Abstract
In a pre-stage of the applications of neural networks to combinatorial problems, the problems are formulated as minimization problems with a quadratic or a bilinear function with respect to binaryvalued variables e. g. {0, 1}, and then a neural network are composed of neurons as many as the variables. The solution to the problem is obtained as one of the stationary states of the network. Here, if another femulation is introduced for the same problem by using ternary-valued variables e. g. {-1, 0, 1} to realize reduction of the number of variables, it would be possible to reduce the number of neurons and the total number of states of the neural network.
From the above motivation, the paper presents the fundamentals of the neural network with ternaryvalued neurons whose states transit deterministicly so as to minimize a bi-quadratic function. In the asynchronous state transition, however, the states of neurons are trapped at one of local optima. So, in order to get off from such a local optimum, a manner of operating the state transition stochastically among the ternary values is proposed, in which increase of the function value to be minimized is accepted in a ratio.
