Global convergence of the Hopfield neural network with nonzero diagonal elements

In this paper we derive stability conditions of local minima and their convergence regions of the Hopfield neural network when the diagonal elements of the coefficient matrix are all nonzero. Then for the traveling salesman problem (TSP) we clarify the ranges of the weight values in the energy function and the range of values of the diagonal elements, so that the feasible solutions become stable and infeasible solutions become unstable. Simulations of the TSP show that the above criteria are valid and, by gradually decreasing diagonal elements, quality of solutions is drastically improved, compared with that of zero diagonal elements