Impulsive Projection Neural Networks for Variational Inequalities and Sparse Signal Reconstruction Application

Variational inequalities (VIs) have become a general framework for efficient problems in various fields such as optimal control and nonlinear programming. Some real problems in application areas such as signal processing and network resource allocation can be transformed into VIs. VIs have been frequently solved using neurodynamic approaches, which have good performance and low computational cost. In this paper, two novel projection neural networks (PNNs) with impulsive effects are presented to deal with VIs. Based on the impulsive system theory and PNN, an impulsive projection neural network (IPNN) and its modified version are proposed. Then the stability properties of the presented IPNN and modified impulsive projection neural network (MIPNN) are analyzed by Lyapunov functions and impulsive system theory. VIs can be addressed with the proposed IPNN and MIPNN. Furthermore, two PNNs are applied to sparse signal reconstruction. It is also shown that the solutions of the proposed IPNN and MIPNN converge to the solution of the corresponding VIs. Meanwhile, sparse signals can also be accurately reconstructed by IPNN and MIPNN. Compared with the classical neural network, the newly designed IPNN and MIPNN have an improvement in the convergence rates because of the introduction of impulsive effects. Theoretical results and simulations show the effectiveness and feasibility of the IPNN and MIPNN.