A New Relaxation Procedure in the Hopfield Network for Solving Optimization Problems

When solving an optimization problem with a Hopfield network, a solution is obtained after the network is relaxed to an equilibrium state. The relaxation process is an important step in achieving a solution. In this paper, a new procedure for the relaxation process is proposed. In the new procedure, the amplified signal received by a neuron from other neurons is treated as the target value for its activation (output) value. The activation of a neuron is updated directly based on the difference between its current activation and the received target value, without using the updating of the input value as an intermediate step. A relaxation rate is applied to control the updating scale for a smooth relaxation process. The new procedure is evaluated and compared with the original procedure in the Hopfield network through simulations based on 200 randomly generated instances of the 10-city traveling salesman problem. The new procedure reduces the error rate by 34.6% and increases the percentage of valid tours by 194.6% as compared with the original procedure.