“Optimal” neural representation of higher order for quadratic combinatorial optimization

On a theoretical basis, the author previously presented an “optimal” neural representation for combinatorial optimization problems with a linear cost function (1998). In this paper, taking traveling salesman problems (TSP) as examples, we present such an “optimal” neural representation of 4th order for combinatorial optimization problems with a quadratic cost function. This representation is compiled into a Hopfield network of 3rd order. It is proved that a vertex of this network state hypercube is asymptotically stable iff it is an optimal solution to the problem. One can always obtain an optimal solution whenever the network converges to a vertex