Solution method using correlated noise for TSP

Abstract

We suggest solution method for optimization problems using correlated noises. The correlated noises are introduced to neural networks to discuss mechanism of synfire chain. Kawamura and Okada have introduced correlated noises to associative memory models and have analyzed those dynamics. In the associative memory models, memory patterns are memorized as attractors in the minimum of the system. They found the correlated noise can make the state transit between the attractors. However, the mechanism of the state transition has not been known enough yet. One the other hand, for combinational optimization problems, the energy function of a problem can be defined. Therefore, finding a optimum solution is finding a minima of the energy function. The steepest descent method searches one of the solutions by going down along the gradient direction. By this method, however, the state is usually trapped in a local minimum of the system. In order to escape from the local minimum, the simulated annealing, i.e. Metropolis method, or chaotic disturbance is introduced. These methods can be represented by adding thermal noises or chaotic inputs to the dynamic equation. In this paper, we show that correlated noises introduced to neural networks can be applied to solve the optimization problems. We solve the TSP that is a typical combinational optimization problem of NP-hard, and evaluated solutions obtained by using the steepest descent method, the simulated annealing and the proposed method with the correlated noises. As results, in the case of ten cities, the proposed method with correlated noises can obtain more optimum solutions than the steepest descent method and the simulated annealing. In the cases of large numbers of cites, where it is hard to find one of the optimum solutions, our method can obtain solutions at least as same level as the simulated annealing. © 2008 Springer-Verlag Berlin Heidelberg.