Skip to main content
SpringerLink
Log in

A Re-initialization Clustering-Based Adaptive Differential Evolution for Nonlinear Equations Systems

  • 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))

  • 938 Accesses

Abstract

Solving nonlinear equations systems is one of the most challenges for evolutionary algorithms, especially to locate multiple roots in a single run. In this paper, a new approach which combines speciation clustering with dynamic cluster sizing, adaptive parameter control and re-initiation mechanism is proposed to deal with this optimization problem. The major advantages are as follows: (1) the speciation clustering with dynamic cluster sizing can alleviate the trivial task to set proper cluster size; (2) to improve the search ability in each species and avoid the trivial work of parameter setting, adaptive parameter control is employed; and (3) re-initialization mechanism motivates the search algorithm to find new roots by increasing population diversity. To verify the performance of our approach, 30 nonlinear equations systems are selected as the test suite. Experiment results indicate that the speciation clustering with dynamic cluster size, adaptive parameter control, and re-initialization mechanism can work effectively in a synergistic manner and locate multiple roots in a single run. Moreover, comparison of other state-of-the-art methods, the proposed method is capable of obtaining better results in terms of peak ratio and success rate.

W. Gong—This work was partly supported by the National Natural Science Foundation of China under Grant Nos. 61573324, and 61673354, in part by the National Natural Science Fund for Distinguished Young Scholars of China under Grant 61525304, and the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) under Grant No. CUG160603.

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.

    A successful run is considered as a run where all known optima of a NES are found.

References

  1. Kastner, M.: Phase transitions and configuration space topology. Rev. Mod. Phys. 80, 167–187 (2008)

    Article  MathSciNet  Google Scholar 

  2. Guo, D., Nie, Z., Yan, L.: The application of noise-tolerant ZD design formula to robots’ kinematic control via time-varying nonlinear equations solving. IEEE Trans. Syst. Man Cybern. Syst. 48, 2188–2197 (2017)

    Article  Google Scholar 

  3. Chiang, H.D., Wang, T.: Novel homotopy theory for nonlinear networks and systems and its applications to electrical grids. IEEE Trans. Control Netw. Syst. 5, 1051–1060 (2017)

    Article  MathSciNet  Google Scholar 

  4. Facchinei, F., Kanzow, C.: Generalized Nash equilibrium problems. 4OR 5(3), 173–210 (2007)

    Article  MathSciNet  Google Scholar 

  5. Sun, Z., Wu, J., Pei, J., Li, Z., Huang, Y., Yang, J.: Inclined geosynchronous spacebornec-airborne bistatic SAR: performance analysis and mission design. IEEE Trans. Geosci. Remote Sens. 54(1), 343–357 (2016)

    Article  Google Scholar 

  6. Das, S., Suganthan, P.N.: Differential evolution: a survey of the state-of-the-art. IEEE Trans. on Evol. Comput. 15(1), 4–31 (2011)

    Article  Google Scholar 

  7. Ramadas, G.C.V., Fernandes, E.M.G.P., Rocha, A.M.A.C.: Multiple roots of systems of equations by repulsion merit functions. In: Murgante, B., et al. (eds.) ICCSA 2014. LNCS, vol. 8580, pp. 126–139. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-09129-7_10

    Chapter  Google Scholar 

  8. Pourjafari, E., Mojallali, H.: Solving nonlinear equations systems with a new approach based on invasive weed optimization algorithm and clustering. Swarm Evol. Comput. 4, 33–43 (2012)

    Article  Google Scholar 

  9. Tsoulos, I.G., Stavrakoudis, A.: On locating all roots of systems of nonlinear equations inside bounded domain using global optimization methods. Nonlinear Anal. Real World Appl. 11(4), 2465–2471 (2010)

    Article  MathSciNet  Google Scholar 

  10. Sacco, W.F., Henderson, N.: Finding all solutions of nonlinear systems using a hybrid metaheuristic with fuzzy clustering means. Appl. Soft Comput. 11(8), 5424–5432 (2011)

    Article  Google Scholar 

  11. Karafotias, G., Hoogendoorn, M., Eiben, A.E.: Parameter control in evolutionary algorithms: trends and challenges. IEEE Trans. Evol. Comput. 19(2), 167–187 (2015)

    Article  Google Scholar 

  12. Brest, J., Greiner, S., Boskovic, B., Mernik, M., Zumer, V.: Self-adapting control parameters in differential evolution: a comparative study on numerical benchmark problems. IEEE Trans. Evol. Comput. 10(6), 646–657 (2006)

    Article  Google Scholar 

  13. Qin, A.K., Huang, V.L., Suganthan, P.N.: Differential evolution algorithm with strategy adaptation for global numerical optimization. IEEE Trans. Evol. Comput. 13(2), 398–417 (2009)

