Canonical duality solution for alternating support vector machine

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July 2012, 8(3): 611-621. doi: 10.3934/jimo.2012.8.611

Canonical duality solution for alternating support vector machine

1.	Department of Computer Science and Technology, School of Information Science and Engineering, East China University of Science and Technology, Shanghai 200237, China

Received June 2011 Revised December 2011 Published June 2012

Abstract
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Support vector machine (SVM) is one of the most popular machine learning methods and is educed from a binary data classification problem. In this paper, the canonical duality theory is used to solve the normal model of SVM. Several examples are illustrated to show that the exact solution can be obtained after the canonical duality problem being solved. Moreover, the support vectors can be located by non-zero elements of the canonical dual solution.

Keywords: support vector machine, Global optimization, classification, data mining., duality theory.

Mathematics Subject Classification: Primary: 68T10; Secondary:90C2.

Citation: Yubo Yuan. Canonical duality solution for alternating support vector machine. Journal of Industrial and Management Optimization, 2012, 8 (3) : 611-621. doi: 10.3934/jimo.2012.8.611

References:

[1]	J. O. Berger, "Statistical Decision Theory and Bayesian Analysis," Second edition, Springer Series in Statistics, Springer-Verlag, New York, 1985.
[2]	P. J. Bickel and K. A. Doksum, "Mathematical Statistics. Basic Ideas and Selected Topics," Second edition, Prentice-Hall, Inc., 2001.
[3]	C. J. C. Burges, A tutorial on support vector machines for pattern recognition, Data Mining and Knowledge Discovery, 2 (1998), 121-167.
[4]	O. Chapelle, V. Vapnik, O. Bousquet and S. Mukherjee, Choosing multiple parameters for support vector machines, Machine Learning, 46 (2002), 131-159.
[5]	B. Chen and P. T. Harker, Smooth approximations to nonlinear complementarity problems, SIAM J. Optimization, 7 (1997), 403-420.
[6]	C. Chen and O. L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems, Computational Optimization and Applications, 5 (1996), 97-138.
[7]	C. Chen and O. L. Mangasarian, Smoothing methods for convex inequalities and linear complementarity problems, Math. Programming, 71 (1995), 51-69. doi: 10.1016/0025-5610(95)00005-4.
[8]	X. Chen, L. Qi and D. Sun, Global and superlinear convergence of the smoothing newton method and its application to general box constrained variational inequalities, Math. of Computation, 67 (1998), 519-540.
[9]	X. Chen and Y. Ye, On homotopy-smoothing methods for variational inequalities, SIAM J. Control and Optimization, 37 (1999), 589-616. doi: 10.1137/S0363012997315907.
[10]	G. P. Crespi, I. Ginchev and M. Rocca, Two approaches toward constrained vector optimization and identity of the solutions, Journal of Industrial and Management Optimization, 1 (2005), 549-563.
[11]	G. W. Flake and L. Steve, Efficient SVM regression training with SMO, Machine Learning, 46 (2002), 271-290.
[12]	K. Fukunaga, "Introduction to Statistical Pattern Recognition," Second edition, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1990.
[13]	G. Fung and O. L. Mangasarian, Finite Newton method for Lagrangian support vector machine classification, Neurocomputing, 55 (2003), 39-55.
[14]	D. Y. Gao, Canonical dual transformation method and generalized triality theory in nonsmooth global optimization, J. Global Optimization, 17 (2000), 127-160.
[15]	D. Y. Gao, Perfect duality theory and complete set of solutions to a class of global optimization, Optimization, 52 (2003), 467-493. doi: 10.1080/02331930310001611501.
[16]	D. Y. Gao, Complete solutions to constrained quadratic optimization problems, Journal of Global Optimisation, 29 (2004), 377-399. doi: 10.1023/B:JOGO.0000048034.94449.e3.
[17]	D. Y. Gao, Sufficient conditions and perfect duality in nonconvex minimization with inequality constraints, Journal of Industrial and Management Optimization, 1 (2005), 53-63.
[18]	D. Y. Gao, Complete solutions and extremality criteria to polynomial optimization problems, Journal of Global Optimization, 35 (2006), 131-143. doi: 10.1007/s10898-005-3068-5.
[19]	L. Gonzalez, C. Angulo, F. Velasco and A. Catala, Dual unification of bi-class support vector machine formulations, Pattern Recognition, 39 (2006), 1325-1332.
