On the analysis of a recurrent neural network for solving nonlinear monotone variational inequality problems

We investigate the qualitative properties of a recurrent neural network (RNN) for solving the general monotone variational inequality problems (VIPs), defined over a nonempty closed convex subset, which are assumed to have a nonempty solution set but need not be symmetric. The equilibrium equation of the RNN system simply coincides with the nonlinear projection equation of the VIP to be solved. We prove that the RNN system has a global and bounded solution trajectory starting at any given initial point in the above closed convex subset which is positive invariant for the RNN system. For general monotone VIPs, we show by an example that the trajectory of the RNN system can converge to a limit cycle rather than an equilibrium in the case that the monotone VIPs are not symmetric. Contrary to this, for the strictly monotone VIPs, it is shown that every solution trajectory of the RNN system starting from the above closed convex subset converges to the unique equilibrium which is also locally asymptotically stable in the sense of Lyapunov, no matter whether the VIPs are symmetric or nonsymmetric. For the uniformly monotone VIPs, the aforementioned solution trajectory of the RNN system converges to the unique equilibrium exponentially.