Soft Computing

, Volume 22, Issue 6, pp 2045–2064 | Cite as

Incremental cooperative coevolution for large-scale global optimization

  • Sedigheh Mahdavi
  • Shahryar Rahnamayan
  • Mohammad Ebrahim Shiri
Methodologies and Application


Cooperative coevolution (CC) is an efficient framework for solving large-scale global optimization (LSGO) problems. It uses a decomposition method to divide the LSGO problems into several low-dimensional subcomponents; then, subcomponents are optimized. Since CC algorithms do not consider any imbalance feature, their performance degrades during solving imbalanced LSGO problems. In this paper, we propose an incremental CC (ICC) algorithm in which the algorithm optimizes an integrated subcomponent which subcomponents are dynamically added to it. Therefore, the search space of the optimizer is grown incrementally toward the original problem search space. Various search spaces are built according to three approaches, namely random-based, sensitivity analysis-based, and random sensitivity analysis-based methods; then, ICC explores these search spaces effectively. Random-based selects a subcomponent randomly for adding it to the current search space and the sensitivity analysis-based method uses a sensitivity analysis strategy to select a subcomponent. The random sensitivity analysis-based strategy is a hybrid of the random and sensitivity analysis-based methods. Theoretical analysis is provided to demonstrate that the proposed ICC-based algorithms are effective for solving imbalanced LSGO problems. Finally, the efficiency of these algorithms is benchmarked on the complex imbalanced LSGO problems. Simulation results confirm that ICC obtains a better performance overall.


Large-scale global optimization (LSGO) Cooperative coevolution (CC) Sensitivity analysis (SA) Incremental problem solving Problem decomposition 