[20]	A. G. Hadigheh and T. Terlaky, Generalized support set invariancy sensitivity analysis in linear optimization, Journal of Industrial and Management Optimization, 2 (2006), 1-18.
[21]	Q. He, Z.-Z. Shi, L.-A. Ren and E. S. Lee, A novel classification method based on hyper-surface, Mathematical and Computer Modelling, 38 (2003), 395-407. doi: 10.1016/S0895-7177(03)90096-3.
[22]	C. W. Hsu and C. J. Lin, A simple decomposition method for support vector machines, Machine Learning, 46 (2002), 291-314.
[23]	T. Joachims, Making large-scale support vector machine learning practical, in "Advanced in Kernel methods: Support Vector Learning," (eds. B. Scholkopf, B. Burges and A. Smola), The MIT Press, Cambridge, Massachusetts, 1999.
[24]	S. S. Keerthi, K. B. Duan, S. K. Shevade and A. N. Poo, A fast dual algorithm for kernel logistic regression, Machine Learning, 61 (2005), 151-165.
[25]	P. Laskov, Feasible direction decomposition algorithms for training support vector machines, Machine Learning, 46 (2002), 315-349.
[26]	Y.-J. Lee, W. F. Hsieh and C. M. Huang, $\epsilon$-SSVR: A smooth support vector machine for $\epsilon$-insensitive regression, IEEE Transaction on Knowledge and Data Engineering, 17 (2005), 678-685. doi: 10.1109/TKDE.2005.77.
[27]	Y.-J. Lee and O. L. Mangarasian, SSVM: A smooth support vector machine for classification, Computational Optimization and Applications, 22 (2001), 5-22.
[28]	O. L. Mangasarian and D. R. Musicant, Successive overrelaxation for support vector machines, IEEE Transactions on Neural Networks, 10 (1999), 1032-1037. doi: 10.1109/72.788643.
[29]	T. M. Mitchell, "Machine Learning," McGraw-Hill Companies, Inc., 1997.
[30]	T. Mitchell, Statistical Approaches to Learning and Discovery, The course of Machine Learning at CMU, 2003.
[31]	D. Montgomery, "Design and Analysis of Experiments," Third edition, John Wiley & Sons, Inc., New York, 1991.
[32]	D. J. Newman, S. Hettich, C. L. Blake and C. J. Merz, "UCI Repository of Machine Learning Databases," University of California, Department of Information and Computer Science, Irvine, CA, 1998. Available from: http://www.ics.uci.edu/~mlearn/MLRepository.html.
[33]	P.-F. Pai, System reliability forecasting by support vector machines with genetic algorithms, Mathematical and Computer Modelling, 43 (2006), 262-274. doi: 10.1016/j.mcm.2005.02.008.
[34]	N. Panda and E. Y. Chang, KDX: An Indexer for support vector machines, IEEE Transaction on Knowledge and Data Engineering, 18 (2006), 748-763. doi: 10.1109/TKDE.2006.101.
[35]	J. Platt, Sequential minimal optimization: A fast algorithm for training support vector machines, Advances in Kernel Methods-Support Vector Learning [R], (1999), 185-208.
[36]	K. Schittkowski, Optimal parameter selection in support vector machines, Journal of Industrial and Management Optimization, 1 (2005), 465-476.
[37]	B. Schölkoft, "Support Vector Learning," R. Oldenbourg Verlag, Munich, 1997.
[38]	V. Vapnik, "The Nature of Statistical Learning Theory," Springer-Verlag, New York, 1995.
[39]	V. Vapnik, The support vector method of function estimation NATO ASI Series, in "Neural Network and Machine Learning," (ed. C. Bishop), Springer, 1998.
[40]	V. Vapnik, An overview of statistical learning theory, in "Advanced in Kernel methods: Support Vector Learning" (eds. B. Scholkopf, B. Burges and A. Smola), The MIT Press, Cambridge, Massachusetts, 1999.
[41]	V. Vapnik, Three remarks on support vector function estimation, IEEE transactions on Neural Networks, 10 (1999), 988-1000.
[42]	Z. Y. Wu, H. W. J. Lee, F. S. Bai and L. S. Zhang, Quadratic smoothing approximation to $l_1$ exact penalty function in global optimization, Journal of Industrial and Management Optimization, 1 (2005), 533-547.
[43]	K. F. C. Yiu, K. L. Mak and K. L. Teo, Airfoil design via optimal control theory, Journal of Industrial and Management Optimization, 1 (2005), 133-148.
[44]	Y. Yuan, J. Yan and C. Xu, Polynomial smooth support vector machine (PSSVM), Chinese Journal Of Computers, 28 (2005), 9-17.
[45]	Y. Yuan and T. Huang, A polynomial smooth support vector machine for classification, Lecture Note in Artificial Intelligence, 3584 (2005), 157-164.
[46]	Y. Yuan and R. Byrd, Non-quasi-Newton updates for unconstrained optimization, J. Comput. Math., 13 (1995), 95-107.
[47]	Y. Yuan, A modified BFGS algorithm for unconstrained optimization, IMA J. Numer. Anal., 11 (1991), 325-332. doi: 10.1093/imanum/11.3.325.

