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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 
References:
[1] 
J. O. Berger, "Statistical Decision Theory and Bayesian Analysis,", Second edition, (1985). Google Scholar 
[2] 
P. J. Bickel and K. A. Doksum, "Mathematical Statistics. Basic Ideas and Selected Topics," Second edition,, PrenticeHall, (2001). Google Scholar 
[3] 
C. J. C. Burges, A tutorial on support vector machines for pattern recognition,, Data Mining and Knowledge Discovery, 2 (1998), 121. Google Scholar 
[4] 
O. Chapelle, V. Vapnik, O. Bousquet and S. Mukherjee, Choosing multiple parameters for support vector machines,, Machine Learning, 46 (2002), 131. Google Scholar 
[5] 
B. Chen and P. T. Harker, Smooth approximations to nonlinear complementarity problems,, SIAM J. Optimization, 7 (1997), 403. Google Scholar 
[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. Google Scholar 
[7] 
C. Chen and O. L. Mangasarian, Smoothing methods for convex inequalities and linear complementarity problems,, Math. Programming, 71 (1995), 51. doi: 10.1016/00255610(95)000054. Google Scholar 
[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. Google Scholar 
[9] 
X. Chen and Y. Ye, On homotopysmoothing methods for variational inequalities,, SIAM J. Control and Optimization, 37 (1999), 589. doi: 10.1137/S0363012997315907. Google Scholar 
[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. Google Scholar 
[11] 
G. W. Flake and L. Steve, Efficient SVM regression training with SMO,, Machine Learning, 46 (2002), 271. Google Scholar 
[12] 
K. Fukunaga, "Introduction to Statistical Pattern Recognition,", Second edition, (1990). Google Scholar 
[13] 
G. Fung and O. L. Mangasarian, Finite Newton method for Lagrangian support vector machine classification,, Neurocomputing, 55 (2003), 39. Google Scholar 
[14] 
D. Y. Gao, Canonical dual transformation method and generalized triality theory in nonsmooth global optimization,, J. Global Optimization, 17 (2000), 127. Google Scholar 
[15] 
D. Y. Gao, Perfect duality theory and complete set of solutions to a class of global optimization,, Optimization, 52 (2003), 467. doi: 10.1080/02331930310001611501. Google Scholar 
[16] 
D. Y. Gao, Complete solutions to constrained quadratic optimization problems,, Journal of Global Optimisation, 29 (2004), 377. doi: 10.1023/B:JOGO.0000048034.94449.e3. Google Scholar 
[17] 
D. Y. Gao, Sufficient conditions and perfect duality in nonconvex minimization with inequality constraints,, Journal of Industrial and Management Optimization, 1 (2005), 53. Google Scholar 
[18] 
D. Y. Gao, Complete solutions and extremality criteria to polynomial optimization problems,, Journal of Global Optimization, 35 (2006), 131. doi: 10.1007/s1089800530685. Google Scholar 
[19] 
L. Gonzalez, C. Angulo, F. Velasco and A. Catala, Dual unification of biclass support vector machine formulations,, Pattern Recognition, 39 (2006), 1325. Google Scholar 
[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. Google Scholar 
[21] 
Q. He, Z.Z. Shi, L.A. Ren and E. S. Lee, A novel classification method based on hypersurface,, Mathematical and Computer Modelling, 38 (2003), 395. doi: 10.1016/S08957177(03)900963. Google Scholar 
[22] 
C. W. Hsu and C. J. Lin, A simple decomposition method for support vector machines,, Machine Learning, 46 (2002), 291. Google Scholar 
[23] 
T. Joachims, Making largescale support vector machine learning practical,, in, (1999). Google Scholar 
[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. Google Scholar 
[25] 
P. Laskov, Feasible direction decomposition algorithms for training support vector machines,, Machine Learning, 46 (2002), 315. Google Scholar 
[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. doi: 10.1109/TKDE.2005.77. Google Scholar 
[27] 
Y.J. Lee and O. L. Mangarasian, SSVM: A smooth support vector machine for classification,, Computational Optimization and Applications, 22 (2001), 5. Google Scholar 
[28] 
O. L. Mangasarian and D. R. Musicant, Successive overrelaxation for support vector machines,, IEEE Transactions on Neural Networks, 10 (1999), 1032. doi: 10.1109/72.788643. Google Scholar 
[29] 
T. M. Mitchell, "Machine Learning,", McGrawHill Companies, (1997). Google Scholar 
[30] 
T. Mitchell, Statistical Approaches to Learning and Discovery,, The course of Machine Learning at CMU, (2003). Google Scholar 
[31] 