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. Arora J (2004) Introduction to optimum design. Academic Press, LondonGoogle Scholar
  2. Auger A, Hansen N (2005) A restart CMA evolution strategy with increasing population size. In: 2005 IEEE congress on evolutionary computation, vol. 2, pp. 1769–1776. IEEEGoogle Scholar
  3. Campolongo F, Cariboni J, Saltelli A (2007) An effective screening design for sensitivity analysis of large models. Environ Model Softw 22(10):1509–1518CrossRefGoogle Scholar
  4. Chen W, Weise T, Yang Z, Tang K (2010) Large-scale global optimization using cooperative coevolution with variable interaction learning. In: Parallel problem solving from nature, PPSN XI. Springer, Berlin, pp 300–309Google Scholar
  5. Doerr B, Sudholt D, Witt C (2013) When do evolutionary algorithms optimize separable functions in parallel? In: Proceedings of the twelfth workshop on foundations of genetic algorithms XII. ACM, pp 51–64Google Scholar
  6. Ekstrom PA (2005) Eikos: a simulation toolbox for sensitivity analysis in matlab. Uppsala University, UppsalaGoogle Scholar
  7. García S, Herrera F. The software for conducting multiple comparison involving all possible pairwise comparisons.
  8. García S, Fernández A, Luengo J, Herrera F (2009a) A study of statistical techniques and performance measures for genetics-based machine learning: accuracy and interpretability. Soft Comput 13(10):959–977CrossRefGoogle Scholar
  9. García S, Molina D, Lozano M, Herrera F (2009b) A study on the use of non-parametric tests for analyzing the evolutionary algorithms behaviour: a case study on the cec2005 special session on real parameter optimization. J Heuristics 15(6):617–644CrossRefzbMATHGoogle Scholar
  10. Hansen N (2016) The CMA evolution strategy: a tutorial. arXiv:1604.00772
  11. Li X, Yao X (2012) Cooperatively coevolving particle swarms for large scale optimization. IEEE Trans Evol Comput 16(2):210–224CrossRefGoogle Scholar
  12. Li X, Tang K, Omidvar MN, Yang Z, Qin K (2013) Benchmark functions for the cec2013 special session and competition on large-scale global optimization. Gene 7:33CrossRefGoogle Scholar
  13. Liu J, Tang K (2013) Scaling up covariance matrix adaptation evolution strategy using cooperative coevolution. In: Intelligent Data Engineering and Automated Learning–IDEAL 2013. Springer, Berlin, pp 350–357Google Scholar
  14. Liu Y, Yao X, Zhao Q, Higuchi T (2001) Scaling up fast evolutionary programming with cooperative coevolution. In: Evolutionary computation, 2001. Proceedings of the 2001 congress on IEEE, vol 2, pp 1101–1108Google Scholar
  15. Luengo J, García S, Herrera F (2009) A study on the use of statistical tests for experimentation with neural networks: analysis of parametric test conditions and non-parametric tests. Exp Syst Appl 36(4):7798–7808CrossRefGoogle Scholar
  16. Mahdavi Sedigheh, Shiri Mohammad Ebrahim, Rahnamayan Shahryar (2014). Cooperative co-evolution with a new decomposition method for large-scale optimization. In: Evolutionary computation (CEC), 2014 IEEE congress on IEEE, pp 1285–1292Google Scholar
  17. Mahdavi S, Shiri ME, Rahnamayan S (2015) Metaheuristics in large-scale global continues optimization: a survey. Inform Sci 295:407–428MathSciNetCrossRefGoogle Scholar
  18. Mei Y, Omidvar MN, Li X, Yao X (2016) A competitive divide-and-conquer algorithm for unconstrained large-scale black-box optimization. ACM Trans Math Softw (TOMS) 42(2):13MathSciNetCrossRefGoogle Scholar
  19. Miller BL, Goldberg DE (1995) Genetic algorithms, tournament selection, and the effects of noise. Complex Syst 9(3):193–212MathSciNetGoogle Scholar
  20. Molina D, Lozano M, Herrera F (2010) Ma-sw-chains: Memetic algorithm based on local search chains for large scale continuous global optimization. In: Evolutionary Computation (CEC), 2010 IEEE Congress on IEEE, pp 1–8Google Scholar
  21. Morris MD (1991) Factorial sampling plans for preliminary computational experiments. Technometrics 33(2):161–174CrossRefGoogle Scholar
  22. Omidvar MN, Li X (2010) A comparative study of CMA-ES on large scale global optimisation. In: Australasian joint conference on artificial intelligence. Springer, BerlinGoogle Scholar
  23. Omidvar MN, Li X (2011) A comparative study of CMA-ES on large scale global optimisation. In: AI 2010: advances in artificial intelligence. Springer, Berlin, pp 303–312Google Scholar
  24. Omidvar MN, Li X, Yang Z, Yao X (2010a) Cooperative co-evolution for large scale optimization through more frequent random grouping. In: Evolutionary computation (CEC), 2010 IEEE Congress on IEEE, pp 1–8Google Scholar
  25. Omidvar MN, Li X, Yao X (2010b) Cooperative co-evolution with delta grouping for large scale non-separable function optimization. In: 2010 IEEE Congress on Evolutionary Computation (CEC), pp 1–8Google Scholar
  26. Omidvar MN, Li X, Yao X (2011) Smart use of computational resources based on contribution for cooperative co-evolutionary algorithms. In: Proceedings of the 13th annual conference on genetic and evolutionary computation, ACM, pp 1115–1122Google Scholar
