A new class of neural networks for NCPs using smooth perturbations of the natural residual function

We present a new class of neural networks for solving nonlinear complementarity problems (NCPs) based on some family of real-valued functions (denoted by ℱ) that can be used to construct smooth perturbations of the level curve defined by Φ_NR(x,y)=0, where Φ_NR is the natural residual function (also called the “min” function). We introduce two important subclasses of ℱ, which deserve particular attention because of their significantly different theoretical and numerical properties. One of these subfamilies yields a smoothing function for Φ_NR, while the other subfamily only yields a smoothing curve for Φ_NR(x,y)=0. We also propose a simple framework for generating functions from these subclasses. Using the smoothing approach, we build two types of neural networks and provide sufficient conditions to guarantee asymptotic and exponential stability of equilibrium solutions. Finally, we present extensive numerical experiments to validate the theoretical results and to illustrate the difference in numerical performance of functions from the two subclasses. Numerical comparisons with existing neural networks for NCPs are also demonstrated.