On the analysis of a recurrent neural network for solving nonlinearmonotone variational inequality problems
Xue-Bin Liang
Neural Networks, IEEE Transactions on
Volume 13, Issue 2, Mar 2002 Page(s):481 - 486
Digital Object Identifier 10.1109/72.991434
Summary: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
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