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Quadratic optimization over a polyhedral cone
Subgradientbased neural network for nonconvex optimization problems in support vector machines with indefinite kernels
1.  Department of Mathematics, Harbin University of Science and Technology, Harbin, 150080, China 
2.  Department of Mathematics, Harbin Institute of Technology, Harbin 150001 
References:
[1] 
J. P. Aubin and A. Cellina, Differential Inclusions, SpringerVerlag, Berlin, 1984. doi: 10.1007/9783642695124. 
[2] 
J. P. Aubin and H. Frankowska, SetValued Analysis, MA: Birkauser, Boston, 1990. 
[3] 
E. K. P. Chong, S. Hui and S. H. Zak, An analysis of a class of neural networks for solving linear programming problems, Automatic Control, IEEE Transactions on, 44 (1999), 19952006. doi: 10.1109/9.802909. 
[4] 
J. Chen and J. Ye, Training SVM with indefinite kernels, In Proceedings of the 25th International Conference on Machine Learning, (2008), 136143. doi: 10.1145/1390156.1390174. 
[5] 
F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1969. 
[6] 
L. Fengqiu and X. Xiaoping, Design of natural classification kernels using prior knowledge, Fuzzy Systems, IEEE Transactions on, 20 (2012), 135152. 
[7] 
A. F. Filippove, Differential Equations with Discontinuous RightHand Side, Mathematics and Its Applications (Soviet Series), Boston, MA: Kluwer, 1988. 
[8] 
M. Forti, P. Nistri and M. Quincampoix, Convergence of neural networks for programming problems via a nonsmooth Łojasiewicz inequality, Neural Networks, IEEE Transactions on, 17 (2006), 14711486. 
[9] 
M. Forti and A. Tesi, Absolute stability of analytic neural networks: An approach based on finite trajectory length, Circuits System I, IEEE Transation on, 51 (2004), 24602469. doi: 10.1109/TCSI.2004.838143. 
[10] 
B. Haasdonk, Feature space interpretation of svms with indefinite kernels, Pattern Analysis and Machine Intelligence, IEEE Transactions on, 27 (2005), 482492. doi: 10.1109/TPAMI.2005.78. 
[11] 
M. Hein and O. Bousquet, Hilbertian metrics and positive definite kernels on probability measures, In Proceedings of the 10th International Workshop on Artificial Intelligence and Statistics, (eds. Z. Ghahramani and R. Cowell), Society for Artificial Intelligence and Statistics, United States, (2005), 136143. 
[12] 
J. J. Hopfield and D. W. Tank, "Neural" computation of decisions in optimization problems, Biological cybernetics, 52 (1985), 141152. 
[13] 
R. Luss and A. d'Aspremon, Support vector machine classification with indefinite kernels, Mathematical Programming Computation, 1 (2009), 97118. doi: 10.1007/s1253200900055. 
[14] 
I. Mierswa, Making Indefinite Kernel Learning Practical, Technical report, Artificial Intelligence Unit Department of Computer Science University of Dortmund, 2006. 
[15] 
J. C. Platt, Fast training of support vector machines using sequential minimal optimization, In Advances in Kernel Methods, (eds. Schölkopf, C. Burges and A. Smola), MIT, (1999), 185208. 
[16] 
T. Rockafellar and R. Wets, Variational Analysis, SpringerVerlag, Germany, 1998. doi: 10.1007/9783642024313. 
[17] 
S. ShalevShwartz, Y. Singer, N. Srebro and A. Cotter, Pegasos: Primal estimated subgradient solver for SVM, Math. Program., 127 (2011), 330. doi: 10.1007/s1010701004204. 
[18] 
H. Saigo, J. P. Vert, N. Ueda and T. Akutsu, Protein homology detection using string alignment kernels, Bioinformatics, 20 (2004), 16821689. doi: 10.1093/bioinformatics/bth141. 
[19] 
B. Schölkopf and A. J. Smola, Learning with Kernels, MIT, Cambridge, MA, 2002. 
[20] 
B. Schölkopf, P. Simard, A. J. Smola and V. N. Vapnik, Prior Knowledge in Support Vector Kernels, MIT, Cambridge, MA, 1998. 
[21] 
V. N. Vapnik, Statistical Learning Theory, Wiley, New York, 1998. 
[22] 
B. Wei and X. Xiaoping, Subgradientbased neural networks for nonsmooth nonconvex optimization problems, Circuits and Systems I: Regular Papers, IEEE Transactions on, 55 (2008), 23782391. doi: 10.1109/TCSI.2008.920131. 
[23] 
Y. Xia and J. Wang, A onelayer recurrent neural network for support vector machine learning, Systems, Man, and Cybernetics, Part B: Cybernetics, IEEE Transactions on, 34 (2004), 16211629. doi: 10.1109/TSMCB.2003.822955. 
