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This paper presents a new neural network model for solving constrained variational inequality problems by converting the necessary and sufficient conditions for the solution into a system of nonlinear projection equations. Five sufficient conditions are provided to ensure that the proposed neural network is stable in the sense of Lyapunov and converges to an exact solution of the original problem by defining a proper convex energy function. The proposed neural network includes an existing model, and can be applied to solve some nonmonotone and nonsmooth problems. The validity and transient behavior of the proposed neural network are demonstrated by some numerical examples. Index Terms--Convergence, neural network, stability, variational inequality.

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. Index Terms--Convergence, neural network, stability, variational inequality.

In this paper, we propose a neural network for solving a class of convex quadratic minimax problems with constraints. Four sufficient conditions are provided to ensure the asymptotic stability of the proposed network. Furthermore, the exponential stability of the proposing network is also proved under certain conditions. The results obtained here can be further extended to the globally projected dynamical system. In addition, some new stability conditions for the system are also obtained. Since our stability conditions can be easily checked in practice, these results becomes more attractive in real applications. Index Terms--Convergence and stability, minimax problem, neural network, saddle point.

A method for exploiting the sparsity structure of a finite-element simulation model in a linked dynamic optimal control model is developed. The linked optimization/simulation model has been used for the computation of time-varying optimal pumping rates for the pump-and-treat remediation of contaminated ground water. The methodology presented reduces the computational effort involved in the determination of time-varying optimal pumping rates by an order of n, from [n.sup.3] to [n.sup.2], where n is the number of non-Dirichlet nodes used in the simulation of the aquifer. The method presented uses the characteristic banded structure of finite element model matrices in derivative computations used by the optimal control algorithm, and also within the algorithmic computations of the optimization method itself. Timing results demonstrating the improvement of this method for problems having n = 100 to n = 1,575 (state dimension 2n = 200 to 2n = 3,150) are presented. It is shown that the efficiency of the sparse algorithm is highest when the length of management periods equals the length of simulation time periods. The results indicate that for a problem with 2n = 1,000, the sparse algorithm is as much as 98% faster than an algorithm neglecting sparsity.