Abstract
This paper is devoted to the study of continuous Hopfield-like neural networks, either in its original version [Hopfield,1984] or in its high order generalization [Samad, 1990], [Kobuchi, 1991], applied to the solution of optimization problems. Main problems affecting the practical application of these networks are brought to light: a) Incoherence between the network dynamics and the associated energy function; b) Error due to discretization of the continuous dynamical equations caused by simulation on a digital computer; c) Existence of local minima. The behavior of this kind of neural networks with respect to these problems is analyzed and simulated, indicating possible mechanisms to avoid them. The last part of the paper we shown that the integral term in the energy function is bounded, in contrast with Hopfield's statement. Using this result, a new local minima avoidance strategy is proposed with an enhanced efficiency.
Preview
Unable to display preview. Download preview PDF.
References
Abe, S., (1989), “Theories on the Hopfield Neural Networks”, Int. Joint Conf. on Neural Network, Vol. 1, pp. 557–564.
Atencia, M. A., (1996), “Entorno de diseño y simulación de redes neuronales artificiales de alto orden”, Internal Report, Dpto Tecnología Electrónica, Univ.Málaga, Spain.
Garey, M. R., y Johnson,D.S., (1979), Computers and intractabily. A guide to the Theory of NP-Completeness, W.H. Freeman and Company, p. 245.
Hertz, J., Krogh, A., Palmer, R.G.,(1991), Introduction to the theory of neural computation, Addison-Wesley.
Hopfield, J.J., (1982), “Neural networks and physical systems with emergent collective computational abilities”, in Proc. National Academic Sciences U.S.A., Vol. 79, April, pp. 2554–2558.
Hopfield, J.J., (1984), “Neurons with graded response have collective computational properties like those of two-state neurons”, in Proc. National Academic Sciences U.S.A., Vol. 81, May, pp. 3088–3092.
Hopfield, J. J. y Tank, D. W., (1985), “Neural computation of decisions in optimization problems”, Biological Cybernetics, No. 52, pp. 141–152.
Joya, G., Atencia M. A. and Sandoval, F., (1991), “Application of high-order Hopfield neural networks to the solution of diophantine equations”, in Artificial Neural Networks, A. Prieto (Ed.), LNCS 540, Springer-Verlag, pp. 395–400.
Joya, G., Atencia, M. A., Sandoval, F., (1997), “Associating arbitrary-order energy functions to an artificial neural network. Implications concerning the resolution of optimization problems”, accepted to be published in Neurocomputing.
Kim, K.H., Lee C.H., Kim, B.Y. y Hwang, H.Y., (1991), “Neural optimization network for minimum-via layer assignment”, Neurocomputing, Vol. 3, pp. 15–27.
Kobuchi, Y., (1991), “State Evaluation Functions and Lyapunov Functions for Neural Networks”, Neural Networks, Vol. 4, pp. 505–510.
Metha, S., y Fulop, L., (1993), “An analog Neural Network to Solve the Hamiltonian Cycle Problem”, Neural Networks, Vol. 6, pp. 869–881.
Samad, T., y Harper, P., (1990), “High-order Hopfield and Tank optimization networks”, Parallel Computing, Vol.16, pp. 287–292.
Takefuji, Y. y Lee, K.C., (1991), “Artificial Neural Networks for Four-Coloring Map Problems and K-Colorability Problems”, IEEE Transactions on Circuits and Systems, Vol. 38, No. 3, pp. 326–333.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Joya, G., Atencia, M.A., Sandoval, F. (1997). Hopfield neural network applied to optimization problems: Some theoretical and simulation results. In: Mira, J., Moreno-Díaz, R., Cabestany, J. (eds) Biological and Artificial Computation: From Neuroscience to Technology. IWANN 1997. Lecture Notes in Computer Science, vol 1240. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0032515
Download citation
DOI: https://doi.org/10.1007/BFb0032515
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-63047-0
Online ISBN: 978-3-540-69074-0
eBook Packages: Springer Book Archive