Delayed neural network based on a new complementarity function for the NCP

Nonlinear complementarity problems (NCP) have been extensively studied in optimization due to its widespread applications. In this paper, we utilize the neural dynamic approach to solve the NCP. By integrating the famous FB function an NR function, we construct a new type of complementarity functions with one parameter p , which is aesthetically pleasing and easy to apply. Combined with the Lagrange multiplier method, a new type of merit function is also developed. Based on the complementarity function and merit function, we transform the NCP into an unconstrained minimization problem. Then, by KKT condition and gradient descent method, we propose a Lagrange neural network method. Under mild conditions, every equilibrium point of the proposed neural network model is a solution of the NCP. More importantly, by throwing a delay factor, we also develop a novel delayed neural network model. Both of these networks are shown to be global convergent, Lyapunov stable and exponential stable. Finally, we give some numerical experiments of the two neural network approaches and also give some applications to the compressed sensing signal reconstruction. Simulation results indicate that the parameter p in the complementarity function plays an important role on the convergence rate of the two neural networks. The delayed neural network outperforms the non-delayed neural network in some specific situations. It also demonstrates that the two networks can efficiently reconstruct the original signals.