Abstract
In this paper, we focus on the all together model, which is one of the support vector machine (SVM) using a piece-wise linear function for multiclass classification. We already proposed a multiobjective hard-margin SVM model as a new all together model for piecewise linearly separable data, which maximizes all of the geometric margins simultaneously for the generalization ability. In addition, we derived a single-objective convex problem and showed that a Pareto optimal solution for the proposed multiobjective SVM is obtained by solving single-objective problems. However, in the real-world classification problem the data are often piecewise linearly inseparable. Therefore, in this paper we extend the hard-margin SVM for the data by using penalty functions for the margin slack variables between outliers and the corresponding discriminant hyperplane. Those functions are incorporated into the objective functions. Moreover, we derive a single-objective second-order cone programming (SOCP) problem based on Benson’s method and some techniques, and show that a Pareto optimal solution for the proposed soft-margin SVM is obtained by solving the SOCP iteratively. Furthermore through numerical experiments we verify that the proposed iterative method maximizes the geometric margins and constructs a classifier with a high generalization ability.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alizadeh, F., Goldfarb, D.: Second-order cone programming. Mathematical Programming, Ser. B 95, 3–51 (2003)
Andersen, E.D., Roos, C., Terlaky, T.: On implementing a primal-dual interior-point method for conic quadratic optimization. Mathematical Programming, Ser. B 95, 249–277 (2003)
Bottou, L., Cortes, C., Denker, J., Drucker, H., Guyon, I., Jackel, L., LeCun, Y., Muller, U., Sackinger, E., Simard, P., Vapnik, V.: Comparison of classifier methods: A case study in handwriting digit recognition. In: Proc. Int. Conf. Pattern Recognition, pp. 77–87 (1994)
Bredensteiner, E.J., Bennett, K.P.: Multicategory classification by support vector machines. Computational Optimization and Applications 12, 53–79 (1999)
Ehrgott, M.: Multicriteria optimization. Springer, Berlin (2005)
Guermeur, Y.: Combining discriminant models with new multi-class SVMs. Neuro COLT2 Technical Report Series (2000)
Hsh, C.W., Lin, C.J.: A comparison of methods for multiclass support vector machines. IEEE Transactions on Neural Networks 13(2), 181–201 (2002)
Kressel, U.: Pairwise classification and support vector machines. In: Schölkopf, B., Burges, C., Smola, A.J. (eds.) Advances in kernel methods – Support vector learning, pp. 255–268. MIT Press, Cambridge (1999)
Mittelmann, H.D.: An independent benchmarking of SDP and SOCP solvers. Mathematical Programming, Ser. B 95, 407–430 (2003)
Tatsumi, K., Hayashida, K., Higashi, H., Tanino, T.: Multi-objective multiclass support vector machine for pattern recognition. In: Proceedings of SICE Annual Conference 2007, pp. 1095–1098 (2007)
Tatsumi, K., Tanino, T., Hayashida, K.: Multiobjective Multiclass Support Vector Machines Maximizing Geometric Margins. Pacific Journal of Optimization (to appear)
Vapnik, Y.: Statistical learning theory. A Wiley-Interscience Publication, Hoboken (1998)
Weston, J., Watkins, C.: Multi-class support vector machines. Technical report CSD-TR-98-04, Univ. London, Royal Holloway (1998)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Tatsumi, K., Kawachi, R., Hayashida, K., Tanino, T. (2009). Multiobjective Multiclass Soft-Margin Support Vector Machine and Its Solving Technique Based on Benson’s Method. In: Torra, V., Narukawa, Y., Inuiguchi, M. (eds) Modeling Decisions for Artificial Intelligence. MDAI 2009. Lecture Notes in Computer Science(), vol 5861. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04820-3_33
Download citation
DOI: https://doi.org/10.1007/978-3-642-04820-3_33
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-04819-7
Online ISBN: 978-3-642-04820-3
eBook Packages: Computer ScienceComputer Science (R0)