Deoptim: global optimization by differential evolution.
  2. Bonyadi MR, Michalewicz Z (2016) Stability analysis of the particle swarm optimization without stagnation assumption. IEEE Trans Evol Comput 20(5):814–819CrossRefGoogle Scholar
  3. Bosman P, Grahl J, Thierens D (2007) Adapted maximum-likelihood Gaussian models for numerical optimization with continuous EDAs. Technical Report, Amsterdam: CWI, AmsterdamGoogle Scholar
  4. Brest J, Greiner S, Boskovic B, Mernik M, Zumer V (2006) Self-adapting control parameters in differential evolution: a comparative study on numerical benchmark problems. IEEE Trans Evol Comput 10(6):646–657CrossRefGoogle Scholar
  5. Budhraja KK, Singh A, Dubey G, Khosla A (2012) Exploration enhanced particle swarm optimization using guided re-initialization. In: Proceedings of seventh international conference on bio-inspired computing: theories and applications (BIC-TA 2012), vol 1, pp 403–416Google Scholar
  6. Cuevas E, Díaz Cortés MA, Oliva Navarro DA (2016) A states of matter algorithm for global optimization. Springer, Cham, pp 35–54Google Scholar
  7. Eberhart R, Kennedy J (1995) A new optimizer using particle swarm theory. In: Proceedings of the sixth international symposium on micro machine and human science, 1995. MHS ’95, pp 39–43Google Scholar
  8. Engelbrecht AP (2006) Fundamentals of computational swarm intelligence. Wiley, ChichesterGoogle Scholar
  9. Ergezer M, Simon D (2015) Probabilistic properties of fitness-based quasi-reflection in evolutionary algorithms. Comput Oper Res 63:114–124MathSciNetCrossRefMATHGoogle Scholar
  10. Gong YJ, Zhou Q, Lin Y, Zhang J (2015) Orthogonal predictive differential evolution, vol 1, pp 141–154Google Scholar
  11. Hansen N, Finck S, Ros R, Auger A (2009) Real-parameter black-box optimization benchmarking 2009: noiseless functions definitions. Research Report RR-6829, INRIAGoogle Scholar
  12. Hui W, Rahnamayan S, Hui S, Omran M (2013) Gaussian bare-bones differential evolution. IEEE Trans Cybern 43(2):634–647CrossRefGoogle Scholar
  13. Jin W (2011) Particle swarm optimization with adaptive parameter control and opposition. J Comput Inf Syst 7(12):4463–4470Google Scholar
  14. Moraglio A (2007) Towards a geometric unification of evolutionary algorithms. PhD Thesis, University of Essex, UKGoogle Scholar
  15. Moraglio A, C Di Chio, R Poli (2007) Geometric particle swarm optimisation. Springer, BerlinCrossRefGoogle Scholar
  16. Moraglio A, Di Chio C, Togelius J, Poli R (2008) Geometric particle swarm optimization. J Artif Evol Appl 2008:11. doi: 10.1155/2008/143624
  17. Moraglio A, Johnson CG (2010) Geometric generalization of the Nelder–Mead algorithm. Springer, BerlinCrossRefGoogle Scholar
  18. Moraglio A, Poli R (2004) Topological interpretation of crossover. Springer, BerlinCrossRefMATHGoogle Scholar
  19. Moraglio A, Togelius J (2009) Geometric differential evolution. In: Proceedings of the 11th annual conference on genetic and evolutionary computation. GECCO ’09ACM, New York, NY, USA, pp 1705–1712Google Scholar
  20. Moraglio A, Togelius J (2009) Inertial geometric particle swarm optimization. In: IEEE congress on evolutionary computation, 2009. CEC’09. IEEE, pp 1973–1980Google Scholar
  21. Moraglio A, Togelius J, Lucas S (2006) Product geometric crossover for the sudoku puzzle. In: IEEE congress on evolutionary computation, 2006. CEC 2006, pp 470–476Google Scholar
  22. Price K, Storn RM, Lampinen JA (2005) Differential evolution: a practical approach to global optimization (natural computing series). Springer, New YorkMATHGoogle Scholar
  23. Rosenhahn B, Sommer G (2005) Pose estimation in conformal geometric algebra. part II: real-time pose estimation using extended feature concepts. J Math Imaging Vis. CiteseerGoogle Scholar
  24. Serrano-Rubio J (2016a) Spherical evolutionary algorithm.
  25. Serrano-Rubio J (2016b) Differential evolution.
  26. Sinha A, Porokka A, Malo P, Deb K (2015) Unconstrained robust optimization using a descent-based crossover operator. In: 2015 IEEE congress on evolutionary computation (CEC), pp 85–92Google Scholar
  27. Uriarte A, Melin P, Valdez F (2016) An improved particle swarm optimization algorithm applied to benchmark functions. In: 2016 IEEE 8th international conference on intelligent systems (IS), pp 128–132Google Scholar
  28. van den Bergh F, Engelbrecht A (2006) A study of particle swarm optimization particle trajectories. Inf Sci 176(8):937–971MathSciNetCrossRefMATHGoogle Scholar
  29. Vesterstrom J, Thomsen R (2004) A comparative study of differential evolution, particle swarm optimization, and evolutionary algorithms on numerical benchmark problems. In: Evolutionary computation, 2004. CEC2004. Congress on Evolutionary Computation, vol 2, pp 1980–1987Google Scholar
  30. Xin Y, Yong L, Guangming L (1999) Evolutionary programming made faster. IEEE Trans Evol Comput 3(2):82–102CrossRefGoogle Scholar
  31. Xinchao Z (2010) A perturbed particle swarm algorithm for numerical optimization. Appl Soft Comput 10(1):119–124CrossRefGoogle Scholar
  32. Xu J, Zhang J (2014) Exploration-exploitation tradeoffs in metaheuristics: survey and analysis. In: Control conference (CCC), 2014 33rd Chinese, pp 8633–8638Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Juan Pablo Serrano-Rubio
    • 1
    • 4
  • Arturo Hernández-Aguirre
    • 2
  • Rafael Herrera-Guzmán
    • 3
  1. 1.Information Technologies LaboratoryTechnological Institute of Irapuato (ITESI)IrapuatoMexico
  2. 2.Computer Science DepartmentCenter for Research in Mathematics (CIMAT)GuanajuatoMexico
  3. 3.Mathematics DepartmentCenter for Research in Mathematics (CIMAT)GuanajuatoMexico
  4. 4.School of Engineering, Computer Science and MathematicsUniversity of ExeterExeterUK

Personalised recommendations