Solving TSP by Transiently Chaotic Neural Networks

Abstract

Inspired by the information processing of human neural systems, the artificial neural networks (ANNs) have been developed and applied to solve problems in various disciplines with varying degrees of success. For example, ANNs have been applied to memory storage, pattern recognition, categorization, error correction, decision making, and machine learning in object oriented machine. Various computational schemes and algorithms have been devised for solving the travelling salesman problem which is a difficult NP-hard combinatorial optimization problem. The use of ANN as a computational machine to solve combinatorial optimization problems, including TSP, dates back to 1985 by Hopfield and Tank (1985). Although the achievement of such an application broadens the capacity of ANNs, there remain several insufficiencies to be improved for such a computational task, cf. (Smith, 1999). They include that the computations can easily get trapped at local minimum of the objective function, feasibility of computational outputs, and suitable choice of parameters. Improvements of feasibility and solution quality for the scheme have been reported subsequently. Among them, there is a success in adding the chaotic ingredient into the network to enhance the global searching ability. Chaotic behavior is an inside essence of stochastic processes in nonlinear deterministic system. Recently, chaotic neural networks have been paid much attention to, and contribute toward solving TSP. Chaotic phenomena arise from nonlinear system, and the discrete-time analogue of Hopfield’s model can admit such a dynamics. Notably, the discrete-time neural network models can also be implemented into analogue circuits, cf. (Hanggi et al., 1999 ; Harrer & Nossek, 1992). The chapter aims at introducing recent progress in discrete-time neural network models, in particular, the transiently chaotic neural network (TCNN) and the advantage of adopting piecewise linear activation function. We shall demonstrate the use of TCNN in solving the TSP and compare the results with other neural networks. The chaotic ingredients improve the shortcoming of the previous ODE models in which the outputs strongly depend on the initial conditions and are easily trapped at the local minimum of objective function. There are transiently chaotic phase and convergent phase for the TCNN. The parameters for convergent phase are confirmed by the nonautonomous discrete-time LaSalle’s invariant principle, whereas the ones for chaotic phase are derived by applying the Marotto’s theorem. The Marotto’s theorem which generalizes the Li-York’s theorem on chaos from one-dimension to multi-dimension has found its best application in the discrete-time neural network model considered herein. O pe n A cc es s D at ab as e w w w .ite ch on lin e. co m