Numerical Algorithm to Hardware Implementation {

| We review the approaches for solving combinatorial optimization problems by chaotic dynamics. We mention both numerical algorithms with chaotic neural networks and hardware implementation. I. Chaos for avoiding local minima A. Mutual Connection Neural Network Dynamics Various methods are proposed for solving NP-hard combinatorial optimization problems, for example, traveling salesman problem (TSP) [2] 1 , quadratic assignment problems (QAP) [3] 2 , and so on. One of the novel approaches, called \modern heuristics," solves TSP with neural network dynamics. The basic concept of this approach was formulated by Hop eld and Tank [1]. They applied dynamics of the mutual connection neural network to TSP, which possesses steepest descent down hill dynamics. Although this approach was very attractive from the theoretical aspects, there are several drawbacks for its applications to real engineering problems. The rst is that there exist many undesirable local minima and the second is that size of problems is so small as real engineering applications. B. Chaotic Neural Networks In order to solve the local minimum problem, a new approach using chaotic neurodynamics was proposed. The basic concept is that chaotic dynamics could play an essential role for searching solutions in state space. For example, self-similarity, which means that attractors of chaotic dynamical systems could have fractal structures, would be e ective for searching solutions in state space. If the optimum solution is embedded in the attractor, chaotic dynamics searches only 1 When a position of N cities is given, TSP is usually described to nd minimum length tour which satis es the condition that each city is visited exactly once. 2 When a distance matrix A and a ow matrix B are given, QAP is described to nd the permutation p which corresponds to the minimum value of the objective function f(p) = P n i=1 P n j=1 a ij b p(i)p(j) ; where a ij and b ij are the (i; j) th elements of A and B, p(i) is the ith element of vector p, and n is size. along such fractal attractors whose Lebesgue measure is zero. Nozawa's work [5] was the rst approach based on the above concept. Nozawa modi ed the Hop eldTank neural network by the Euler's method to the neural network model with negative self-feedback connections, which is equivalent to the chaotic neural network proposed by Aihara et al.[4]. In Ref.[5], 10-city TSPs are solved and it was reported that chaotic approach exhibits better performance than stochastic ones. It should be noted that Inoue and Nagayoshi have also tried to solve TSP in a similar way[6], but their basic unit is not the chaotic neuron [4] but a coupled logistic map. Yamada and Aihara studied the solving abilities of the chaotic neural networks for TSP by computing the largest Lyapunov exponents [7]. They showed that the solving abilities are very high when largest Lyapunov exponents are near zero, which fact implies that \an edge of chaos" could have high performance to solve combinatorial problems [7]. As for real engineering applications, Chen and Aihara showed e ectiveness of chaotic simulated annealing on solving maintenance scheduling problems [8]. Recently, Chen and Aihara proved the existence of chaotic dynamics in the case of solving combinatorial optimization problems by the chaotic neural networks [9] and possible existence of the optimum solution [10]. Ishii and Sato also investigated potential abilities of chaotic dynamics. They proposed the new neural network model called chaotic potts spin [11]. By this neural network, the constraint term is always satis ed, therefore feasible solutions can be always obtained. They reported that good solutions of TSP are obtained with this method [11]. For analyzing why these approaches are good for solving combinatorial optimization problems, it is very natural to introduce bifurcation theories based on dynamical systems' theory. Tokuda, Nagashima and Aihara show the bifurcation scenario of the chaotic neural network under solving TSP [12]. They proposed effective cooling schedule by adaptive chaotic simulated annealing based on its bifurcation scenario.