By Kroese B., van der Smagt P.

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**Sample text**

As before, this system can be proved to be stable when a symmetric weight matrix is used (Hopfield, 1984). 2. THE HOPFIELD NETWORK 53 Hopfield networks for optimisation problems An interesting application of the Hopfield network with graded response arises in a heuristic solution to the NP-complete travelling salesman problem (Garey & Johnson, 1979). In this problem, a path of minimal distance must be found between n cities, such that the begin- and end-points are the same. Hopfield and Tank (Hopfield & Tank, 1985) use a network with n × n neurons.

Output activation values are fed back to the input layer, to a set of extra neurons called the state units. output units are fed back into the input layer through a set of extra input units called the state units. There are as many state units as there are output units in the network. The connections between the output and state units have a fixed weight of +1; learning takes place only in the connections between input and hidden units as well as hidden and output units. Thus all the learning rules derived for the multi-layer perceptron can be used to train this network.

The resulting cost is O(n) which is significantly better than the linear convergence 4 of steepest descent. 2 A matrix A is called positive definite if ∀y = 0, yT Ay > 0. ) However, line minimisation methods exist with super-linear convergence (see footnote 4). 4 A method is said to converge linearly if E i+1 = cE i with c < 1. , E i+1 = c(E i )m with m > 1 are called super-linear. 42 CHAPTER 4. 6: Slow decrease with conjugate gradient in non-quadratic systems. The hills on the left are very steep, resulting in a large search vector ui .