    Article  Google Scholar 

  14. Zhang, J., Sanderson, A.C.: JADE: adaptive differential evolution with optional external archive. IEEE Trans. Evol. Comput. 13(5), 945–958 (2009)

    Article  Google Scholar 

  15. Tanabe, R., Fukunaga, A.: Evaluating the performance of SHADE on CEC 2013 benchmark problems. In: IEEE Congress on Evolutionary Computation (CEC), pp. 1952–1959 (2013)

    Google Scholar 

  16. Mallipeddi, R., Suganthan, P.N., Pan, Q.K., Tasgetiren, M.F.: Differential evolution algorithm with ensemble of parameters and mutation strategies. Appl. Soft Comput. 11(2), 1679–1696 (2011)

    Article  Google Scholar 

  17. Jain, A.K., Murty, M.N., Flynn, P.J.: Data clustering: a review. ACM Comput. Surv. 31(3), 264–323 (1999)

    Article  Google Scholar 

  18. Freitas, L., Platt, G., Henderson, N.: Novel approach for the calculation of critical points in binary mixtures using global optimization. Fluid Phase Equilib. 225, 29–37 (2004)

    Article  Google Scholar 

  19. Henderson, N., Sacco, W.F., Platt, G.M.: Finding more than one root of nonlinear equations via a polarization technique: an application to double retrograde vaporization. Chem. Eng. Res. Des. 88(5–6), 551–561 (2010)

    Article  Google Scholar 

  20. Hirsch, M.J., Pardalos, P.M., Resende, M.G.C.: Solving systems of nonlinear equations with continuous grasp. Nonlinear Anal. Real World Appl. 10(4), 2000–2006 (2009)

    Article  MathSciNet  Google Scholar 

  21. Silva, R.M.A., Resende, M.G.C., Pardalos, P.M.: Finding multiple roots of a box-constrained system of nonlinear equations with a biased random-key genetic algorithm. J. Global Optim. 60(2), 289–306 (2013). https://doi.org/10.1007/s10898-013-0105-7

    Article  MathSciNet  MATH  Google Scholar 

  22. Gong, W., Wang, Y., Cai, Z., Wang, L.: Finding multiple roots of nonlinear equation systems via a repulsion-based adaptive differential evolution. IEEE Trans. Syst. Man Cybern. Syst. PP(99), 1–15 (2018)

    Google Scholar 

  23. Liao, Z., Gong, W., Yan, X., Wang, L., Hu, C.: Solving nonlinear equations system with dynamic repulsion-based evolutionary algorithms. IEEE Trans. Syst. Man Cybern. Syst. 1–12 (2018)

    Google Scholar 

  24. Grosan, C., Abraham, A.: A new approach for solving nonlinear equations systems. IEEE Trans. Syst. Man Cybern. Part A Syst. Hum. 38(3), 698–714 (2008)

    Article  Google Scholar 

  25. Song, W., Wang, Y., Li, H.-X., Cai, Z.: Locating multiple optimal solutions of nonlinear equation systems based on multiobjective optimization. IEEE Trans. Evol. Comput. 19(3), 414–431 (2015)

    Article  Google Scholar 

  26. Gong, W., Wang, Y., Cai, Z., Yang, S.: A weighted biobjective transformation technique for locating multiple optimal solutions of nonlinear equation systems. IEEE Trans. Evol. Comput. 21(5), 697–713 (2017)

    Article  Google Scholar 

  27. Naidu, Y.R., Ojha, A.K.: Solving multiobjective optimization problems using hybrid cooperative invasive weed optimization with multiple populations. IEEE Trans. Syst. Man Cybern. Syst. 48, 821–832 (2016)

    Article  Google Scholar 

  28. Gao, W., Yen, G.G., Liu, S.: A cluster-based differential evolution with self-adaptive strategy for multimodal optimization. IEEE Trans. Cybern. 44(8), 1314–1327 (2014)

    Article  Google Scholar 

  29. Yang, Q., Chen, W.N., Li, Y., Chen, C.L., Xu, X.M., Zhang, J.: Multimodal estimation of distribution algorithms. IEEE Trans. Cybern. 47(3), 636–650 (2017)

    Article  Google Scholar 

  30. Qu, B.Y., Suganthan, P.N., Liang, J.J.: Differential evolution with neighborhood mutation for multimodal optimization. IEEE Trans. Evol. Comput. 16(5), 601–614 (2012)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Computer Science, China University of Geosciences, Wuhan, 430074, China

    Zuowen Liao, Wenyin Gong & Zhihua Cai

Authors
  1. Zuowen Liao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wenyin Gong
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhihua Cai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wenyin Gong .

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

Liao, Z., Gong, W., Cai, Z. (2020). A Re-initialization Clustering-Based Adaptive Differential Evolution for Nonlinear Equations Systems. 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_33

Download citation

  • DOI: https://doi.org/10.1007/978-981-15-3425-6_33

  • 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