We analyze additive effects of nonlinear dynamics for conbinatorial optimization. We apply chaotic time series as noise sequence to neural networks for 10-city and 20-city traveling salesman problems and compare the performance with stochastic processes, such as Gaussian random numbers, uniform random numbers, 1/fα noise and surrogate data sets which preserve several statistics of the original chaotic data. In result, it is shown that not only chaotic noise but also surrogates with similar autocorrelation as chaotic noise exhibit high solving abilities. It is also suggested that since temporal structure of chaotic noise characterized by autocorrelation affects abilities for combinatorial optimization problems, effects of chaotic sequence as additive noise for escaping from undesirable local minima in case of solving combinatorial optimization problems can be replaced by stochastic noise with similar autocorrelation.

Analog computation is a processing method that solves problems utilizing an analogy of a physical system to the problem. As it is based on actual physical effects and not on symbolic operations, it is therefore a promising architecture for quantum processors. This paper presents an idea for relating quantum structures with analog computation. As an instance, a method is proposed for solving an NP-complete (nondeterminis-tic polynomial time complete) problem, the three-color-map problem, by using a quantum-cell circuit. The computing process is parallel and instantaneous, so making it possible to obtain the solution in a short time regardless of the size of the problem.

We propose a novel segmentation algorithm which combines an image segmentation method into small regions with chaotic neurodynamics that has already been clarified to be effective for solving some combinatorial optimization problems. The basic algorithm of an image segmentation is the variable-shape-bloch-segmentation (VB) which searches an opti-mal state of the segmentation by moving the vertices of quadran-gular regions. However, since the algorithm for moving vertices is based upon steepest descent dynamics, this segmentation method has a local minimum problem that the algorithm gets stuck at undesirable local minima. In order to treat such a problem of the VB and improve its performance, we introduce chaotic neurodynamics for optimization. The results of our novel method are compared with those of conventional stochastic dynamics for escaping from undesirable local minima. As a result, the better results are obtained with the chaotic neurodynamical image segmentation.

The p-collection problem is where to locate p sinks in a flow network such that the value of a maximum flow is maximum. In this paper we show complexity results of the p-collection problem. We prove that the decision problem corresponding to the p-collection problem is strongly NP-complete. Although location problems (the p-center problem and the p-median problem) in networks and flow networks with tree structure is solvable in polynomial time, we prove that the decision problem of the p-collection problem in networks with tree structure, is weakly NP-complete. And we show a polynomial time algorithm for the subproblem of the p-collection problem such that the degree sum of vertices with degree3 in a network, is bound to some constant K0.

The mean field theory has been recognized as offering an efficient computational framework in solving discrete optimization problems by neural networks. This paper gives a formulation based on the information geometry to the mean field theory, and makes clear from the information-theoretic point of view the meaning of the mean field theory as a method of approximating a given probability distribution. The geometrical interpretation of the phase transition observed in the mean field annealing is shown on the basis of this formulation. The discussion of the standard mean field theory is extended to introduce a more general computational framework, which we call the generalized mean field theory.

It is well known that the Hopfield Model (HM) for neural networks to solve the TSP suffers from three major drawbacks: (D1) it can converge to non-optimal local minimum solutions; (D2) it can also converge to non-feasible solutions; (D3) results are very sensitive to the careful tuning of its parameters. A number of methods have been proposed to overcome (D1) well. In contrast, work on (D2) and (D3) has not been sufficient; techniques have not been generalized to larger classes of optimization problems with constraint including the TSP. We first construct Extended HMs (E-HMs) that overcome both (D2) and (D3). The extension of the E-HM lies in the addition of a synapse dynamical system cooperated with the corrent HM unit dynamical system. It is this synapse dynamical system that makes the TSP constraint hold at any final states for whatever choices of the HM parameters and an initial state. We then generalize the E-HM further into a network that can solve a larger class of continuous optimization problems with a constraint equation where both of the objective function and the constraint function are non-negative and continuously differentiable.

Genetic algorithms have been shown to be very useful in a variety of search and optimization problems. In this paper, we propose a modified genetic channel router. We adopt the compatible crossover operator and newly designed compatible mutation operator in order to search solution space more effectively, where vertical constraints are integrated. By carefully selected fitness function forms and optimized genetic parameters, the current version speeds up benchmarks on average about 5.83 times faster than that of our previous version. Moreover the total convergence to optimal solutions for benchmarks can be always obtained.

