Convergence of the synaptic weights for the elastic net method, and its application

Solution procedures for the traveling salesman problem (TSP), i.e. the problem of finding the minimum Hamiltonian circuit in a network of cities, can be divided into two categories: exact methods and approximate (or heuristic) methods. Since TSP is an NP hard problem, good heuristic approaches are of interest. The neural networks heuristic solutions of TSP was initiated by Hopfield and Tank. One such heuristic called the elastic net method is illustrated by the following, an imaginary rubber band is placed at the centroid of the distribution of n cities. Then some finite number (m greater than n) of points (nodes) on this rubber band changes their positions according to the dynamics of the method. Eventually they describe a tour around the cities. We express the dynamics and stability of the elastic net algorithm. We show that if a unique node is converging to a city, then the synaptic strength between them approaches one. Then we generalize to the case where more than one node converges to a city. Furthermore, a typical application that could make use of the elastic net method (e.g. multi-target tracking) will be pointed out for later studies. In order to verify the proof of the concept and the associated theorems, computer simulations were conducted for a reasonable number of cities.