In general these algorithms have been shown to be not very competitive
with the aforementioned methods or other classical approaches when
tackling combinatorial optimization problems. For the TSP, the first
neural network approach was due to Hopfield and Tank (HT) (please see
[1] for more details about the model). Their approach was based
on the integer programming formulation of the TSP shown in
Figure 10. Here is taken to mean that
city
is the kth city in the tour, in which case the sum being
minimized is the tour length. The first constraint says that each
position contains precisely one city and the second says that each
city is in precisely one position on the tour.
Figure 10: An integer programming formulation for the TSP.
Hopfield and Tank's algorithm attempted to find a good feasible
solution to the formulation of Figure 10 by viewing
the as neurons that could take arbitrary values in
the interval
. The neurons were connected up with an inhibitory
network that tried simultaneously to impose the constraints and to
lower the cost of an energy function. Local optima for this
energy function were found using a local optimization method where
individual neurons changed state so as to lower their contribution to
the total energy. Multiple random starts were allowed.
In comparison to other methods summarized in this paper and tried on problems such as the TSP, Hopfield and Tank's results were not at all promising. The main reason for this is the inability of the algorithm to constrain the network into valid solutions, that is, the network's partition function sums over a vast number of configurations which are nothing like valid tours, and even though these offenders have small Boltzmann weights their large number inevitably affects the thermal average quantities.
Next, we would like to summarize the overall Hopfield continuous model.