Connections between fuzzy theory, simulated annealing, and convex duality

Simulated annealing is a well-known heuristic method in which the basic notions of statistical mechanics are used as an analogy for treating optimization problems. The physical term of energy is regarded as an objective criterion which has to be minimized. In this context also the terms of temperature, free energy, and entropy have been involved. The present paper generalizes this formalism in order to make it available as a basic tool for fuzzy theory and multicriteria analysis. Therefore the formula of free energy is generalized in order to introduce a new aggregation operator psi. The main feature of this operator is the compensatory property of aggregation. It fills the gap between the maximum operator, the minimum operator, and the weighted sum by a coherent mathematical formalism which is connected with a large mathematical background. The aggregation operator psi plays the same peculiar part as the Shannon entropy does in information theory. This is verified by means of convex duality. In this formalism the fuzzy decisions that correspond to the Boltzmann distribution are involved. From there we derive a fuzzy extension principle which can be applied to developing new algorithms. Moreover, we discover a bridge between the Boltzmann distribution and the fuzzy c-means approach. Across this bridge some help arises for both of the two sides. Thus we derive new and generalized versions of the fuzzy c-means algorithm. And we establish a fuzzy version of simulated annealing, which is more efficient than the original simulated annealing. (C) 1998 Elsevier Science B.V. All rights reserved.