Experimental analysis of chaotic neural network models for combinatorial optimization under a unifying framework

Abstract

The aim of this paper is to study both the theoretical and experimental properties of chaotic neural network (CNN) models for solving combinatorial optimization problems. Previously we have proposed a unifying framework which encompasses the three main model types, namely, Chen and Aihara's chaotic simulated annealing (CSA) with decaying self-coupling, Wang and Smith's CSA with decaying timestep, and the Hopfield network with chaotic noise. Each of these models can be represented as a special case under the framework for certain conditions. This paper combines the framework with experimental results to provide new insights into the effect of the chaotic neurodynamics of each model. By solving the N-queen problem of various sizes with computer simulations, the CNN models are compared in different parameter spaces, with optimization performance measured in terms of feasibility, efficiency, robustness and scalability. Furthermore, characteristic chaotic neurodynamics crucial to effective optimization are identified, together with a guide to choosing the corresponding model parameters.

Introduction

Combinatorial optimization problems (COP) are often encountered in engineering and business, and since many COP are NP-hard and hence difficult to solve, heuristic algorithms like simulated annealing and tabu search, etc. have been widely used to provide near-optimal solutions for a reasonable computational effort. However, for engineering applications that require fast solution and robust hardware implementation, neural networks are becoming a powerful tool for their inherent parallel computational architecture and fault tolerance (Hopfield, 1984). Hopfield and Tank (1985) first proposed a recurrent Hopfield network to solve COP, and used the famous Travelling Salesman Problem (TSP) to illustrate the approach. However subsequent research showed that, in order to obtain feasible solutions, appropriate values of the penalty parameters are required, often by trial-and-error (Wilson & Pawley, 1988). Moreover, the steepest descent dynamics of the method often leads to local minima of the energy landscape, which limits the network's potential to solve more complex or larger scale problems where the ratio of local to global minima is typically increased.

Recently, chaotic neural networks (CNN) exploiting the rich behaviors of nonlinear dynamics have been developed as a new approach to extend the problem solving ability of standard Hopfield neural networks (Aihara, 1994). Aihara, Takabe, and Toyoda (1990) first proposed a general neuron model with chaotic dynamics, which constituted a CNN that encompassed various associative and back-propagation networks. The model was applied to solve the TSP with a higher efficiency and solution quality than the traditional Hopfield network (Aihara et al., 1990, Yamada et al., 1993). Although chaotic dynamics were found to improve optimization, the unstable neuron outputs can be difficult to interpret, and a convergent network is more desirable for practical purposes. To meet both ends, a deterministic simulated annealing algorithm was proposed by Chen and Aihara (1995), where the self-feedback strength acts as the bifurcation parameter of the network dynamics. As the parameter is gradually reduced, a reverse bifurcation process known as chaotic simulated annealing (CSA) results, in contrast with stochastic simulated annealing (Kirkpatrick, Gelatt, & Vecchi, 1983). The process starts with an unstable phase for searching global minima, followed by a stable, convergent phase. To show the effectiveness of the algorithm, the TSP, a maintenance scheduling problem (Chen & Aihara, 1995) and the N-queen problem (Kwok, Smith, & Wang, 1998a) were computationally solved with high efficiency and solution quality. Methods combining conventional heuristic techniques like the 2-opt algorithm, tabu search, etc. with CNN were investigated by Hasegawa et al., 1997a, Hasegawa et al., 1998 with good performance even on larger problems. To better understand the theoretical aspects of CNN's, the existence of chaotic structure and stability of discrete-time neural networks underlying the CSA algorithm was proved by Chen and Aihara (1997), and a dynamical mechanism explaining the efficiency and novel properties of CNN for optimization was described by Tokuda, Nagashima, & Aihara (1997). This research suggests a crisis-induced intermittent switching phenomenon to be the dynamical mechanism of chaotic search for minima in the Hopfield energy landscape.

Another general approach to CNN is from the Euler discretization of the continuous Hopfield network. In the framework of globally coupled map (GCM) dynamics, Nozawa (1992) proposed an alternative approach to construct a CNN by using Euler discretization on the Hopfield network with negative self-couplings, which was equivalent to the simplest version of Aihara et al.'s model (1990). The TSP was solved computationally with this GCM model, and was found to be more efficient than the Hopfield network and some stochastic networks like the Boltzmann machine and Gaussian machine. More recently, Wang and Smith (1998) proposed a CSA scheme with the timestep of the Euler discretization as the bifurcation parameter that controls the reverse bifurcation process, which provides chaotic minima searching as well as convergence to a stable solution. For some parameter values, this model is equivalent to the one proposed by Chen and Aihara (1995) but, as will be seen later in this paper, the underlying chaotic mechanism is quite different.

The ability of various CNN models to improve optimization raises questions of which chaotic properties most benefit optimization performance and how they arise. One way to approach the problem is to add external noise, correlated or not, into the network, and compare its response and optimization performance with CNN's that generate chaotic dynamics internally. Hayakawa, Marumoto, & Sawada (1995) compared the effects of adding random noise and the logistic map time series into the Euler discretized Hopfield network, and found that short time correlation of the chaotic time series is effective for the search of global minima. Asai, Onodera, Kamio, & Ninomiya (1995) also experimentally studied how autocorrelation in various chaotic time series improves the tracing of optimal solutions when solving the TSP. A more detailed study along this line by Hasegawa, Ikeguchi, Matozaki, & Aihara (1997b) added surrogates of the logistic map time series to a Hopfield-like network and compared the TSP optimization performance to using random and 1/f^α noise. Other research includes using the Henon map time series as noise (Zhou, Yasuda, & Yokoyama, 1997), and solving the N-queen problem with added logistic map noise (Kwok, Smith, & Wang, 1998b). It should be noted that although various chaotic noises have been found to improve optimization, there is no strong evidence of random noise being less effective in general. Also lacking is a detailed account of the underlying mechanism for chaotic noise to improve optimization performance.

From the brief outline given above, we can see there are currently two major classes of CNN. One is the internal approach, where chaotic dynamics is generated within the network controlled by some bifurcation parameters. Examples of this type include Chen and Aihara's decaying self-feedback CSA, Wang and Smith's decaying timestep CSA, and Nozawa's GCM model. The other class contains CNN models employing an external approach, where an externally generated chaotic signal is added to the network as perturbation. All CNN's utilizing externally generated chaotic noise belong to this class. To seek a better understanding of the functional aspects of chaotic dynamics existing in various CNN's, a unified framework was recently proposed (Kwok & Smith, 1999a). It allows us to compare and highlight important common features among the many CNN models, as well as to draw new classifications and insights, thereby providing a basis for constructing new models.

In the next section, the unified framework is first described, followed by the formulations of various CNN models under this framework. In Section 3, implementation of the models to solve the N-queen problem is presented, with the corresponding computational results in Section 4. Finally, we interpret the experimental results under the framework and discuss their implications, followed by the conclusion.