Graded rescaling in Hopfield networks

In this work we propose a method with the capability of improving the performance of the Hopfield network for solving optimization problems by using a graded rescaling scheme on the distance matrix of the energy function. This method controls the magnitude of rescaling by adjusting a parameter (scaling factor) in order to explore the optimal range for performance. We have evaluated different scaling factors through 20,000 simulations, based on 200 randomly generated city distributions of the 10-city traveling salesman problem. The results show that the graded rescaling can improve the performance significantly for a wide range of scaling factors. It increases the percentage of valid tours by 72.2%, reduces the error rate of tour length by 10.2%, and increases the chance of finding optimal tours by 39.0%, as compared to the original Hopfield network without rescaling.