Harmonic competition is a learning strategy based upon winner-take-all or winner-take-quota with respect to a composite of heterogeneous subcosts. This learning is unsupervised and organizes itself. The subcosts may conflict with each other. Thus, the total learning system realizes a self-organizing multiple criteria optimization. The subcosts are combined additively and multiplicatively using adjusting parameters. For such a total cost, a general successive learning algorithm is derived first. Then, specific problems in the Euclidian space are addressed. Vector quantization with various constraints and traveling salesperson problems are selected as test problems. The former is a typical class of problems where the number of neurons is less than that of the data. The latter is an opposite case. Duality exists in these two classes. In both cases, the combination parameters of the subcosts show wide dynamic ranges in the course of learning. It is possible, however, to decide the parameter control from the structure of the total cost. This method finds a preferred solution from the Pareto optimal set of the multiple object optimization. Controlled mutations motivated by genetic algorithms are proved to be effective in finding near-optimal solutions.
SUMMARYBy using competitive learning, which causes just one or a group of a small number of neurons to respond to a given input, self-organization of entire neural networks can be achieved. When this self-organization process is applied to various kinds of travelling salesman problems in a Euclidean space, a good approximation or the true solution is obtained. We use a sequential update which looks at the position vector of each city one at a time as the training method for a neural network arranged as a closed loop. In this case, we use symmetrical connections between neurons. The number of neurons required is approximately linear in the number of cities. In the first experiment, we carried out a quantitative comparison with the simulated annealing method using 500 sets of 30 cities and demonstrated this method's superiority. Next, we obtained a good approximation on a set of 532 U.S. cities and demonstrate its superiority with respect to the increase in the number of cities in actual (realistic) data. Further, for a generalized constrained multiple-salesman problem, we explain this method's compactness and efficiency and give an experimental example. The computation can be adequately performed by a common workstation with a serial processor.
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