Neural Dynamics and Newton-Raphson Iteration for Nonlinear Optimization

In this paper, a special type of neural dynamics (ND) is generalized and investigated for time-varying and static scalar-valued nonlinear optimization. In addition, for comparative purpose, the gradient-based neural dynamics (or termed gradient dynamics (GD)) is studied for nonlinear optimization. Moreover, for possible digital hardware realization, discrete-time ND (DTND) models are developed. With the linear activation function used and with the step size being 1, the DTND model reduces to Newton-Raphson iteration (NRI) for solving the static nonlinear optimization problems. That is, the well-known NRI method can be viewed as a special case of the DTND model. Besides, the geometric representation of the ND models is given for time-varying nonlinear optimization. Numerical results demonstrate the efficacy and advantages of the proposed ND models for time-varying and static nonlinear optimization.