Abstract
This article presents a general six-step discrete-time Zhang neural network (ZNN) for time-varying tensor absolute value equations. Firstly, based on the Taylor expansion theory, we derive a general Zhang et al. discretization (ZeaD) formula, i.e., a general Taylor-type 1-step-ahead numerical differentiation rule for the first-order derivative approximation, which contains two free parameters. Based on the bilinear transform and the Routh–Hurwitz stability criterion, the effective domain of the two free parameters is analyzed, which can ensure the convergence of the general ZeaD formula. Secondly, based on the general ZeaD formula, we design a general six-step discrete-time ZNN (DTZNN) for time-varying tensor absolute value equations (TVTAVEs), whose steady-state residual error changes in a higher order manner than those presented in the literature. Meanwhile, the feasible region of its step size, which determines its convergence, is also studied. Finally, experiment results corroborate that the general six-step DTZNN model is quite efficient for TVTAVE solving.
Highlights
Tensors are higher order generalizations of matrices
discrete-time ZNN (DTZNN) model (43), which is compared with the simplest discrete-time Zhang neural network, i.e., the DTZN model in [23]
We have proposed a general six-step DTZNN model for the time-varying tensor absolute equations, which is an NP-hard problem. e steady-state residual error of the proposed DTZNN model changes in anO 5 manner
Summary
Tensors are higher order generalizations of matrices. Oneorder tensor is vector, and two-order tensor is matrix. We are concerned with the following time-varying tensor absolute value equations (TVTAVEs):. Due to the dynamic change in the market, the payment matrices/tensors of -person in noncooperative games dynamically change, and the corresponding dynamic Nash equilibrium can be obtained by solving a system of time-varying matrix/ tensor absolute value equations. We are going to apply a special recurrent neural network, i.e., the Zhang neural network (ZNN), to solve TVTAVEs (1) and propose a general six-step discrete-time Zhang neural network, which includes many existing DTZNN models as its special cases. (2) A general six-step DTZNN model is designed to solve the time-varying tensor absolute value equations based on the above rst-order derivative approximation. (4) e e ciency of the general six-step DTZNN model is substantiated by the numerical simulations
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