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A SUBSPACE APPROACH TO SOLVING COMBINATORIAL OPTIMIZATION PROBLEMS WITH HOPFIELD NETWORKS
Sreeram V. B. Aiyer and Frank Fallside
This paper extends and generalizes the subspace analysis of the Hopfield Network presented in our earlier work [IEEE trans. NN. 6/90]. In [IEEE trans. NN. 6/90] it was shown that the ability of the Hopfield Network to confine a vector within a particular subspace was essential to the network's ability to reach valid solutions to the Travelling Salesman problem. Through the use of Kronecker (Tensor) products, this paper shows how an analogous subspace can be constructed for a much larger class of combinatorial optimization problems. By using the form of this subspace as the basis for determining the elements of the network connection matrix, convergence to a valid solution can be guaranteed. Further, the quality of the final solution is significantly improved. This is confirmed by benchmark experiments using 30 and 50 city Travelling Salesman problems as an illustrative example. These indicate that the network can reliably and efficiently achieve solutions within 2\% of the global optimum.
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