A novel neural network for variational inequalities with linear and nonlinear constraints

Variational inequality is a uniform approach for many important optimization and equilibrium problems. Based on the sufficient and necessary conditions of the solution, this paper presents a novel neural network model for solving variational inequalities with linear and nonlinear constraints. Three sufficient conditions are provided to ensure that the proposed network with an asymmetric mapping is stable in the sense of Lyapunov and converges to an exact solution of the original problem. Meanwhile, the proposed network with a gradient mapping is also proved to be stable in the sense of Lyapunov and to have a finite-time convergence under some mild conditions by using a new energy function. Compared with the existing neural networks, the new model can be applied to solve some nonmonotone problems, has no adjustable parameter, and has lower complexity. Thus, the structure of the proposed network is very simple. Since the proposed network can be used to solve a broad class of optimization problems, it has great application potential. The validity and transient behavior of the proposed neural network are demonstrated by several numerical examples.