  27. Omidvar MN, Li X, Mei Y, Yao X (2014a) Cooperative co-evolution with differential grouping for large scale optimization. IEEE Trans Evol Comput 18(3):378–393CrossRefGoogle Scholar
  28. Omidvar MN, Mei Y, Li X (2014b) Effective decomposition of large-scale separable continuous functions for cooperative co-evolutionary algorithms. In: 2014 IEEE congress on evolutionary computation (CEC), pp 1305 – 1312. IEEEGoogle Scholar
  29. Potter MA (1997) The design and analysis of a computational model of cooperative coevolution. PhD thesis, CiteseerGoogle Scholar
  30. Potter MA, De Jong KA (1994) A cooperative coevolutionary approach to function optimization. In: Parallel Problem Solving from NaturePPSN III. Springer, Berlin, pp 249–257Google Scholar
  31. Rabitz H, Aliş ÖF (1999) General foundations of high-dimensional model representations. J Math Chem 25(2–3):197–233MathSciNetCrossRefzbMATHGoogle Scholar
  32. Rao SS, Rao SS (2009) Engineering optimization: theory and practice. Wiley, New YorkCrossRefGoogle Scholar
  33. Ray T, Yao X (2009) A cooperative coevolutionary algorithm with correlation based adaptive variable partitioning. In: IEEE congress on evolutionary computation, 2009. CEC’09, pp 983–989. IEEEGoogle Scholar
  34. Saltelli A, Chan K, Scott EM et al (2000) Sensitivity analysis, vol 134. Wiley, New YorkzbMATHGoogle Scholar
  35. Saltelli A, Ratto M, Andres T, Campolongo F, Cariboni J, Gatelli D, Saisana M, Tarantola S (2008) Global sensitivity analysis: the primer. Wiley, New YorkzbMATHGoogle Scholar
  36. Sayed E, Essam D, Sarker R (2012a) Dependency identification technique for large scale optimization problems. In: 2012 IEEE Congress on Evolutionary computation (CEC), pp 1–8. IEEEGoogle Scholar
  37. Sayed E, Essam D, Sarker R (2012b) Using hybrid dependency identification with a memetic algorithm for large scale optimization problems. In: Simulated evolution and learning. Springer, Berlin, pp 168–177Google Scholar
  38. Shan S, Wang GG (2010) Metamodeling for high dimensional simulation-based design problems. J Mech Des 132(5):051009CrossRefGoogle Scholar
  39. Shi Y, Teng H, Li Z (2005) Cooperative co-evolutionary differential evolution for function optimization. In: Proceedings of the first international conference on advances in natural computation. Springer, Berlin, vol Part II, pp 1080–1088Google Scholar
  40. Singh HK, Ray T (2010). Divide and conquer in coevolution: a difficult balancing act. In Agent-based evolutionary search. Springer, Berlin, pp 117–138Google Scholar
  41. Sun L, Yoshida S, Cheng X, Liang Y (2012) A cooperative particle swarm optimizer with statistical variable interdependence learning. Inform Sci 186(1):20–39MathSciNetCrossRefGoogle Scholar
  42. Tang K, Li X, Suganthan PN, Yang Z, Weise T (2010) Benchmark functions for the CEC’2010 special session and competition on large-scale global optimization. Technical report, Nature inspired computation and applications laboratory (NICAL), USTC, China.
  43. Van den Bergh F, Engelbrecht AP (2004) A cooperative approach to particle swarm optimization. IEEE Trans Evol Comput 8(3):225–239CrossRefGoogle Scholar
  44. Wang H, Rahnamayan S, Wu Z (2013a) Parallel differential evolution with self-adapting control parameters and generalized opposition-based learning for solving high-dimensional optimization problems. J Parallel Distrib Comput 73(1):62–73CrossRefGoogle Scholar
  45. Wang Y, Huang J, Dong WS, Yan JC, Tian CH, Li M, Mo WT (2013b) Two-stage based ensemble optimization framework for large-scale global optimization. Eur J Oper Res 228(2):308–320MathSciNetCrossRefzbMATHGoogle Scholar
  46. Weicker K, Weicker N (1999) On the improvement of coevolutionary optimizers by learning variable interdependencies. In: Evolutionary computation, 1999. CEC 99. Proceedings of the 1999 congress on IEEE, vol 3Google Scholar
  47. Yang Z, Tang K, Yao X (2008a) Large scale evolutionary optimization using cooperative coevolution. Inform Sci 178(15):2985–2999MathSciNetCrossRefzbMATHGoogle Scholar
  48. Yang Z, Tang K, Yao X (2008b) Multilevel cooperative coevolution for large scale optimization. In: Evolutionary computation, 2008. CEC 2008. (IEEE World Congress on computational intelligence). IEEE congress on IEEE, pp 1663–1670Google Scholar
  49. Yang Zhenyu, Tang Ke, Yao Xin (2008c) Self-adaptive differential evolution with neighborhood search. In: Evolutionary Computation, 2008. CEC 2008. (IEEE World Congress on Computational Intelligence). IEEE Congress on IEEE, pp 1110–1116Google Scholar
  50. Zhao SZ, Suganthan PN, Das S (2011) Self-adaptive differential evolution with multi-trajectory search for large-scale optimization. Soft Comput 15(11):2175–2185CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Sedigheh Mahdavi
    • 1
  • Shahryar Rahnamayan
    • 1
  • Mohammad Ebrahim Shiri
    • 2
  1. 1.Department of Electrical, Computer, and Software EngineeringUniversity of Ontario Institute of Technology (UOIT)OshawaCanada
  2. 2.Department of Mathematics and Computer ScienceAmirkabir University of TechnologyTehranIran

Personalised recommendations