Harmonic competition: a self-organizing multiple criteria optimization

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.