Skip to main content
SpringerLink
Log in

Estimating Approximation Errors of Elitist Evolutionary Algorithms

  • Conference paper
  • First Online:
Bio-inspired Computing: Theories and Applications (BIC-TA 2019)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 1159))

  • 935 Accesses

Abstract

When evolutionary algorithms (EAs) are unlikely to locate precise global optimal solutions with satisfactory performances, it is important to substitute alternative theoretical routine for the analysis of hitting time/running time. In order to narrow the gap between theories and applications, this paper is dedicated to perform an analysis on approximation error of EAs. First, we proposed a general result on upper bound and lower bound of approximation errors. Then, several case studies are performed to present the routine of error analysis, and theoretical results show the close connections between approximation errors and eigenvalues of transition matrices. The analysis validates applicability of error analysis, demonstrates significance of estimation results, and then, exhibits its potential to be applied for theoretical analysis of elitist EAs.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    In linear algebra, a nilpotent matrix is a square matrix M such that \(N^{k}=0\) for some positive integer k. The smallest such k is called the index of M [26].

References

  1. Oliveto, P., He, J., Yao, X.: Time complexity of evolutionary algorithms for combinatorial optimization: a decade of results. Int. J. Autom. Comput. 4(3), 281–293 (2007). https://doi.org/10.1007/s11633-007-0281-3

    Article  Google Scholar 

  2. He, J., Yao, X.: Towards an analytic framework for analysing the computation time of evolutionary algorithms. Artif. Intell. 145(1–2), 59–97 (2003)

    Article  MathSciNet  Google Scholar 

  3. Ding, L., Yu, J.: Some techniques for analyzing time complexity of evolutionary algorithms. Trans. Inst. Meas. Control. 34(6), 755–766 (2012)

    Article  Google Scholar 

  4. He, J., Yao, X.: Drift analysis and average time complexity of evolutionary algorithms. Artif. Intell. 127(1), 57–85 (2001)

    Article  MathSciNet  Google Scholar 

  5. Doerr, B., Johannsen, D., Winzen, C.: Multiplicative drift analysis. Algorithmica 64(4), 673–697 (2012). https://doi.org/10.1007/s00453-012-9622-x

    Article  MathSciNet  MATH  Google Scholar 

  6. Yu, Y., Qian, C., Zhou, Z.H.: Switch analysis for running time analysis of evolutionary algorithms. IEEE Trans. Evol. Comput. 19(6), 777–792 (2014)

    Article  Google Scholar 

  7. Droste, S., Jansen, T., Wegener, I.: On the analysis of the (1+1) evolutionary algorithm. Theor. Comput. Sci. 276(1–2), 51–81 (2002)

    Article  MathSciNet  Google Scholar 

  8. Chen, Y., Zou, X., He, J.: Drift conditions for estimating the first hitting times of evolutionary algorithms. Int. J. Comput. Math. 88(1), 37–50 (2011)

    Article  MathSciNet  Google Scholar 

  9. Huang, H., Xu, W., Zhang, Y., Lin, Z., Hao, Z.: Runtime analysis for continuous (1+1) evolutionary algorithm based on average gain model. Scientia Sinica Informationis 44(6), 811–824 (2014)

    Article  Google Scholar 

  10. Yushan, Z., Han, H., Zhifeng, H., Guiwu, H.: First hitting time analysis of continuous evolutionary algorithms based on average gain. Clust. Comput. 19(3), 1323–1332 (2016). https://doi.org/10.1007/s10586-016-0587-4

    Article  Google Scholar 

  11. Akimoto, Y., Auger, A., Glasmachers, T.: Drift theory in continuous search spaces: expected hitting time of the (1+1)-ES with 1/5 success rule. In: Proceedings of the Genetic and Evolutionary Computation Conference, pp. 801–808. ACM (2018)

    Google Scholar 

  12. Yu, Y., Yao, X., Zhou, Z.H.: On the approximation ability of evolutionary optimization with application to minimum set cover. Artif. Intell. 180–181, 20–33 (2012)

    Article  MathSciNet  Google Scholar 

  13. Lai, X., Zhou, Y., He, J., Zhang, J.: Performance analysis of evolutionary algorithms for the minimum label spanning tree problem. IEEE Trans. Evol. Comput. 18(6), 860–872 (2014)

    Article  Google Scholar 

  14. Zhou, Y., Lai, X., Li, K.: Approximation and parameterized runtime analysis of evolutionary algorithms for the maximum cut problem. IEEE Trans. Cybern. 45(8), 1491–1498 (2015)

