Journal of Applied Mathematics

Linear Simultaneous Equations’ Neural Solution and Its Application to Convex Quadratic Programming with Equality-Constraint

Yuhuan Chen, Chenfu Yi, and Jian Zhong

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A gradient-based neural network (GNN) is improved and presented for the linear algebraic equation solving. Then, such a GNN model is used for the online solution of the convex quadratic programming (QP) with equality-constraints under the usage of Lagrangian function and Karush-Kuhn-Tucker (KKT) condition. According to the electronic architecture of such a GNN, it is known that the performance of the presented GNN could be enhanced by adopting different activation function arrays and/or design parameters. Computer simulation results substantiate that such a GNN could obtain the accurate solution of the QP problem with an effective manner.

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J. Appl. Math., Volume 2013 (2013), Article ID 695647, 6 pages.

First available in Project Euclid: 14 March 2014

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Chen, Yuhuan; Yi, Chenfu; Zhong, Jian. Linear Simultaneous Equations’ Neural Solution and Its Application to Convex Quadratic Programming with Equality-Constraint. J. Appl. Math. 2013 (2013), Article ID 695647, 6 pages. doi:10.1155/2013/695647.

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  • P. Liu, S. Zhang, and Q. Li, “On the positive definite solutions of a nonlinear matrix equation,” Journal of Applied Mathematics, vol. 2013, Article ID 676978, 6 pages, 2013.
  • Y. Zhang, D. Jiang, and J. Wang, “A recurrent neural network for solving sylvester equation with time-varying coefficients,” IEEE Transactions on Neural Networks, vol. 13, no. 5, pp. 1053–1063, 2002.
  • Y. Li, M. Reisslein, and C. Chakrabarti, “Energy-efficient video transmission over a wireless link,” IEEE Transactions on Vehicular Technology, vol. 58, no. 3, pp. 1229–1244, 2009.
  • C. Yi, Y. Chen, and X. Lan, “Comparison on neural solvers for the Lyapunov matrix equation with stationary & nonstationary coefficients,” Applied Mathematical Modelling, vol. 37, no. 4, pp. 2495–2502, 2013.
  • F. Ding and T. Chen, “Gradient based iterative algorithms for solving a class of matrix equations,” IEEE Transactions on Automatic Control, vol. 50, no. 8, pp. 1216–1221, 2005.
  • M. A. Ghorbani, O. Kisi, and M. Aalinezhad, “A probe into the chaotic nature of daily streamflow time series by correlation dimension and largest Lyapunov methods,” Applied Mathematical Modelling, vol. 34, no. 12, pp. 4050–4057, 2010.
  • Y. Zhang and W. E. Leithead, “Exploiting Hessian matrix and trust-region algorithm in hyperparameters estimation of Gaussian process,” Applied Mathematics and Computation, vol. 171, no. 2, pp. 1264–1281, 2005.
  • X. Zou, Y. Tang, S. Bu, Z. Luo, and S. Zhong, “Neural-network-based approach for extracting eigenvectors and eigenvalues of real normal matrices and some extension to real matrices,” Journal of Applied Mathematics, vol. 2013, Article ID 597628, 13 pages, 2013.
  • D. W. Tank and J. J. Hopfield, “Simple neural optimization networks: an A/D converter, signal decision circuit, and a linear programming circuit,” IEEE Transactions on Circuits and Systems, vol. 33, no. 5, pp. 533–541, 1986.
  • J. Wang, “Electronic realisation of recurrent neural network for solving simultaneous linear equations,” Electronics Letters, vol. 28, no. 5, pp. 493–495, 1992.
  • Y. Zhang, C. Yi, and W. Ma, “Simulation and verification of Zhang neural network for online time-varying matrix inversion,” Simulation Modelling Practice and Theory, vol. 17, no. 10, pp. 1603–1617, 2009.
  • C. Yi and Y. Zhang, “Analogue recurrent neural network for linear algebraic equation solving,” Electronics Letters, vol. 44, no. 18, pp. 1078–1080, 2008.
  • K. Chen, “Robustness analysis of Wang neural network for online linear equation solving,” Electronic Letters, vol. 48, no. 22, pp. 1391–1392, 2012.
  • Y. Zhang, “Dual neural networks: design, analysis, and application to redundant robotics,” in Progress in Neurocomputing Research, pp. 41–81, Nova Science Publishers, New York, NY, USA, 2008.
  • Y. Zhang and J. Wang, “Global exponential stability of recurrent neural networks for synthesizing linear feedback control systems via pole assignment,” IEEE Transactions on Neural Networks, vol. 13, no. 3, pp. 633–644, 2002.
  • N. Petrot, “Some existence theorems for nonconvex variational inequalities problems,” Abstract and Applied Analysis, vol. 2010, Article ID 472760, 9 pages, 2010.
  • S. Burer and D. Vandenbussche, “A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations,” Mathematical Programming, vol. 113, no. 2, pp. 259–282, 2008.
  • Z. Dostál and R. Kučera, “An optimal algorithm for minimization of quadratic functions with bounded spectrum subject to separable convex inequality and linear equality constraints,” SIAM Journal on Optimization, vol. 20, no. 6, pp. 2913–2938, 2010.