The stability of the solution in Hopfield neural network

Taking travelling salesman problem as an example for solving the combinatorial optimization problem by Hopfield neural network, the stability condition of the solution satisfying constraints of the problem, and the unstability condition of nonsolution are shown. By setting weights among the constraints and optimization requirement to satisfy these conditions, best solution can be obtained very easily. It is shown that, using these conditions, many properties of the network, e.g., the theoretical limitation of the network without self-connections (w_ii=0), can be derived and theoretical explanations can also be given to many phenomena.