    Article  Google Scholar 

  15. Zhou, Y., Zhang, J., Wang, Y.: Performance analysis of the (1+1) evolutionary algorithm for the multiprocessor scheduling problem. Algorithmica 73(1), 21–41 (2015). https://doi.org/10.1007/s00453-014-9898-0

    Article  MathSciNet  MATH  Google Scholar 

  16. Xia, X., Zhou, Y., Lai, X.: On the analysis of the (1+1) evolutionary algorithm for the maximum leaf spanning tree problem. Int. J. Comput. Math. 92(10), 2023–2035 (2015)

    Article  MathSciNet  Google Scholar 

  17. Peng, X., Zhou, Y., Xu, G.: Approximation performance of ant colony optimization for the TSP (1, 2) problem. Int. J. Comput. Math. 93(10), 1683–1694 (2016)

    Article  MathSciNet  Google Scholar 

  18. Rudolph, G.: Convergence rates of evolutionary algorithms for a class of convex objective functions. Control. Cybern. 26, 375–390 (1997)

    MathSciNet  MATH  Google Scholar 

  19. He, J., Lin, G.: Average convergence rate of evolutionary algorithms. IEEE Trans. Evol. Comput. 20(2), 316–321 (2016)

    Article  Google Scholar 

  20. Jansen, T., Zarges, C.: Fixed budget computations: a different perspective on run time analysis. In: Proceedings of the 14th Annual Conference on Genetic and Evolutionary Computation, pp. 1325–1332. ACM (2012)

    Google Scholar 

  21. Jansen, T., Zarges, C.: Performance analysis of randomised search heuristics operating with a fixed budget. Theor. Comput. Sci. 545, 39–58 (2014)

    Article  MathSciNet  Google Scholar 

  22. He, J.: An analytic expression of relative approximation error for a class of evolutionary algorithms. In: Proceedings of 2016 IEEE Congress on Evolutionary Computation (CEC 2016), pp. 4366–4373, July 2016

    Google Scholar 

  23. He, J., Jansen, T., Zarges, C.: Unlimited budget analysis. In: Proceedings of the Genetic and Evolutionary Computation Conference Companion, pp. 427–428. ACM (2019)

    Google Scholar 

  24. He, J., Chen, Y., Zhou, Y.: A theoretical framework of approximation error analysis of evolutionary algorithms. arXiv preprint arXiv:1810.11532 (2018)

  25. Aigner, M.: Combinatorial Theory. Springer, New York (2012)

    MATH  Google Scholar 

  26. Herstein, I.N.: Topics in Algebra. Wiley, Hoboken (2006)

    MATH  Google Scholar 

  27. Lay, D.C.: Linear Algebra and Its Applications. Pearson Education, London (2003)

    Google Scholar 

Download references

Acknowledgements

This work was supported in part by the National Nature Science Foundation of China under Grants 61303028 and 61763010, in part by the Guangxi “BAGUI Scholar” Program, and in part by the Science and Technology Major Project of Guangxi under Grant AA18118047.

Author information

Authors and Affiliations

  1. School of Science, Wuhan University of Technology, Wuhan, 430070, China

    Cong Wang & Yu Chen

  2. School of Science and Technology, Nottingham Trent University, Nottingham, NG11 8NS, UK

    Jun He

  3. School of Computer and Information Engineering, Nanning Normal University, Nanning, 530299, China

    Chengwang Xie

Authors
  1. Cong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jun He
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Chengwang Xie
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Yu Chen or Chengwang Xie .

Editor information

Editors and Affiliations

  1. Huazhong University of Science and Technology, Wuhan, China

    Linqiang Pan

  2. Zhengzhou University, Zhengzhou, China

    Jing Liang

  3. Zhongyuan University of Technology, Zhengzhou, China

    Boyang Qu

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Singapore Pte Ltd.

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Wang, C., Chen, Y., He, J., Xie, C. (2020). Estimating Approximation Errors of Elitist Evolutionary Algorithms. In: Pan, L., Liang, J., Qu, B. (eds) Bio-inspired Computing: Theories and Applications. BIC-TA 2019. Communications in Computer and Information Science, vol 1159. Springer, Singapore. https://doi.org/10.1007/978-981-15-3425-6_26

Download citation

  • DOI: https://doi.org/10.1007/978-981-15-3425-6_26

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-15-3424-9

  • Online ISBN: 978-981-15-3425-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation