Quadratic Kernel-Free Least Square Twin Support Vector Machine for Binary Classification Problems

  • Qian-Qian Gao
  • Yan-Qin BaiEmail author
  • Ya-Ru Zhan


In this paper, a new quadratic kernel-free least square twin support vector machine (QLSTSVM) is proposed for binary classification problems. The advantage of QLSTSVM is that there is no need to select the kernel function and related parameters for nonlinear classification problems. After using consensus technique, we adopt alternating direction method of multipliers to solve the reformulated consensus QLSTSVM directly. To reduce CPU time, the Karush-Kuhn-Tucker (KKT) conditions is also used to solve the QLSTSVM. The performance of QLSTSVM is tested on two artificial datasets and several University of California Irvine (UCI) benchmark datasets. Numerical results indicate that the QLSTSVM may outperform several existing methods for solving twin support vector machine with Gaussian kernel in terms of the classification accuracy and operation time.


Twin support vector machine Quadratic kernel-free Least square Binary classification 

Mathematics Subject Classification

68T99 90C20 



We are very grateful to the editor and the anonymous reviewers for their helpful and valuable comments of this paper.


  1. 1.
    Cortes, C., Vapnik, V.: Support-vector networks. Mach. Learn. 20(3), 273–297 (1995)zbMATHGoogle Scholar
  2. 2.
    Deng, N., Tian, Y., Zhang, C.: Support Vector Machines: Optimization Based Theory, Algorithms, and Extensions, pp. 1–28. Chapman and Hall/CRC, London (2012)Google Scholar
  3. 3.
    Mingheng, Z., Yaobao, Z., Ganglong, H., Gang, C.: Accurate multisteps traffic flow prediction based on SVM. Math. Probl. Eng. 2013, 1–8 (2013)CrossRefGoogle Scholar
  4. 4.
    Wei, L., Wei, B., Wang, B.: Text classification using support vector machine with mixture of kernel. J. Softw. Eng. Appl. 05(12), 55–58 (2012)CrossRefGoogle Scholar
  5. 5.
    Tian, Y., Ju, X.: Nonparallel support vector machine based on one optimization problem for pattern recognition. J. Oper. Res. Soc. China 3(4), 499–519 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Narayana, K.V., Manoj, V.V.R., Swathi, K.: Enhanced face recognition based on PCA and SVM. Int. J. Comput. Appl. 117(2), 40–42 (2015)Google Scholar
  7. 7.
    Cao, L., Tay, F.E.H.: Support vector machine with adaptive parameters in financial time series forecasting. IEEE Trans. Neural Netw. 14(6), 1506–1518 (2003)CrossRefGoogle Scholar
  8. 8.
    Suykens, J.A.K., Vandewalle, J.: Least squares support vector machine classifiers. Neural Process. Lett. 9(3), 293–300 (1999)CrossRefGoogle Scholar
  9. 9.
    Mangasarian, O.L., Wild, E.W.: Multisurface proximal support vector machine classification via generalized eigenvalues. IEEE Trans. Patttern Anal. Mach. Intell. 28(1), 69–74 (2006)CrossRefGoogle Scholar
  10. 10.
    Bai, Y., Shen, Y., Shen, K.: Consensus proximal support vector machine for classification problems with sparse solutions. J. Oper. Res. Soc. China 2(1), 57–74 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bai, Y., Zhu, Z., Yan, W.: Sparse proximal support vector machine with a specialized interior-point method. J. Oper. Res. Soc. China 3(1), 1–15 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jayadeva, Khemchandani, R., Chandra, S: Twin support vector machines for pattern classification. IEEE Trans. Pattern Anal. Mach. Intell. 29(5), 905–910 (2007)Google Scholar
  13. 13.
    Arun Kumar, M., Gopal, M.: Least squares twin support vector machines for pattern classification. Exp. Syst. Appl. 36(4), 7535–7543 (2009)CrossRefGoogle Scholar
  14. 14.
    Yang, Z.X., Shao, Y.H., Zhang, X.S.: Multiple birth support vector machine for multi-class classification. Neural Comput. Appl. 22(1), 153–161 (2013)CrossRefGoogle Scholar
  15. 15.
    Chen, S.G., Wu, X.J.: Multiple birth least squares support vector machine for multi-class classification. Int. J. Mach. Learn. Cybern. 8(6), 1731–1742 (2017)CrossRefGoogle Scholar
  16. 16.
    Shao, Y., Wang, Z., Chen, W., Deng, N.: Least squares twin parametric-margin support vector machine for classification. Appl. Intell. 39(3), 451–464 (2013)CrossRefGoogle Scholar
  17. 17.
    Shao, Y., Deng, N.: A novel margin-based twin support vector machine with unity norm hyperplanes. Neural Comput. Appl. 22, 1627–1635 (2013)CrossRefGoogle Scholar
  18. 18.
    Tomar, D., Agarwal, S.: Twin support vector machine: a review from 2007 to 2014. Egypt. Inform. J. 16(1), 55–69 (2015)CrossRefGoogle Scholar
  19. 19.
    Tian, Y., Qi, Z., Ju, X., Shi, Y., Liu, X.: Nonparallel support vector machines for pattern classification. IEEE Trans. Syst. Man. Cybern. 44(7), 1067–1079 (2014)Google Scholar
  20. 20.
    Shao, Y., Zhang, C., Wang, X., Deng, N.: Improvements on twin support vector machines. IEEE Trans. Neural Netw. 22(6), 962–968 (2011)CrossRefGoogle Scholar
  21. 21.
    Tian, Y., Ping, Y.: Large-scale linear nonparallel support vector machine solver. Neural Netw. 50, 166–174 (2014)CrossRefzbMATHGoogle Scholar
  22. 22.
    Dagher, I.: Quadratic kernel-free non-linear support vector machine. J. Glob. Optim. 41(1), 15–30 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Luo, J., Fang, S., Deng, Z., Guo, X.: Soft quadratic surface support vector machine for binary classification. Asia Pac. J. Oper. Res. 33(6), 1–22 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Bai, Y., Han, X., Chen, T., Yu, H.: Quadratic kernel-free least squares support vector machine for target diseases classification. J. Comb. Optim. 30(4), 850–870 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Yan, X., Bai, Y., Fang, S.C., Luo, J.: A kernel-free quadratic surface support vector machine for semi-supervised learning. J. Oper. Res. Soc. 67(7), 1001–1011 (2016)CrossRefGoogle Scholar
  26. 26.
    Zhan, Y., Bai, Y., Zhang, W., Ying, S.: A P-ADMM for sparse quadratic kernel-free least squares semi-supervised support vector machine. Neurocomputing 306, 37–50 (2018)CrossRefGoogle Scholar
  27. 27.
    Boyd, S.P., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 3(1), 1–122 (2011)CrossRefzbMATHGoogle Scholar

Copyright information

© Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsShanghai UniversityShanghaiChina

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