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-Łojasiewicz (PL) condition. Moreover, for the nonsmooth case, DNM-III's globally finite-time convergence is proved under the Kurdyka-Łojasiewicz (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.