Theoretical analysis of quantized Hopfield network for integer programming

Quantized Hopfield networks, where each neuron takes quantized values (e.g, integers), need fewer neurons and connections than the binary or continuous Hopfield networks with the counting method, when applied to integer optimization problems, so they can obtain feasible solutions much more quickly. In spite of this practical effectiveness, although it was shown that the quantized networks converge to an energy minimum state, their dynamics have not well been analyzed, for example, it is unknown which states are energy minimum or not. We clarify the dynamics of quantized Hopfield networks, and show the energy minimum and nonminimum conditions of the feasible and infeasible solutions to the problems in terms of the values of network coefficients. This gives the insight into the practical network tuning