Neural Networks pp 233-244 | Cite as

# Solution of the Traveling-Salesman Problem

Chapter

## Abstract

As we know from the spin-glass analogy neural networks can model a complicated ‘energy landscape’ and hence can be employed for solving optimization problems by searching for minima in this energy surface. In Sect. 10.3.1 we have discussed an approach for solving the paragon combinatorial optimization task, the traveling salesman problem (TSP), using a neural network. Hopfield and Tank [Ho85] have mapped the
with formal temperature

*N*-city TSP onto a network with*N*×*N*formal neurons*n*_{ iα}. These are assumed to have a graded response with a continuous output value in the range 0 ... 1 determined by the local field*u*_{ iα}through the Fermi function$$
{n_{i\alpha }}\,\, = \,\,\frac{1}{{1\,\, + \,\,{e^{ - 2{u_{i\alpha }}/T}}}}
$$

(24.1)

*T*. The neurons are arranged in a square array, the row*i*denoting the number of the city and the column α denoting the station of the tour on which this city is visited. A valid tour is therefore characterized by an activation pattern with exactly*N*neurons ‘firing’ and*N*(*N*− 1) ‘quiescent’. There must be exactly one entry of 1 in each row and each column of the neural matrix*n*_{ iα}.### Keywords

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## Copyright information

© Springer-Verlag Berlin Heidelberg 1990