×

zbMATH — the first resource for mathematics

A dynamical model for solving degenerate quadratic minimax problems with constraints. (English) Zbl 1229.90262
Summary: This paper presents a new neural network model for solving degenerate quadratic minimax (DQM) problems. On the basis of the saddle point theorem, optimization theory, convex analysis theory, Lyapunov stability theory and LaSalle invariance principle, the equilibrium point of the proposed network is proved to be equivalent to the optimal solution of the DQM problems. It is also shown that the proposed network model is stable in the sense of Lyapunov and it is globally convergent to an exact optimal solution of the original problem. Several illustrative examples are provided to show the feasibility and the efficiency of the proposed method in this paper.

MSC:
90C47 Minimax problems in mathematical programming
90C20 Quadratic programming
92B20 Neural networks for/in biological studies, artificial life and related topics
37B25 Stability of topological dynamical systems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Agrawal, S.K.; Fabien, B.C., Optimization of dynamic systems, (1999), Kluwer Academic Publishers Netherlands
[2] Avriel, M., Nonlinear programming: analysis and methods, (1976), Prentice-Hall Englewood Cliffs, NJ · Zbl 0361.90035
[3] Bazaraa, M.S.; Sherali, H.D.; Shetty, C.M., Nonlinear programming – theory and algorithms, (1993), Wiley New York · Zbl 0774.90075
[4] Bertsekas, D.P., Parallel and distributed computation: numerical methods, (1989), Prentice-Hall Englewood Cliffs, NJ · Zbl 0743.65107
[5] Boyd, S.; Vandenberghe, L., Convex optimization, (2004), Cambridge University Press Cambridge · Zbl 1058.90049
[6] Fletcher, R., Practical methods of optimization, (1981), Wiley New York · Zbl 0474.65043
[7] He, B.S., Solutions and application of a class of general linear variational inequalities, Science in China series A, 39, 395-404, (1996) · Zbl 0851.49010
[8] Rockafellar, R.T., Linear – quadratic programming and optimal control, SIAM journal on control and optimization, 25, 781-814, (1987) · Zbl 0617.49010
[9] Liao, L.; Qi, H.; Qi, L., Solving nonlinear complementarity problems with neural networks: a reformulation method approach, Journal of computational and applied mathematics, 131, 343-359, (2001) · Zbl 0985.65072
[10] Xia, Y.S.; Wang, G., An improved network for convex quadratic optimization with application to real-time beamforming, Neurocomputing, 64, 359-374, (2005)
[11] Cichocki, A.; Unbehauer, R., Neural networks for optimization with bounded constraints, IEEE transactions on neural networks, 4, 293-304, (1993)
[12] Effati, S.; Nazemi, A.R., Neural network models and its application for solving linear and quadratic programming problems, Applied mathematics and computation, 172, 305-331, (2006) · Zbl 1093.65059
[13] Effati, S.; Ghomashi, A.; Nazemi, A.R., Application of projection neural network in solving convex programming problems, Applied mathematics and computation, 188, 1103-1114, (2007) · Zbl 1121.65066
[14] Gao, X., A novel neural network for nonlinear convex programming, IEEE transactions on neural networks, 15, 613-621, (2004)
[15] Hu, X.; Wang, J., Design of general projection neural network for solving monotone linear variational inequalities and linear and quadratic programming problems, IEEE transactions on systems, man, and cybernetics, part B: cybernetics, 37, 1414-1421, (2007)
[16] Leung, Y.; Chen, K.; Gao, X., A high-performance feedback neural network for solving convex nonlinear programming problems, IEEE transactions on neural networks, 14, 1469-1477, (2003)
[17] Liang, X.B.; Wang, J., A recurrent neural network for nonlinear optimization with a continuously differentiable objective function and bounded constraints, IEEE transactions on neural networks, 11, 1251-1262, (2000)
[18] Malek, A.; Hosseinipour-Mahani, N.; Ezazipour, S., Efficient recurrent neural network model for the solution of general nonlinear optimization problems, Optimization methods and software, 25, 1-18, (2009) · Zbl 1225.90129
[19] Xia, Y.; Wang, J., A recurrent neural network for solving linear projection equations, Neural networks, 13, 337-350, (2000)
[20] Xia, Y.; Wang, J., A general projection neural network for solving monotone variational inequality and related optimization problems, IEEE transactions on neural networks, 15, 318-328, (2004)
[21] Xia, Y.; Wang, J., A recurrent neural network for nonlinear convex optimization subject to nonlinear inequality constraints, IEEE transactions on circuits and systems, 51, 447-458, (2004)
[22] Xia, Y.; Feng, G., A new neural network for solving nonlinear projection equations, Neural networks, 20, 577-589, (2007) · Zbl 1123.68110
[23] Xue, X.; Bian, W., A project neural network for solving degenerate convex quadratic program, Neural networks, 70, 2449-2459, (2007)
[24] Yang, Y.; Cao, J., A feedback neural network for solving convex constraint optimization problems, Applied mathematics and computation, 201, 340-350, (2008) · Zbl 1152.90566
[25] Gao, X.B.; Liao, L.Z., A novel neural network for a class of convex quadratic minimax problems, Neural computation, 18, 1818-1846, (2006) · Zbl 1185.90205
[26] Gao, X.B.; Liao, L.Z.; Xue, W., A neural network for a class of convex quadratic minimax problems with constraints, IEEE transactions on neural networks, 15, 318-328, (2004)
[27] Xue, X.; Bian, W., A project neural network for solving degenerate quadratic minimax problem with linear constraints, Neurocomputing, 72, 1826-1838, (2009)
[28] Ortega, T.M.; Rheinboldt, W.C., Iterative solution of nonlinear equations in several variables, (1970), Academic Press New York · Zbl 0241.65046
[29] Quarteroni, A.; Sacco, R.; Saleri, F., ()
[30] Miller, R.K.; Michel, A.N., Ordinary differential equations, (1982), Academic Press New York · Zbl 0499.34024
[31] Hale, J.K., Ordinary differential equations, (1969), Wiley-Interscience New York · Zbl 0186.40901
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.