[24] 
S. Ziye and J. Qingwei, Second order optimality conditions and reformulations for nonconvex quadratically constrained quadratic programming problems, Journal of Industrial and Management Optimization, 10 (2014), 871882. doi: 10.3934/jimo.2014.10.871. 
show all references
References:
[1] 
J. P. Aubin and A. Cellina, Differential Inclusions, SpringerVerlag, Berlin, 1984. doi: 10.1007/9783642695124. 
[2] 
J. P. Aubin and H. Frankowska, SetValued Analysis, MA: Birkauser, Boston, 1990. 
[3] 
E. K. P. Chong, S. Hui and S. H. Zak, An analysis of a class of neural networks for solving linear programming problems, Automatic Control, IEEE Transactions on, 44 (1999), 19952006. doi: 10.1109/9.802909. 
[4] 
J. Chen and J. Ye, Training SVM with indefinite kernels, In Proceedings of the 25th International Conference on Machine Learning, (2008), 136143. doi: 10.1145/1390156.1390174. 
[5] 
F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1969. 
[6] 
L. Fengqiu and X. Xiaoping, Design of natural classification kernels using prior knowledge, Fuzzy Systems, IEEE Transactions on, 20 (2012), 135152. 
[7] 
A. F. Filippove, Differential Equations with Discontinuous RightHand Side, Mathematics and Its Applications (Soviet Series), Boston, MA: Kluwer, 1988. 
[8] 
M. Forti, P. Nistri and M. Quincampoix, Convergence of neural networks for programming problems via a nonsmooth Łojasiewicz inequality, Neural Networks, IEEE Transactions on, 17 (2006), 14711486. 
[9] 
M. Forti and A. Tesi, Absolute stability of analytic neural networks: An approach based on finite trajectory length, Circuits System I, IEEE Transation on, 51 (2004), 24602469. doi: 10.1109/TCSI.2004.838143. 
[10] 
B. Haasdonk, Feature space interpretation of svms with indefinite kernels, Pattern Analysis and Machine Intelligence, IEEE Transactions on, 27 (2005), 482492. doi: 10.1109/TPAMI.2005.78. 
[11] 
M. Hein and O. Bousquet, Hilbertian metrics and positive definite kernels on probability measures, In Proceedings of the 10th International Workshop on Artificial Intelligence and Statistics, (eds. Z. Ghahramani and R. Cowell), Society for Artificial Intelligence and Statistics, United States, (2005), 136143. 
[12] 
J. J. Hopfield and D. W. Tank, "Neural" computation of decisions in optimization problems, Biological cybernetics, 52 (1985), 141152. 
[13] 
R. Luss and A. d'Aspremon, Support vector machine classification with indefinite kernels, Mathematical Programming Computation, 1 (2009), 97118. doi: 10.1007/s1253200900055. 
[14] 
I. Mierswa, Making Indefinite Kernel Learning Practical, Technical report, Artificial Intelligence Unit Department of Computer Science University of Dortmund, 2006. 
[15] 
J. C. Platt, Fast training of support vector machines using sequential minimal optimization, In Advances in Kernel Methods, (eds. Schölkopf, C. Burges and A. Smola), MIT, (1999), 185208. 
[16] 
T. Rockafellar and R. Wets, Variational Analysis, SpringerVerlag, Germany, 1998. doi: 10.1007/9783642024313. 
[17] 
S. ShalevShwartz, Y. Singer, N. Srebro and A. Cotter, Pegasos: Primal estimated subgradient solver for SVM, Math. Program., 127 (2011), 330. doi: 10.1007/s1010701004204. 
[18] 
H. Saigo, J. P. Vert, N. Ueda and T. Akutsu, Protein homology detection using string alignment kernels, Bioinformatics, 20 (2004), 16821689. doi: 10.1093/bioinformatics/bth141. 
[19] 
B. Schölkopf and A. J. Smola, Learning with Kernels, MIT, Cambridge, MA, 2002. 
[20] 
B. Schölkopf, P. Simard, A. J. Smola and V. N. Vapnik, Prior Knowledge in Support Vector Kernels, MIT, Cambridge, MA, 1998. 
[21] 
V. N. Vapnik, Statistical Learning Theory, Wiley, New York, 1998. 
[22] 
B. Wei and X. Xiaoping, Subgradientbased neural networks for nonsmooth nonconvex optimization problems, Circuits and Systems I: Regular Papers, IEEE Transactions on, 55 (2008), 23782391. doi: 10.1109/TCSI.2008.920131. 
[23] 
Y. Xia and J. Wang, A onelayer recurrent neural network for support vector machine learning, Systems, Man, and Cybernetics, Part B: Cybernetics, IEEE Transactions on, 34 (2004), 16211629. doi: 10.1109/TSMCB.2003.822955. 
[24] 
S. Ziye and J. Qingwei, Second order optimality conditions and reformulations for nonconvex quadratically constrained quadratic programming problems, Journal of Industrial and Management Optimization, 10 (2014), 871882. doi: 10.3934/jimo.2014.10.871. 
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