D. Montgomery, "Design and Analysis of Experiments,", Third edition, (1991). Google Scholar 
[32] 
D. J. Newman, S. Hettich, C. L. Blake and C. J. Merz, "UCI Repository of Machine Learning Databases,", University of California, (1998). Google Scholar 
[33] 
P.F. Pai, System reliability forecasting by support vector machines with genetic algorithms,, Mathematical and Computer Modelling, 43 (2006), 262. doi: 10.1016/j.mcm.2005.02.008. Google Scholar 
[34] 
N. Panda and E. Y. Chang, KDX: An Indexer for support vector machines,, IEEE Transaction on Knowledge and Data Engineering, 18 (2006), 748. doi: 10.1109/TKDE.2006.101. Google Scholar 
[35] 
J. Platt, Sequential minimal optimization: A fast algorithm for training support vector machines,, Advances in Kernel MethodsSupport Vector Learning [R], (1999), 185. Google Scholar 
[36] 
K. Schittkowski, Optimal parameter selection in support vector machines,, Journal of Industrial and Management Optimization, 1 (2005), 465. Google Scholar 
[37] 
B. Schölkoft, "Support Vector Learning,", R. Oldenbourg Verlag, (1997). Google Scholar 
[38] 
V. Vapnik, "The Nature of Statistical Learning Theory,", SpringerVerlag, (1995). Google Scholar 
[39] 
V. Vapnik, The support vector method of function estimation NATO ASI Series,, in, (1998). Google Scholar 
[40] 
V. Vapnik, An overview of statistical learning theory,, in, (1999). Google Scholar 
[41] 
V. Vapnik, Three remarks on support vector function estimation,, IEEE transactions on Neural Networks, 10 (1999), 988. Google Scholar 
[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. Google Scholar 
[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. Google Scholar 
[44] 
Y. Yuan, J. Yan and C. Xu, Polynomial smooth support vector machine (PSSVM),, Chinese Journal Of Computers, 28 (2005), 9. Google Scholar 
[45] 
Y. Yuan and T. Huang, A polynomial smooth support vector machine for classification,, Lecture Note in Artificial Intelligence, 3584 (2005), 157. Google Scholar 
[46] 
Y. Yuan and R. Byrd, NonquasiNewton updates for unconstrained optimization,, J. Comput. Math., 13 (1995), 95. Google Scholar 
[47] 
Y. Yuan, A modified BFGS algorithm for unconstrained optimization,, IMA J. Numer. Anal., 11 (1991), 325. doi: 10.1093/imanum/11.3.325. Google Scholar 
show all references
References:
[1] 
J. O. Berger, "Statistical Decision Theory and Bayesian Analysis,", Second edition, (1985). Google Scholar 
[2] 
P. J. Bickel and K. A. Doksum, "Mathematical Statistics. Basic Ideas and Selected Topics," Second edition,, PrenticeHall, (2001). Google Scholar 
[3] 
C. J. C. Burges, A tutorial on support vector machines for pattern recognition,, Data Mining and Knowledge Discovery, 2 (1998), 121. Google Scholar 
[4] 
O. Chapelle, V. Vapnik, O. Bousquet and S. Mukherjee, Choosing multiple parameters for support vector machines,, Machine Learning, 46 (2002), 131. Google Scholar 
[5] 
B. Chen and P. T. Harker, Smooth approximations to nonlinear complementarity problems,, SIAM J. Optimization, 7 (1997), 403. Google Scholar 
[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. Google Scholar 
[7] 
C. Chen and O. L. Mangasarian, Smoothing methods for convex inequalities and linear complementarity problems,, Math. Programming, 71 (1995), 51. doi: 10.1016/00255610(95)000054. Google Scholar 
[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. Google Scholar 
[9] 
X. Chen and Y. Ye, On homotopysmoothing methods for variational inequalities,, SIAM J. Control and Optimization, 37 (1999), 589. doi: 10.1137/S0363012997315907. Google Scholar 
[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. Google Scholar 
[11] 
G. W. Flake and L. Steve, Efficient SVM regression training with SMO,, Machine Learning, 46 (2002), 271. Google Scholar 
[12] 
K. Fukunaga, "Introduction to Statistical Pattern Recognition,", Second edition, (1990). Google Scholar 
[13] 
G. Fung and O. L. Mangasarian, Finite Newton method for Lagrangian support vector machine classification,, Neurocomputing, 55 (2003), 39. Google Scholar 
[14] 
D. Y. Gao, Canonical dual transformation method and generalized triality theory in nonsmooth global optimization,, J. Global Optimization, 17 (2000), 127. Google Scholar 
[15] 
D. Y. Gao, Perfect duality theory and complete set of solutions to a class of global optimization,, Optimization, 52 (2003), 467. doi: 10.1080/02331930310001611501. Google Scholar 
[16] 
