A dynamic system model for solving convex nonlinear optimization problems.

*(English)*Zbl 1250.90067Summary: This paper proposes a feedback neural network model for solving convex nonlinear programming (CNLP) problems. Under the condition that the objective function is convex and all constraint functions are strictly convex or that the objective function is strictly convex and the constraint function is convex, the proposed neural network is proved to be stable in the sense of Lyapunov and globally convergent to an exact optimal solution of the original problem. The validity and transient behavior of the neural network are demonstrated by using some examples.

##### MSC:

90C25 | Convex programming |

37N40 | Dynamical systems in optimization and economics |

92B20 | Neural networks for/in biological studies, artificial life and related topics |

PDF
BibTeX
XML
Cite

\textit{A. R. Nazemi}, Commun. Nonlinear Sci. Numer. Simul. 17, No. 4, 1696--1705 (2012; Zbl 1250.90067)

Full Text:
DOI

##### References:

[1] | Avriel, M., Nonlinear programming: analysis and methods, (1976), Prentice-Hall Englewood Cliffs, NJ · Zbl 0361.90035 |

[2] | Bazaraa, M.S.; Sherali, H.D.; Shetty, C.M., Nonlinear programming – theory and algorithms, (1993), Wiley New York · Zbl 0774.90075 |

[3] | Fletcher, R., Practical methods of optimization, (1981), Wiley New York · Zbl 0474.65043 |

[4] | He, B.S.; Zhou, J., A modified alternating direction method for convex minimization problems, Appl math lett, 13, 123-130, (2000) · Zbl 0988.90020 |

[5] | Agrawal, S.K.; Fabien, B.C., Optimization of dynamic systems, (1999), Kluwer Academic Publishers Netherlands |

[6] | Liao, L.; Qi, H.; Qi, L., Solving nonlinear complementarity problems with neural networks: a reformulation method approach, J comput appl math, 131, 343-359, (2001) · Zbl 0985.65072 |

[7] | Effati, S.; Ghomashi, A.; Nazemi, A.R., Application of projection neural network in solving convex programming problems, Appl math comput, 188, 1103-1114, (2007) · Zbl 1121.65066 |

[8] | Effati, S.; Nazemi, A.R., Neural network models and its application for solving linear and quadratic programming problems, Appl math comput, 172, 305-331, (2006) · Zbl 1093.65059 |

[9] | Gao, X., A novel neural network for nonlinear convex programming, IEEE trans neural netw, 15, 613-621, (2004) |

[10] | Hu, X.; Wang, J., Design of general projection neural network for solving monotonoe linear variational inequalities and linear and quadratic programming problems, IEEE trans syst man cybernet part B: cybernet, 37, 1414-1421, (2007) |

[11] | Kennedy, M.P.; Chua, L.O., Neural networks for nonlinear programming, IEEE trans circuits syst, 35, 554-562, (1988) |

[12] | Leung, Y.; Chen, K.; Gao, X., A high-performance feedback neural network for solving convex nonlinear programming problems, IEEE trans neural netw, 14, 1469-1477, (2003) |

[13] | Liang, X.B.; Wang, J., A recurrent neural network for nonlinear optimization with a continuously differentiable objective function and bounded constraints, IEEE trans neural netw, 11, 1251-1262, (2000) |

[14] | X. Hu, P. Balasubramaniam, Recurrent neural networks, ISBN: 978-953-7619-08-4, pp. 255-388, 2008, I-Tech, Vienna, Austria. |

[15] | Huang, Y.C., A novel method to handle inequality constraints for convex programming neural network, Neural process lett, 16, 17-27, (2002) · Zbl 1008.68757 |

[16] | Malek, A.; Hosseinipour-Mahani, N.; Ezazipour, S., Efficient recurrent neural network model for the solution of general nonlinear optimization problems, Opt methods softw, 25, 1-18, (2009) · Zbl 1225.90129 |

[17] | Popescua, M.; Dumitracheb, A., Stabilization of feedback control and stabilizability optimal solution for nonlinear quadratic problems, Commun nonlinear sci numer simul, 16, 2319-2327, (2011) · Zbl 1221.93228 |

[18] | Xia, Y.; Wang, J., A general methodology for designing globally convergent optimization neural networks, IEEE trans neural netw, 9, 1331-1343, (1998) |

[19] | Xia, Y.; Wang, J., A general projection neural network for solving monotone variational inequality and related optimization problems, IEEE trans neural netw, 15, 318-328, (2004) |

[20] | Xia, Y.; Wang, J., A recurrent neural network for solving linear projection equations, Neural netw, 13, 337-350, (2000) |

[21] | Xia, Y.; Feng, G., A new neural network for solving nonlinear projection equations, Neural netw, 20, 577-589, (2007) · Zbl 1123.68110 |

[22] | Xia, Y.; Leng, H.; Wang, J., A projection neural network and its application to constrained optimization problems, IEEE trans circuits syst, 49, 447-458, (2002) · Zbl 1368.92019 |

[23] | Xia, Y.; Wang, J., A recurrent neural network for nonlinear convex optimization subject to nonlinear inequality constraints, IEEE trans circuits syst, 51, 447-458, (2004) |

[24] | Yang, Y.; Xu, X., The projection neural network for solving convex nonlinear programming, (), p. 174-81 |

[25] | Yang, Y.; Cao, J., A feedback neural network for solving convex constraint optimization problems, Appl math comput, 201, 340-350, (2008) · Zbl 1152.90566 |

[26] | Ortega, T.M.; Rheinboldt, W.C., Iterative solution of nonlinear equations in several variables, (1970), Academic New York · Zbl 0241.65046 |

[27] | Miller, R.K.; Michel, A.N., Ordinary differential equations, (1982), Academic Press New York |

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.