One of the modifications to the Hopfield and Tank model for dealing with optimization problems is the use of simulated annealing to perform a discretized version of the local optimization as in the Boltzmann machine approach of Aarts and Korst [11]. A key factor holding these variants back is the integer programming formulation of Figure 10, which shows that a large portion of a neural net's work be spent merely getting a feasible solution.
There are also two variants of neural nets that can be classified as
``geometric'' neural networks. These are the elastic net of
Durbin and Willshaw [36] and the self-organizing map which is
based on ideas of Kohonen [37]. When applied to the TSP, these
network can be viewed as a string connecting a set of points on the plane, as shown in sketch A of Figure
13, such that
where N is the number of
cities. The idea here is to move these points iteratively toward the
cities, thus deforming the geometric figure shaped by the connecting
string as shown in sketch B of Figure 13. The
objective is to continue this process until the geometric shape of the
connecting string and the X points looks like a tour, with each city
matched with one of the X points. See also the work of Durbin,
Szeliski and Yuille [38] for more details on these networks.
Johnson and colleagues experimented with these geometric models on the TSP and the best tours they obtained are far worse than the ``average'' tours obtained using the 2-Opt and Lin-Kernighan heuristics.
Figure 13: A geometric-based neural network. The empty circles each
denotes a city. Sketch A shows the start of the execution and sketch B
shows an intermediate stage in the execution.