show all references

References:

[1]	J. O. Berger, "Statistical Decision Theory and Bayesian Analysis," Second edition, Springer Series in Statistics, Springer-Verlag, New York, 1985.
[2]	P. J. Bickel and K. A. Doksum, "Mathematical Statistics. Basic Ideas and Selected Topics," Second edition, Prentice-Hall, Inc., 2001.
[3]	C. J. C. Burges, A tutorial on support vector machines for pattern recognition, Data Mining and Knowledge Discovery, 2 (1998), 121-167.
[4]	O. Chapelle, V. Vapnik, O. Bousquet and S. Mukherjee, Choosing multiple parameters for support vector machines, Machine Learning, 46 (2002), 131-159.
[5]	B. Chen and P. T. Harker, Smooth approximations to nonlinear complementarity problems, SIAM J. Optimization, 7 (1997), 403-420.
[6]	C. Chen and O. L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems, Computational Optimization and Applications, 5 (1996), 97-138.
[7]	C. Chen and O. L. Mangasarian, Smoothing methods for convex inequalities and linear complementarity problems, Math. Programming, 71 (1995), 51-69. doi: 10.1016/0025-5610(95)00005-4.
[8]	X. Chen, L. Qi and D. Sun, Global and superlinear convergence of the smoothing newton method and its application to general box constrained variational inequalities, Math. of Computation, 67 (1998), 519-540.
[9]	X. Chen and Y. Ye, On homotopy-smoothing methods for variational inequalities, SIAM J. Control and Optimization, 37 (1999), 589-616. doi: 10.1137/S0363012997315907.
[10]	G. P. Crespi, I. Ginchev and M. Rocca, Two approaches toward constrained vector optimization and identity of the solutions, Journal of Industrial and Management Optimization, 1 (2005), 549-563.
[11]	G. W. Flake and L. Steve, Efficient SVM regression training with SMO, Machine Learning, 46 (2002), 271-290.
[12]	K. Fukunaga, "Introduction to Statistical Pattern Recognition," Second edition, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1990.
[13]	G. Fung and O. L. Mangasarian, Finite Newton method for Lagrangian support vector machine classification, Neurocomputing, 55 (2003), 39-55.
[14]	D. Y. Gao, Canonical dual transformation method and generalized triality theory in nonsmooth global optimization, J. Global Optimization, 17 (2000), 127-160.
[15]	D. Y. Gao, Perfect duality theory and complete set of solutions to a class of global optimization, Optimization, 52 (2003), 467-493. doi: 10.1080/02331930310001611501.
[16]	D. Y. Gao, Complete solutions to constrained quadratic optimization problems, Journal of Global Optimisation, 29 (2004), 377-399. doi: 10.1023/B:JOGO.0000048034.94449.e3.
[17]	D. Y. Gao, Sufficient conditions and perfect duality in nonconvex minimization with inequality constraints, Journal of Industrial and Management Optimization, 1 (2005), 53-63.
[18]	D. Y. Gao, Complete solutions and extremality criteria to polynomial optimization problems, Journal of Global Optimization, 35 (2006), 131-143. doi: 10.1007/s10898-005-3068-5.
[19]	L. Gonzalez, C. Angulo, F. Velasco and A. Catala, Dual unification of bi-class support vector machine formulations, Pattern Recognition, 39 (2006), 1325-1332.
[20]	A. G. Hadigheh and T. Terlaky, Generalized support set invariancy sensitivity analysis in linear optimization, Journal of Industrial and Management Optimization, 2 (2006), 1-18.
[21]	Q. He, Z.-Z. Shi, L.-A. Ren and E. S. Lee, A novel classification method based on hyper-surface, Mathematical and Computer Modelling, 38 (2003), 395-407. doi: 10.1016/S0895-7177(03)90096-3.
[22]	C. W. Hsu and C. J. Lin, A simple decomposition method for support vector machines, Machine Learning, 46 (2002), 291-314.
[23]	T. Joachims, Making large-scale support vector machine learning practical, in "Advanced in Kernel methods: Support Vector Learning," (eds. B. Scholkopf, B. Burges and A. Smola), The MIT Press, Cambridge, Massachusetts, 1999.
