Hopfield neural networks for optimization: study of the different dynamics

In this paper the application of arbitrary order Hopfield-like neural networks to optimization problems is studied. These networks are classified in three categories according to their dynamics, expliciting the energy function for each category. The main problems affecting practical applications of these networks are brought to light: (a) Incoherence between the network dynamics and the associated energy function; (b) Error due to discrete simulation on a digital computer of the continuous dynamics equations; (c) Existence of local minima; (d) Convergence depends on the coefficients weighting the cost function terms. The effect of these problems on each network is analysed and simulated, indicating possible solutions. Finally, the called continuous dynamics II is dealt with, proving that the integral term in the energy function is bounded, in contrast with Hopfield's statement, and proposing an efficient local minima avoidance strategy. Experimental results are obtained solving Diophantine equation, Hamiltonian cycle and k-colorability problems, (C) 2002 Elsevier Science B.V. All rights reserved.