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Journal of Global Optimization

, Volume 48, Issue 4, pp 657–669 | Cite as

DC models for spherical separation

  • A. Astorino
  • A. Fuduli
  • M. GaudiosoEmail author
Article

Abstract

We propose two different approaches for spherical separation of two sets. Both methods are based on minimizing appropriate nonconvex nondifferentiable error functions, which can be both expressed in a DC (Difference of two Convex) form. We tackle the problem by adopting the DC-Algorithm. Some numerical results on classical binary datasets are reported.

Keywords

Spherical separation DC functions DCA 

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References

  1. 1.
    An L.T.H., Tao P.D.: The DC (Difference of Convex Functions) programming and DCA revisited with DC models of real world nonconvex optimization problems. Ann. Oper. Res. 133, 23–46 (2005)CrossRefGoogle Scholar
  2. 2.
    Astorino A., Gaudioso M.: Polyhedral separability through successive LP. J. Optim. Theory Appl. 112, 265–293 (2002)CrossRefGoogle Scholar
  3. 3.
    Astorino A., Gaudioso M.: Ellipsoidal separation for classification problems. Optim. Methods Softw. 20, 261–270 (2005)CrossRefGoogle Scholar
  4. 4.
    Astorino A., Gaudioso M.: A fixed-center spherical separation algorithm with kernel transformations for classification problems. Computat. Manage. Sci. 6, 357–372 (2009)CrossRefGoogle Scholar
  5. 5.
    Chapelle, O., Zien, A.: Semi-supervised classification by low density separation. In: Proceedings of the tenth international workshop on artificial intelligence and statistics, pp. 57–64. (2005)Google Scholar
  6. 6.
    Cristianini N., Shawe-Taylor J.: An introduction to support vector machines and other kernel-based learning methods. Cambridge University Press, Cambridge (2000)Google Scholar
  7. 7.
    Fuduli A., Gaudioso M., Giallombardo G.: Minimizing nonconvex nonsmooth functions via cutting planes and proximity control. SIAM J. Optim. 14, 743–756 (2004)CrossRefGoogle Scholar
  8. 8.
    Fung, G., Mangasarian, O.L.: Proximal support vector machine classifiers. in Proceedings KDD-2001, pp. 77–86. San Francisco (2001)Google Scholar
  9. 9.
    Mangasarian O.: Linear and nonlinear separation of patterns by linear programming. Oper. Res. 13, 444–452 (1965)CrossRefGoogle Scholar
  10. 10.
    Mangasarian O.L., Musicant D.R.: Lagrangian support vector machines. J. Mach. Learn. Res. 1, 161–177 (2001)CrossRefGoogle Scholar
  11. 11.
    Murphy, P.M., Aha, D.W.: UCI repository of machine learning databases. In http://www.ics.uci.edu/~mlearn/MLRepository.html (1992)
  12. 12.
    Rosen J.B.: Pattern separation by convex programming. J. Math. Anal. Appl. 10, 123–134 (1965)CrossRefGoogle Scholar
  13. 13.
    Odewahn S., Stockwell E., Pennington R., Humphreys R., Zumach W.: Automated star/galaxy discrimination with neural networks. Astron. J. 103, 318–331 (1992)CrossRefGoogle Scholar
  14. 14.
    Schölkopf B., Smola A.J.: Learning with kernels. MIT Press, Cambridge (2002)Google Scholar
  15. 15.
    Schölkopf B., Burges C.J.C., Smola A.J.: Advances in kernel methods. Support vector learning. MIT Press, Cambridge (1999)Google Scholar
  16. 16.
    Shawe-Taylor J., Cristianini N.: Kernel methods for pattern analysis. Cambridge University Press, Cambridge (2004)Google Scholar
  17. 17.
    Tao P.D., An L.T.H.: A D.C. optimization algorithm for solving the trust-region subproblem. SIAM J. Con. Opt. 8, 476–505 (1998)CrossRefGoogle Scholar
  18. 18.
    Tax, D.M.J., Duin, R.P.W.: Data domain description using support vectors. In: ESANN’1999 proceedings Bruges, pp. 251–256. Belgium (1999)Google Scholar
  19. 19.
    Tax D.M.J., Duin R.P.W.: Uniform object generation for optimizing one-class classifiers. J. Mach. Learn. Res. 2, 155–173 (2001)CrossRefGoogle Scholar
  20. 20.
    Vapnik V.: The nature of the statistical learning theory. Springer, New York (1995)Google Scholar

Copyright information

© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

  1. 1.Istituto di Calcolo e Reti ad Alte Prestazioni C.N.R., c/o Dipartimento di Elettronica Informatica e SistemisticaUniversità della CalabriaRende (CS)Italy
  2. 2.Dipartimento di MatematicaUniversità della CalabriaRende (CS)Italy
  3. 3.Dipartimento di Elettronica Informatica e SistemisticaUniversità della CalabriaRende (CS)Italy

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