[24]	S. S. Keerthi, K. B. Duan, S. K. Shevade and A. N. Poo, A fast dual algorithm for kernel logistic regression, Machine Learning, 61 (2005), 151-165.
[25]	P. Laskov, Feasible direction decomposition algorithms for training support vector machines, Machine Learning, 46 (2002), 315-349.
[26]	Y.-J. Lee, W. F. Hsieh and C. M. Huang, $\epsilon$-SSVR: A smooth support vector machine for $\epsilon$-insensitive regression, IEEE Transaction on Knowledge and Data Engineering, 17 (2005), 678-685. doi: 10.1109/TKDE.2005.77.
[27]	Y.-J. Lee and O. L. Mangarasian, SSVM: A smooth support vector machine for classification, Computational Optimization and Applications, 22 (2001), 5-22.
[28]	O. L. Mangasarian and D. R. Musicant, Successive overrelaxation for support vector machines, IEEE Transactions on Neural Networks, 10 (1999), 1032-1037. doi: 10.1109/72.788643.
[29]	T. M. Mitchell, "Machine Learning," McGraw-Hill Companies, Inc., 1997.
[30]	T. Mitchell, Statistical Approaches to Learning and Discovery, The course of Machine Learning at CMU, 2003.
[31]	D. Montgomery, "Design and Analysis of Experiments," Third edition, John Wiley & Sons, Inc., New York, 1991.
[32]	D. J. Newman, S. Hettich, C. L. Blake and C. J. Merz, "UCI Repository of Machine Learning Databases," University of California, Department of Information and Computer Science, Irvine, CA, 1998. Available from: http://www.ics.uci.edu/~mlearn/MLRepository.html.
[33]	P.-F. Pai, System reliability forecasting by support vector machines with genetic algorithms, Mathematical and Computer Modelling, 43 (2006), 262-274. doi: 10.1016/j.mcm.2005.02.008.
[34]	N. Panda and E. Y. Chang, KDX: An Indexer for support vector machines, IEEE Transaction on Knowledge and Data Engineering, 18 (2006), 748-763. doi: 10.1109/TKDE.2006.101.
[35]	J. Platt, Sequential minimal optimization: A fast algorithm for training support vector machines, Advances in Kernel Methods-Support Vector Learning [R], (1999), 185-208.
[36]	K. Schittkowski, Optimal parameter selection in support vector machines, Journal of Industrial and Management Optimization, 1 (2005), 465-476.
[37]	B. Schölkoft, "Support Vector Learning," R. Oldenbourg Verlag, Munich, 1997.
[38]	V. Vapnik, "The Nature of Statistical Learning Theory," Springer-Verlag, New York, 1995.
[39]	V. Vapnik, The support vector method of function estimation NATO ASI Series, in "Neural Network and Machine Learning," (ed. C. Bishop), Springer, 1998.
[40]	V. Vapnik, An overview of statistical learning theory, in "Advanced in Kernel methods: Support Vector Learning" (eds. B. Scholkopf, B. Burges and A. Smola), The MIT Press, Cambridge, Massachusetts, 1999.
[41]	V. Vapnik, Three remarks on support vector function estimation, IEEE transactions on Neural Networks, 10 (1999), 988-1000.
[42]	Z. Y. Wu, H. W. J. Lee, F. S. Bai and L. S. Zhang, Quadratic smoothing approximation to $l_1$ exact penalty function in global optimization, Journal of Industrial and Management Optimization, 1 (2005), 533-547.
[43]	K. F. C. Yiu, K. L. Mak and K. L. Teo, Airfoil design via optimal control theory, Journal of Industrial and Management Optimization, 1 (2005), 133-148.
[44]	Y. Yuan, J. Yan and C. Xu, Polynomial smooth support vector machine (PSSVM), Chinese Journal Of Computers, 28 (2005), 9-17.
[45]	Y. Yuan and T. Huang, A polynomial smooth support vector machine for classification, Lecture Note in Artificial Intelligence, 3584 (2005), 157-164.
[46]	Y. Yuan and R. Byrd, Non-quasi-Newton updates for unconstrained optimization, J. Comput. Math., 13 (1995), 95-107.
[47]	Y. Yuan, A modified BFGS algorithm for unconstrained optimization, IMA J. Numer. Anal., 11 (1991), 325-332. doi: 10.1093/imanum/11.3.325.

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