D. Y. Gao, Complete solutions to constrained quadratic optimization problems,, Journal of Global Optimisation, 29 (2004), 377. doi: 10.1023/B:JOGO.0000048034.94449.e3. Google Scholar 
[17] 
D. Y. Gao, Sufficient conditions and perfect duality in nonconvex minimization with inequality constraints,, Journal of Industrial and Management Optimization, 1 (2005), 53. Google Scholar 
[18] 
D. Y. Gao, Complete solutions and extremality criteria to polynomial optimization problems,, Journal of Global Optimization, 35 (2006), 131. doi: 10.1007/s1089800530685. Google Scholar 
[19] 
L. Gonzalez, C. Angulo, F. Velasco and A. Catala, Dual unification of biclass support vector machine formulations,, Pattern Recognition, 39 (2006), 1325. Google Scholar 
[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. Google Scholar 
[21] 
Q. He, Z.Z. Shi, L.A. Ren and E. S. Lee, A novel classification method based on hypersurface,, Mathematical and Computer Modelling, 38 (2003), 395. doi: 10.1016/S08957177(03)900963. Google Scholar 
[22] 
C. W. Hsu and C. J. Lin, A simple decomposition method for support vector machines,, Machine Learning, 46 (2002), 291. Google Scholar 
[23] 
T. Joachims, Making largescale support vector machine learning practical,, in, (1999). Google Scholar 
[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. Google Scholar 
[25] 
P. Laskov, Feasible direction decomposition algorithms for training support vector machines,, Machine Learning, 46 (2002), 315. Google Scholar 
[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. doi: 10.1109/TKDE.2005.77. Google Scholar 
[27] 
Y.J. Lee and O. L. Mangarasian, SSVM: A smooth support vector machine for classification,, Computational Optimization and Applications, 22 (2001), 5. Google Scholar 
[28] 
O. L. Mangasarian and D. R. Musicant, Successive overrelaxation for support vector machines,, IEEE Transactions on Neural Networks, 10 (1999), 1032. doi: 10.1109/72.788643. Google Scholar 
[29] 
T. M. Mitchell, "Machine Learning,", McGrawHill Companies, (1997). Google Scholar 
[30] 
T. Mitchell, Statistical Approaches to Learning and Discovery,, The course of Machine Learning at CMU, (2003). Google Scholar 
[31] 
D. Montgomery, "Design and Analysis of Experiments,", Third edition, (1991). Google Scholar 
[32] 
D. J. Newman, S. Hettich, C. L. Blake and C. J. Merz, "UCI Repository of Machine Learning Databases,", University of California, (1998). Google Scholar 
[33] 
P.F. Pai, System reliability forecasting by support vector machines with genetic algorithms,, Mathematical and Computer Modelling, 43 (2006), 262. doi: 10.1016/j.mcm.2005.02.008. Google Scholar 
[34] 
N. Panda and E. Y. Chang, KDX: An Indexer for support vector machines,, IEEE Transaction on Knowledge and Data Engineering, 18 (2006), 748. doi: 10.1109/TKDE.2006.101. Google Scholar 
[35] 
J. Platt, Sequential minimal optimization: A fast algorithm for training support vector machines,, Advances in Kernel MethodsSupport Vector Learning [R], (1999), 185. Google Scholar 
[36] 
K. Schittkowski, Optimal parameter selection in support vector machines,, Journal of Industrial and Management Optimization, 1 (2005), 465. Google Scholar 
[37] 
B. Schölkoft, "Support Vector Learning,", R. Oldenbourg Verlag, (1997). Google Scholar 
[38] 
V. Vapnik, "The Nature of Statistical Learning Theory,", SpringerVerlag, (1995). Google Scholar 
[39] 
V. Vapnik, The support vector method of function estimation NATO ASI Series,, in, (1998). Google Scholar 
[40] 
V. Vapnik, An overview of statistical learning theory,, in, (1999). Google Scholar 
[41] 
V. Vapnik, Three remarks on support vector function estimation,, IEEE transactions on Neural Networks, 10 (1999), 988. Google Scholar 
[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. Google Scholar 
[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. Google Scholar 
[44] 
Y. Yuan, J. Yan and C. Xu, Polynomial smooth support vector machine (PSSVM),, Chinese Journal Of Computers, 28 (2005), 9. Google Scholar 
[45] 
Y. Yuan and T. Huang, A polynomial smooth support vector machine for classification,, Lecture Note in Artificial Intelligence, 3584 (2005), 157. Google Scholar 
[46] 
Y. Yuan and R. Byrd, NonquasiNewton updates for unconstrained optimization,, J. Comput. Math., 13 (1995), 95. Google Scholar 
[47] 
Y. Yuan, A modified BFGS algorithm for unconstrained optimization,, IMA J. Numer. Anal., 11 (1991), 325. doi: 10.1093/imanum/11.3.325. Google Scholar 
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