Distributed Neurodynamic Models for Solving a Class of System of Nonlinear Equations

This article investigates a class of systems of nonlinear equations (SNEs). Three distributed neurodynamic models (DNMs), namely a two-layer model (DNM-I) and two single-layer models (DNM-II and DNM-III), are proposed to search for such a system's exact solution or a solution in the sense of least-squares. Combining a dynamic positive definite matrix with the primal-dual method, DNM-I is designed and it is proved to be globally convergent. To obtain a concise model, based on the dynamic positive definite matrix, time-varying gain, and activation function, DNM-II is developed and it enjoys global convergence. To inherit DNM-II's concise structure and improved convergence, DNM-III is proposed with the aid of time-varying gain and activation function, and this model possesses global fixed-time consensus and convergence. For the smooth case, DNM-III's globally exponential convergence is demonstrated under the Polyak-Lojasiewicz (PL) condition. Moreover, for the nonsmooth case, DNM-III's globally finite-time convergence is proved under the Kurdyka-Lojasiewicz (KL) condition. Finally, the proposed DNMs are applied to tackle quadratic programming (QP), and some numerical examples are provided to illustrate the effectiveness and advantages of the proposed models.

Automated summary

This article investigates a class of systems of nonlinear equations and proposes three distributed neurodynamic models for solving such equations. These models have global convergence and certain advantages, are applicable to both smooth and nonsmooth cases, and have good performance in quadratic programming problems.