In this letter an SR-latch circuit using Hopfield neural networks is introduced. An energy function suited for a neural SR-latch circuit is defined for which the global convergence is guaranteed. We also demonstrate how to compose master-slave (M/S) SR- and JK-flip flops of novel SR-latch circuits, and further an asynchronous binary counter of M/S JK-flip flops. Computer simulations are included to illustrate how each presented circuit operates.

This paper describes a novel technique to realize high performance digital sequential circuits by using Hopfield neural networks. For an example of applications of neural networks to digital circuits, a novel gate circuit, full adder circuit and latch circuit using neural networks, which have the global convergence property, are proposed. Here, global convergence means that the energy function is monotonically decreasing and each circulit always operates correctly independently of the initial values. Finally the several digital sequential circuits such as shift register and asynchronous binary counter are designed.

We develop a convergence theory of the simple genetic algorithm (SGA) for two-bit problems (Type I TBP and Type II TBP). SGA consists of two operations, reproduction and crossover. These are imitations of selection and recombination in biological systems. TBP is the simplest optimization problem that is devised with an intention to deceive SGA into deviating from the maximum point. It has been believed that, empirically, SGA can deviate from the maximum point for Type II while it always converges to the maximum point for Type I. Our convergence theory is a first mathematical achievement to ensure that the belief is true. Specifically, we demonstrate the following. (a) SGA always converges to the maximum point for Type I, starting from any initial point. (b) SGA converges either to the maximum or second maximum point for Type II, depending upon its initial points. Regarding Type II, we furthermore elucidate a typical sufficient initial condition under which SGA converges either to the maximum or second maximum point. Consequently, our convergence theory establishes a solid foundation for more general GA convergence theory that is in its initial stage of research. Moreover, it can bring powerful analytical techniques back to the research of original biological systems.

Genetic algorithms have been shown to be very useful in a variety of search and optimization problems. In this paper, we describe the implementation of genetic algorithms for channel routing problems and identify the key points which are essential to making full use of the population of potential solutions, that is one of the characteristics of genetic algorithms. Three efficient crossover techniques which can be divided further into 13 kinds of crossover operators have been compared. We also extend our previous work with ability to deal with dogleg case by simply splitting multi-terminal nets into a series of 2-terminal subnets. It routes the Deutsch's difficult example with 21 tracks without any detours.

Evolution programs have been shown to be very useful in a variety of search and optimization problems, however, until now, there has been little attempt to apply evolution programs to channel routing problem. In this paper, we present an exolution program and identify the key points which are essential to successfully applying evolution programs to channel routing problem. We also indicate how integrating heuristic information related to the problem under consideration helps in convergence on final solutions and illustrate the validity of out approach by providing experimental results obtained for the benchmark tests. compared with the optimal solutions.

This article analyzes the property of the fully interconnected neural networks as a method of solving combinatorial optimization problems in general. In particular, in order to escape local minimums in this model, we analyze theoretically the relation between the diagonal elements of the connection matrix and the stability of the networks. It is shown that the position of the global minimum point of the energy function on the hyper sphere in n dimensional space is given by the eigen vector corresponding the maximum eigen value of the connection matrix. Then it is shown that the diagonal elements of the connection matrix can be improved without loss of generality. The equilibrium points of the improved networks are classified according to their properties, and their stability is investigated. In order to show that the change of the diagonal elements improves the potential for the global minimum search, computer simulations are carried out by using the theoretical values. In according to the simulation result on 10 neurons, the success rate to get the optimum solution is 97.5%. The result shows that the improvement of the diagonal elements has potential for minimum search.

Optimal static load balancing problems in open BCMP queueing networks with state-independent arrival and service rates are studied. Their examples include optimal static load balancing in distributed computer systems and static routing in communication networks. We refer to the load balancing policy of minimizing the overall mean response (or sojourn) time of a job as the overall optimal policy. We show the conditions that the solutions of the overall optimal policy satisfy and show that the policy uniquely determines the utilization of each service center, the mean delay for each class and each path class, etc., although the solution, the utilization for each class, the mean delay for all classes at each service center, etc., may not be unique. Then we give tha linear relations that characterize the set whose elements are the optimal solutions, and discuss the condition wherein the overall optimal policy has a unique solution. In parametric analysis and numerical calculation of optimal values of performance variables we must ensure whether they can be uniquely determined.